IMO ISL 1968-92 186p

geometry problems from IMO Shortlist (IMO ISL) with aops links

1968 - 1992

authors and proposing countries shall be added in the future

IMO Shortlist 1968

IMO ISL 1968 p2
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

IMO ISL 1968 p3   - 1968 IMO Problem 4 (POL) 

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

IMO ISL 1968 p5
Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality
$(n + 1)h_n+1 - nh_n > r.$
Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$

IMO ISL 1968 p7
Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $A=\frac{3\sqrt 3}{8}$ times the product of the side lengths of the triangle. When does equality hold?

IMO ISL 1968 p8
Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E,F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B,C,D$ of trapezoids that satisfy the following:
(i)  $|AB| \leq a$ ($a$ fixed);
(ii)  $|EF| = l$ ($l$ fixed);
(iii) the sum of squares of the nonparallel sides of the trapezoid is constant.

IMO ISL 1968 p9
Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality $abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.$
Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

IMO ISL 1968 p10
Consider two segments of length $a, b \ (a > b)$ and a segment of length $c = \sqrt{ab}$.
(a) For what values of $a/b$ can these segments be sides of a triangle ?
(b) For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled ?

IMO ISL 1968 p13
Given two congruent triangles $A_1A_2A_3$ and $B_1B_2B_3$ ($A_iA_k = B_iB_k$), prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented.

IMO ISL 1968 p14
A line in the plane of a triangle $ABC$ intersects the sides $AB$ and $AC$ respectively at points $X$ and $Y$ such that $BX = CY$ . Find the locus of the center of the circumcircle of triangle $XAY .$

IMO ISL 1968 p17
Given a point $O$ and lengths $x, y, z$, prove that there exists an equilateral triangle $ABC$ for which $OA = x, OB = y, OC = z$, if and only if $x+y \geq  z, y+z \geq x, z+x \geq y$ (the points $O,A,B,C$ are coplanar).

IMO ISL 1968 p18
If an acute-angled triangle $ABC$ is given, construct an equilateral triangle $A'B'C'$ in space such that lines $AA',BB', CC'$ pass through a given point.

IMO ISL 1968 p19
We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0, 1, 2, \ldots$ from it we obtain points with abscisas $n = 0, 1, 2, .\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $\frac 15$ apart ?

IMO ISL 1968 p20
Given $n \ (n \geq 3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1,A_2, \ldots,A_n$ in such a way that for each $i, j, k, 1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or equal to $120^\circ .$

IMO ISL 1968 p25
Given $k$ parallel lines $l_1, \ldots, l_k$ and $n_i$ points on the line $l_i, i = 1, 2, \ldots, k$, find the maximum possible number of triangles with vertices at these points.


IMO Shortlist 1970


IMO ISL 1970 p1
Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that
$\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.$

IMO ISL 1970 p3  - 1970 IMO Problem 5 (BUL) 

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: $(AB+BC+CA)^2\le6(AD^2+BD^2+CD^2).$ When do we have equality?

IMO ISL 1970 p5
Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that
$\begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}$
($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

IMO ISL 1970 p6
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$

IMO ISL 1970 p8  - 1970 IMO Problem 1 (POL)

$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

IMO ISL 1970 p12
Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.


IMO Shortlist 1971


IMO ISL 1971 p2
Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.

IMO ISL 1971 p4
We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.

IMO ISL 1971 p7  - 1971 IMO Problem 4 (NET) 

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$.
a.) If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length;
b.) If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.
IMO ISL 1971 p12
Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.

IMO ISL 1971 p14
A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$

IMO ISL 1971 p16  - 1971 IMO Problem 2 (USS) 

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.


IMO Shortlist 1972


IMO ISL 1972 p2
We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$

IMO ISL 1972 p4
Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1  \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

IMO ISL 1972 p5
Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if $AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.$

IMO ISL 1972 p7  - 1972 IMO Problem 6 (GBR) 

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

IMO ISL 1972 p10  - 1972 IMO Problem 2 (NET) 

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

IMO ISL 1972 p11
Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc.
(a) Prove the relation
$r_1  \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2  \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right)$ where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that $r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1$
(b) Prove that the sequence of circles $K_1,K_2, \ldots$ is periodic.


IMO Shortlist 1973


IMO ISL 1973 p1
Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality $\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4$ holds, where $A_1,B_1, C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$, and $DP$, respectively.

IMO ISL 1973 p2
Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.

IMO ISL 1973 p3
Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.

IMO ISL 1973 p5
A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.

IMO ISL 1973 p6  - 1973 IMO Problem 2 (POL) 

Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.

IMO ISL 1973 p7
Given a tetrahedron $ABCD$, let $x = AB \cdot CD$, $y = AC \cdot BD$, and $z = AD \cdot  BC$. Prove that there exists a triangle with edges $x, y, z.$

IMO ISL 1973 p9
Let $Ox, Oy, Oz$ be three rays, and $G$ a point inside the trihedron $Oxyz$. Consider all planes passing through $G$ and cutting $Ox, Oy, Oz$ at points $A,B,C$, respectively. How is the plane to be placed in order to yield a tetrahedron $OABC$ with minimal perimeter ?

IMO ISL 1973 p13
Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to $1.$

IMO ISL 1973 p14  - 1973 IMO Problem 4 (YUG) 

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

by Ðorde Dugošija

IMO Shortlist 1974


IMO ISL 1974 p2
Prove that the squares with sides $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots$ may be put into the square with side $\frac{3}{2}$ in such a way that no two of them have any interior point in common.

IMO ISL 1974 p5
Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1}$on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$ are congruent.

IMO ISL 1974 p10
Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$.


IMO Shortlist 1975


IMO ISL 1975 p8  -  1975 IMO Problem 3 (NET) 

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$.
Prove that
a.) $\angle QRP = 90\,^{\circ},$ and
b.) $QR = RP.$

by Jan van de Craats

IMO ISL 1975 p12
Consider on the first quadrant of the trigonometric circle the arcs $AM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v$ , such that $x_1 < x_2 < x_3 < \cdots < x_v$. Prove that
$\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})$

IMO ISL 1975 p13
Let $A_0,A_1, \ldots , A_n$ be points in a plane such that
(i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2  \leq  \cdots  \leq  \frac{1}{2^{n-1} } A_{n-1}A_n$ and
(ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$
where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$

IMO ISL 1975 p15
Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?


IMO Shortlist 1976


IMO ISL 1976 p1
Let $ABC$ be a triangle with bisectors $AA_1,BB_1, CC_1$ ($A_1 \in  BC$, etc.) and $M$ their common point. Consider the triangles $MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $AB = BC = CA.$

IMO ISL 1976 p3  - 1976 IMO Problem 1 (CZS) 

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

IMO ISL 1976 p6
A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.


IMO Shortlist 1977


IMO ISL 1977 p2
A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

IMO ISL 1977 p4
Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.

IMO ISL 1977 p8
Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.

IMO ISL 1977 p12  - 1977 IMO Problem 2 (NET) 

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

by Jan van de Craats

IMO ISL 1977 p14
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:
(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$
(ii) some plane contains exactly three points from $E.$

IMO ISL 1977 p16
Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$


IMO Shortlist 1978

IMO ISL 1978 p2
Two identically oriented equilateral triangles, $ABC$ with center $S$ and $A'B'C$, are given in the plane. We also have $A'  \neq  S$ and $B' \neq S$. If $M$ is the midpoint of $A'B$ and $N$ the midpoint of $AB'$, prove that the triangles $SB'M$ and $SA'N$ are similar.

IMO ISL 1978 p4
Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that
$16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).$
When does equality hold?

IMO ISL 1978 p7
We consider three distinct half-lines $Ox, Oy, Oz$ in a plane. Prove the existence and uniqueness of three points $A \in Ox, B \in Oy, C \in Oz$ such that the perimeters of the triangles $OAB,OBC,OCA$ are all equal to a given number $2p > 0.$

IMO ISL 1978 p12  - 1978 IMO Problem 4 (USA) 

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

by Murray Klamkin

IMO ISL 1978 p13  - 1978 IMO Problem 2 (USA) 

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

by Murray Klamkin

IMO ISL 1978 p14
Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times  2 \times  (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$.

Remark.
It is assumed that the edges of the pieces of soap are parallel to the edges of the box.


IMO Shortlist 1979


IMO ISL 1979 p1
Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

IMO ISL 1979 p4
We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots,5$ is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

IMO ISL 1979 p17
Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that
$\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.$
Determine the angles of triangle $PQR.$

IMO ISL 1979 p22  - 1979 IMO Problem 3 (USS) 

Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

by Nikolai Vasil'ev and Igor F. Sharygin

IMO ISL 1979 p24
A circle $C$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB \times CQ = (\frac{BC}{2})^2$, prove that line segment $PQ$ is tangent to circle $C$, and prove the converse.

IMO ISL 1979 p25  - 1979 IMO Problem 4 (USA) 

We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which $\frac{QP+PR}{QR}$ is maximum.

by Murray Klamkin

IMO Shortlist 1980


IMO ISL 1980 p1
Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.

IMO ISL 1980 p4
Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides
$(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})$ are parallel, then the sides $A_n A_{n+1}, A_{2n} A_1$ are parallel as well.

IMO ISL 1980 p5
In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation $y = x^4 + px^3 + qx^2 + rx + s$ in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

IMO ISL 1980 p8
Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$. Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$.

IMO ISL 1980 p10
Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$. The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$.

IMO ISL 1980 p15
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

IMO ISL 1980 p17
Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

IMO ISL 1980 p20
Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$

IMO ISL 1980 p21
Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$


IMO Shortlist 1981


IMO ISL 1981 p2
A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$, then it is also tangent to the edge $BD.$

IMO ISL 1981 p11
On a semicircle with unit radius four consecutive chords $AB,BC, CD,DE$ with lengths $a, b, c, d$, respectively, are given. Prove that $a^2 + b^2 + c^2 + d^2 + abc + bcd < 4.$

IMO ISL 1981 p14
Prove that a convex pentagon (a five-sided polygon) $ABCDE$ with equal sides and for which the interior angles satisfy the condition $\angle A \geq \angle B \geq \angle C \geq \angle D \geq \angle E$ is a regular pentagon.

IMO ISL 1981 p15  - 1981 IMO Problem 1 (GBR) 

Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum ${BC\over PD}+{CA\over PE}+{AB\over PF}.$

by David Monk

IMO ISL 1981 p17  - 1981 IMO Problem 5 (USS) 

Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.

IMO ISL 1981 p18
Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.

IMO ISL 1981 p19
A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\frac{2S}{9}.$


IMO Shortlist 1982


IMO ISL 1982 p2
Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that $\text{area } (K \cap Q_i) =\frac 14  \text{area } K \  (i = 1, 2, 3, 4, ),$ where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$

IMO ISL 1982 p5 - 1982 IMO Problem 5 (NET) 

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that ${AM\over AC}={CN\over CE}=r.$ Determine $r$ if $B,M$ and $N$ are collinear.

by Jan van de Craats

IMO ISL 1982 p6
Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

IMO ISL 1982 p8
A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.

IMO ISL 1982 p9
Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$

IMO ISL 1982 p12
Four distinct circles $C,C_1, C_2$, C3 and a line L are given in the plane such that $C$ and $L$ are disjoint and each of the circles $C_1, C_2, C_3$ touches the other two, as well as $C$ and $L$. Assuming the radius of $C$ to be $1$, determine the distance between its center and $L.$

IMO ISL 1982 p13  - 1982 IMO Problem 2 (NET) 

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

by Jan van de Craats

IMO ISL 1982 p14
Let $ABCD$ be a convex plane quadrilateral and let $A_1$ denote the circumcenter of $\triangle BCD$. Define $B_1, C_1,D_1$ in a corresponding way.
(a) Prove that either all of $A_1,B_1, C_1,D_1$ coincide in one point, or they are all distinct. Assuming the latter case, show that $A_1$, C1 are on opposite sides of the line $B_1D_1$, and similarly,$B_1,D_1$ are on opposite sides of the line $A_1C_1$. (This establishes the convexity of the quadrilateral $A_1B_1C_1D_1$.)
(b) Denote by $A_2$ the circumcenter of $B_1C_1D_1$, and define $B_2, C_2,D_2$ in an analogous way. Show that the quadrilateral $A_2B_2C_2D_2$ is similar to the quadrilateral $ABCD.$

IMO ISL 1982 p17
The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $\angle CAB = \angle C_1AB_1$. Let  $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.

IMO ISL 1982 p18
Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?

IMO ISL 1982 p20
Let $ABCD$ be a convex quadrilateral and draw regular triangles $ABM, CDP, BCN, ADQ$, the first two outward and the other two inward. Prove that $MN = AC$. What can be said about the quadrilateral $MNPQ$?

IMO Shortlist 1983


IMO ISL 1983 p3  - 1983 IMO Problem 4 (BEL) 

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

IMO ISL 1983 p4
On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.

IMO ISL 1983 p9
Let $a$, $b$ and $c$ be the lengths of the sides of a triangle. Prove that
$a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\ge 0.$
Determine when equality occurs.

IMO ISL 1983 p17
Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq  2$. Prove that
$\max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j$

IMO ISL 1983 p23  - 1983 IMO Problem 2 (USS) 

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of  $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.

by Igor F. Sharygin

IMO ISL 1983 p25
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$


IMO Shortlist 1984


IMO ISL 1984 p4
Let $d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $n$ vertices (where $n>3$). Let $p$ be its perimeter. Prove that:
$n-3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n+1\over 2}\Bigr]-2,$
where $[x]$ denotes the greatest integer not exceeding $x$.

IMO ISL 1984 p8
Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

IMO ISL 1984 p13
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$

IMO ISL 1984 p14  - 1984 IMO Problem 4 (ROM) 

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter  $CD$ if and only if the lines $BC$ and $AD$ are parallel.

by Laurentiu Panaitopol

IMO ISL 1984 p15
Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$.
(a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$
(b) Prove that $SD + SE + SF = 2(SA + SB + SC).$

IMO ISL 1984 p18
Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$

IMO Shortlist 1985


IMO ISL 1985 p2
A polyhedron has $12$ faces and is such that:
(i) all faces are isosceles triangles,
(ii) all edges have length either $x$ or $y$,
(iii) at each vertex either $3$ or $6$ edges meet, and
(iv) all dihedral angles are equal.
Find the ratio $x/y.$

IMO ISL 1985 p5
Let $D$ be the interior of the circle $C$ and let $A \in C$. Show that the function $f : D \to \mathbb R, f(M)=\frac{|MA|}{|MM'|}$ where $M' = AM \cap C$, is strictly convex; i.e., $f(P) <\frac{f(M_1)+f(M_2)}{2}, \forall M_1,M_2 \in D, M_1 \neq M_2$ where $P$ is the midpoint of the segment $M_1M_2.$

IMO ISL 1985 p9
Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$

IMO ISL 1985 p10
Prove that for every point $M$ on the surface of a regular tetrahedron there exists a point $M'$ such that there are at least three different curves on the surface joining $M$ to $M'$ with the smallest possible length among all curves on the surface joining $M$ to $M'$.

IMO ISL 1985 p16
If possible, construct an equilateral triangle whose three vertices are on three given circles.

IMO ISL 1985 p19
For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?

IMO ISL 1985 p20  - 1985 IMO Problem 4 (GBR) 

A circle whose center is on the side $ED$ of the cyclic quadrilateral $BCDE$ touches the other three sides. Prove that $EB+CD = ED.$

by Frank Budden

IMO ISL 1985 p21
The tangents at $B$ and $C$ to the circumcircle of the acute-angled triangle $ABC$ meet at $X$. Let $M$ be the midpoint of $BC$. Prove that
(a) $\angle BAM = \angle CAX$, and
(b) $\frac{AM}{AX} = \cos\angle BAC.$

IMO ISL 1985 p22  - 1985 IMO Problem 5 (USS) 

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

by Igor F. Sharygin

IMO Shortlist 1986

IMO ISL 1986 p1
Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

IMO ISL 1986 p3
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

IMO ISL 1986 p11
Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$

Simplified version.
Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$

IMO ISL 1986 p14
The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

IMO ISL 1986 p15
Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$
(a) Prove that $ABCD$ and $A''B''C''D''$ are similar.
(b) The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.

IMO ISL 1986 p16  - 1986 IMO Problem 4 (ICE) 

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

by Sven Sigurðsson

IMO ISL 1986 p17  - 1986 IMO Problem 2 (CHN) 

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define  $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of  $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

 by Gengzhe Chang and Dongxu Qi

MO ISL 1986 p18
Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.

IMO ISL 1986 p19
A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$

IMO ISL 1986 p20
Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

IMO ISL 1986 p21
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
$r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}$ where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.


IMO Shortlist 1987


IMO ISL 1987 p4   (France)
Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality $AF + AH + AC \leq  AB + AD + AE + AG.$ 
In what cases does equality hold?

IMO ISL 1987 p5
Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively.

IMO ISL 1987 p6  (Greece)
Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then
$\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N$

IMO ISL 1987 p9  (Hungary)
Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ?

IMO ISL 1987 p10  (Iceland)
Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$?

IMO ISL 1987 p12  (Poland)
Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear.
IMO ISL 1987 p13  - 1987 IMO Problem 5 (Germany, DR)
Is it possible to put $1987$ points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a non-degenerate triangle with rational area? (IMO Problem 5)

IMO ISL 1987 p19  (Soviet Union)
Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$\beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma$ we can construct a triangle and that its area is not greater than $A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).$

IMO ISL 1987 p21  - 1987 IMO Problem 2 (USS) 

In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.

by I.A. Kushnir

IMO Shortlist 1988


IMO ISL 1988 p3
The triangle $ABC$ is inscribed in a circle. The interior bisectors of the angles $A,B$ and $C$ meet the circle again at $A', B'$ and $C'$ respectively. Prove that the area of triangle $A'B'C'$ is greater than or equal to the area of triangle $ABC.$

IMO ISL 1988 p6
In a given tedrahedron $ABCD$ let $K$ and $L$ be the centres of edges $AB$ and $CD$ respectively. Prove that every plane that contains the line $KL$ divides the tedrahedron into two parts of equal volume.

IMO ISL 1988 p8
Let $u_1, u_2, \ldots, u_m$ be $m$ vectors in the plane, each of length $\leq 1,$ with zero sum. Show that one can arrange $u_1, u_2, \ldots, u_m$ as a sequence $v_1, v_2, \ldots, v_m$ such that each partial sum $v_1, v_1 + v_2, v_1 + v_2 + v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $\sqrt {5}.$

IMO ISL 1988 p12
In a triangle $ABC,$ choose any points $K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $F \in KL.$ If $E_1, E_2, E_3, E_4, E_5, E_6$ and $E$ denote the areas of the triangles $AMR, CKR, BKF, ALF, BNM, CLN$ and $ABC$ respectively, show that
$ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.$

IMO ISL 1988 p13  - 1988 IMO Problem 5 (GRE)

In a right-angled triangle $ABC$ let $AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ABD, ACD$ intersect the sides $AB, AC$ at the points $K,L$ respectively. If $E$ and $E_1$ dnote the areas of triangles $ABC$ and $AKL$ respectively, show that $ \frac {E}{E_1} \geq 2.$

by Dimitris Kontogiannis

IMO ISL 1988 p15
Let $ABC$ be an acute-angled triangle. The lines $L_{A}$, $L_{B}$ and $L_{C}$ are constructed through the vertices $A$, $B$ and $C$ respectively according the following prescription: Let $H$ be the foot of the altitude drawn from the vertex $A$ to the side $BC$; let $S_{A}$ be the circle with diameter $AH$; let $S_{A}$ meet the sides $AB$ and $AC$ at $M$ and $N$ respectively, where $M$ and $N$ are distinct from $A$; then let $L_{A}$ be the line through $A$ perpendicular to $MN$. The lines $L_{B}$ and $L_{C}$ are constructed similarly. Prove that the lines $L_{A}$, $L_{B}$ and $L_{C}$ are concurrent.

IMO ISL 1988 p17
In the convex pentagon $ABCDE,$ the sides $BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($AC$ is parallel to $DE$, $BD$ is parallel to $AE$ etc.). Prove that $ABCDE$ is a regular pentagon.

IMO ISL 1988 p18  - 1988 IMO Problem 1 (LUX) 

Consider 2 concentric circle radii $R$ and  $r$ ($R > r$) with centre $O.$ Fix $P$ on the small circle and consider the variable chord $PA$ of the small circle. Points $B$ and $C$ lie on the large circle; $B,P,C$ are collinear and $BC$ is perpendicular to $AP.$
i.) For which values of $\angle OPA$ is the sum $BC^2 + CA^2 + AB^2$ extremal?
ii.) What are the possible positions of the midpoints $U$ of $BA$ and $V$ of $AC$ as $\angle OPA$ varies?

by Lucien Kieffer

IMO ISL 1988 p23
Let $Q$ be the centre of the inscribed circle of a triangle $ABC.$ Prove that for any point $P,$
$a(PA)^2 + b(PB)^2 + c(PC)^2 = a(QA)^2 + b(QB)^2 + c(QC)^2 + (a + b + c)(QP)^2,$
where $a = BC, b = CA$ and $c = AB.$

IMO ISL 1988 p27
Let $ABC$ be an acute-angled triangle. Let $L$ be any line in the plane of the triangle $ABC$. Denote by $u$, $v$, $w$ the lengths of the perpendiculars to $L$ from $A$, $B$, $C$ respectively. Prove the inequality $u^2\cdot\tan A + v^2\cdot\tan B + w^2\cdot\tan C\geq 2\cdot S$, where $S$ is the area of the triangle $ABC$. Determine the lines $L$ for which equality holds.

IMO ISL 1988 p30
A point $M$ is chosen on the side $AC$ of the triangle $ABC$ in such a way that the radii of the circles inscribed in the triangles $ABM$ and $BMC$ are equal. Prove that
$ BM^{2} = X \cot \left( \frac {B}{2}\right)$
where X is the area of triangle $ABC.$


IMO Shortlist 1989


IMO ISL 1989 p1  - 1989 IMO Problem 2 (AUS) 

$ABC$ is a triangle, the bisector of angle $A$ meets the circumcircle of triangle $ABC$ in $A_1$, points $B_1$ and $C_1$ are defined similarly. Let $AA_1$ meet the lines that bisect the two external angles at $B$ and $C$ in $A_0$. Define $B_0$ and $C_0$ similarly. Prove that the area of triangle $A_0B_0C_0 = 2 \cdot$ area of hexagon $AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ABC$.

by Esther Szekeres

IMO ISL 1989 p6
For a triangle $ABC,$ let $k$ be its circumcircle with radius $r.$ The bisectors of the inner angles $A, B,$ and $C$ of the triangle intersect respectively the circle $k$ again at points $A', B',$ and $C'.$ Prove the inequality $16Q^3 \geq 27 r^4 P,$
where $Q$ and $P$ are the areas of the triangles $A'B'C'$ and $ABC$ respectively.

IMO ISL 1989 p7
Show that any two points lying inside a regular $n-$gon $E$ can be joined by two circular arcs lying inside $E$ and meeting at an angle of at least $\left(1 - \frac{2}{n} \right) \cdot \pi.$

IMO ISL 1989 p8
Let $R$ be a rectangle that is the union of a finite number of rectangles $R_i,$ $1 \leq i \leq n,$ satisfying the following conditions:
(i) The sides of every rectangle $R_i$ are parallel to the sides of $R.$
(ii) The interiors of any two different rectangles $R_i$ are disjoint.
(iii) Each rectangle $R_i$ has at least one side of integral length.
Prove that $R$ has at least one side of integral length.

Variant
Same problem but with rectangular parallelepipeds having at least one integral side.

IMO ISL 1989 p13  - 1989 IMO Problem 4 (ICE) 

Let $ABCD$ be a convex quadrilateral such that the sides $AB, AD, BC$ satisfy $AB = AD + BC.$ There exists a point $P$ inside the quadrilateral at a distance $h$ from the line $CD$ such that $AP = h + AD$ and $BP = h + BC.$ Show that:
$ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} + \frac {1}{\sqrt {BC}}$

by Eggert Briem

IMO ISL 1989 p14
A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle, i.e. both the incircle and circumcircle exists. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.

IMO ISL 1989 p17
Given seven points in the plane, some of them are connected by segments such that:
(i) among any three of the given points, two are connected by a segment;
(ii) the number of segments is minimal.
How many segments does a figure satisfying (i) and (ii) have? Give an example of such a figure.

IMO ISL 1989 p18
Given a convex polygon $A_1A_2 \ldots A_n$ with area $S$ and a point $M$ in the same plane, determine the area of polygon $M_1M_2 \ldots M_n,$ where $M_i$ is the image of $M$ under rotation $R^{\alpha}_{A_i}$ around $A_i$ by $\alpha_i, i = 1, 2, \ldots, n.$

IMO ISL 1989 p21
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $120^{\circ}.$

IMO ISL 1989 p24
For points $A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $\{A_1, \ldots ,A_5\}$ for which this maximum is attained.)

IMO ISL 1989 p28
Consider in a plane $P$ the points $O,A_1,A_2,A_3,A_4$ such that $\sigma(OA_iA_j) \geq 1 \quad \forall i, j = 1, 2, 3, 4, i \neq j.$ where $\sigma(OA_iA_j)$ is the area of triangle $OA_iA_j.$ Prove that there exists at least one pair $i_0, j_0 \in \{1, 2, 3, 4\}$ such that $\sigma(OA_iA_j) \geq \sqrt{2}.$

IMO ISL 1989 p29
155 birds $P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $C.$ Two birds $P_i, P_j$ are mutually visible if the angle at centre $m(\cdot)$ of their positions $m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $\{x,y\}$ of mutually visible pairs of birds with $x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $C$ can be occupied simultaneously by several birds, e.g. all possible birds.

IMO ISL 1989 p32
The vertex $A$ of the acute triangle $ABC$ is equidistant from the circumcenter $O$ and the orthocenter $H.$ Determine all possible values for the measure of angle  $A.$


IMO Shortlist 1990


IMO ISL 1990 p3
Let $n \geq 3$ and consider a set $E$ of $2n - 1$ distinct points on a circle. Suppose that exactly $k$ of these points are to be colored black. Such a coloring is good if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $n$ points from $E$. Find the smallest value of $k$ so that every such coloring of $k$ points of $E$ is good.

IMO ISL 1990 p5
Given a triangle $ABC$. Let $G$, $I$, $H$ be the centroid, the incenter and the orthocenter of triangle $ABC$, respectively. Prove that $\angle GIH > 90^{\circ}$.

IMO ISL 1990 p9
The incenter of the triangle $ABC$ is $K.$ The midpoint of $AB$ is $C_1$ and that of $AC$ is $B_1.$ The lines $C_1K$ and $AC$ meet at $B_2,$ the lines $B_1K$ and $AB$ at $C_2.$ If the areas of the triangles $AB_2C_2$ and $ABC$ are equal, what is the measure of angle $\angle CAB?$

IMO ISL 1990 p10
A plane cuts a right circular cone of volume $V$ into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the volume of the smaller part.

Original formulation:
A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

IMO ISL 1990 p11  - 1990 IMO Problem 1 (IND) 

Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $EB$. The tangent line at $E$ to the circle through $D$, $E$, and $M$ intersects the lines $BC$ and $AC$ at $F$ and $G$, respectively. If $\frac {AM}{AB} = t,$ find $\frac {EG}{EF}$ in terms of $t$.

by C.R. Pranesachar

IMO ISL 1990 p12
Let $ABC$ be a triangle, and let the angle bisectors of its angles $CAB$ and $ABC$ meet the sides $BC$ and $CA$ at the points $D$ and $F$, respectively. The lines $AD$ and $BF$ meet the line through the point $C$ parallel to $AB$ at the points $E$ and $G$ respectively, and we have $FG = DE$. Prove that $CA = CB$.

Original formulation:
Let $ABC$ be a triangle and $L$ the line through $C$ parallel to the side $AB.$ Let the internal bisector of the angle at $A$ meet the side $BC$ at $D$ and the line $L$ at $E$ and let the internal bisector of the angle at $B$ meet the side $AC$ at $F$ and the line $L$ at $G.$ If $GF = DE,$ prove that $AC = BC.$

IMO ISL 1990 p16
Prove that there exists a convex 1990-gon with the following two properties :
a.) All angles are equal.
b.) The lengths of the 1990 sides are the numbers $1^2$, $2^2$, $3^2$, $\cdots$, $1990^2$ in some order.

IMO ISL 1990 p19
Let $P$ be a point inside a regular tetrahedron $T$ of unit volume. The four planes passing through $P$ and parallel to the faces of $T$ partition $T$ into 14 pieces. Let $f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $f(P)$ as $P$ varies over $T.$

IMO ISL 1990 p28
Prove that on the coordinate plane it is impossible to draw a closed broken line such that
(i) the coordinates of each vertex are rational;
(ii) the length each of its edges is 1;
(iii) the line has an odd number of vertices.


IMO Shortlist 1991


IMO ISL 1991 p1
Given a point $P$ inside a triangle $\triangle ABC$. Let $D$, $E$, $F$ be the orthogonal projections of the point $P$ on the sides $BC$, $CA$, $AB$, respectively. Let the orthogonal projections of the point $A$ on the lines $BP$ and $CP$ be $M$ and $N$, respectively. Prove that the lines $ME$, $NF$, $BC$ are concurrent.

Original formulation:
Let $ABC$ be any triangle and $P$ any point in its interior. Let $P_1, P_2$ be the feet of the perpendiculars from $P$ to the two sides $AC$ and $BC.$ Draw $AP$ and $BP,$ and from $C$ drop perpendiculars to $AP$ and $BP.$ Let $Q_1$ and $Q_2$ be the feet of these perpendiculars. Prove that the lines $Q_1P_2,Q_2P_1,$ and $AB$ are concurrent.

IMO ISL 1991 p2
$ABC$ is an acute-angled triangle. $M$ is the midpoint of $BC$ and $P$ is the point on $AM$ such that $MB = MP$. $H$ is the foot of the perpendicular from $P$ to $BC$. The lines through $H$ perpendicular to $PB$, $PC$ meet $AB, AC$ respectively at $Q, R$. Show that $BC$ is tangent to the circle through $Q, H, R$ at $H$.

Original Formulation: 
For an acute triangle $ABC, M$ is the midpoint of the segment $BC, P$ is a point on the segment $AM$ such that $PM = BM, H$ is the foot of the perpendicular line from $P$ to $BC, Q$ is the point of intersection of segment $AB$ and the line passing through $H$ that is perpendicular to $PB,$ and finally, $R$ is the point of intersection of the segment $AC$ and the line passing through $H$ that is perpendicular to $PC.$ Show that the circumcircle of $QHR$ is tangent to the side $BC$ at point $H.$

IMO ISL 1991 p3
Let $S$ be any point on the circumscribed circle of $PQR.$ Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by $l(S, PQR).$ Suppose that the hexagon $ABCDEF$ is inscribed in a circle. Show that the four lines $l(A,BDF),$ $l(B,ACE),$ $l(D,ABF),$ and $l(E,ABC)$ intersect at one point if and only if $CDEF$ is a rectangle.

IMO ISL 1991 p4  - 1991 IMO Problem 5 (FRA) 

Let $\,ABC\,$ be a triangle and $\,P\,$ an interior point of  $\,ABC\,$. Show that at least one of the angles $\,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $30^{\circ }$.

by Johan Yebbou

IMO ISL 1991 p5
In the triangle $ABC,$ with $\angle A = 60 ^{\circ},$ a parallel $IF$ to $AC$ is drawn through the incenter $I$ of the triangle, where $F$ lies on the side $AB.$ The point $P$ on the side $BC$ is such that $3BP = BC.$ Show that $\angle BFP = \frac{\angle B}{2}.$

iSL p6 missing from aops post collection

IMO ISL 1991 p6 (probably) - 1991 IMO Problem 1 (USS) 

Given a triangle ABC, let I be the center of its inscribed circle. The internal bisectors of the angles A,B,C  meet the opposite sides in A’, B’ , C’ respectively. Prove that $\frac{1}{4}<\frac{AI\cdot BI\cdot CI}{AA'\cdot BB'\cdot CC'}\le \frac{8}{27}$

by Arkadii Skopenkov

IMO ISL 1991 p7
$ABCD$ is a terahedron: $AD+BD=AC+BC,$ $BD+CD=BA+CA,$ $CD+AD=CB+AB,$  $M,N,P$ are the mid points of $BC,CA,AB.$ $OA=OB=OC=OD.$ Prove that  $\angle MOP = \angle NOP =\angle NOM.$

IMO ISL 1991 p8
$S$ be a set of $n$ points in the plane. No three points of $S$ are collinear. Prove that there exists a set $P$ containing $2n - 5$ points satisfying the following condition: In the interior of every triangle whose three vertices are elements of $S$ lies a point that is an element of $P.$

IMO ISL 1991 p9
In the plane we are given a set $E$ of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of $E,$ there exist at least 1593 other points of $E$ to which it is joined by a path. Show that there exist six points of $E$ every pair of which are joined by a path.

Alternative version
Is it possible to find a set $E$ of 1991 points in the plane and paths joining certain pairs of the points in $E$ such that every point of $E$ is joined with a path to at least 1592 other points of $E,$ and in every subset of six points of $E$ there exist at least two points that are not joined?

IMO ISL 1991 p22
Real constants $a, b, c$ are such that there is exactly one square all of whose vertices lie on the cubic curve $y = x^3 + ax^2 + bx + c.$ Prove that the square has sides of length $\sqrt[4]{72}.$


IMO Shortlist 1992


IMO ISL 1992 p3
The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC \perp BD.$ Four squares, $ABEF,BCGH,CDIJ,DAKL,$ are erected externally on its sides. The intersection points of the pairs of straight lines $CL, DF, AH, BJ$ are denoted by $P_1,Q_1,R_1, S_1,$ respectively (left figure), and the intersection points of the pairs of straight lines $AI, BK, CE DG$ are denoted by $P_2,Q_2,R_2, S_2,$ respectively (right figure). Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2$ where $P_1,Q_1,R_1, S_1$ and $P_2,Q_2,R_2, S_2$ are the two quadrilaterals.

Alternative formulation
Outside a convex quadrilateral $ABCD$ with perpendicular diagonals, four squares $AEFB, BGHC, CIJD, DKLA,$ are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals $Q_1$ and $Q_2$ formed by the lines $AG, BI, CK, DE$ and $AJ, BL, CF, DH,$ respectively, are congruent.

IMO ISL 1992 p5
A convex quadrilateral has equal diagonals. An equilateral triangle is constructed on the outside of each side of the quadrilateral. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular.

Alternative formulation. 
Given a convex quadrilateral $ABCD$ with congruent diagonals $AC = BD.$ Four regular triangles are errected externally on its sides. Prove that the segments joining the centroids of the triangles on the opposite sides are perpendicular to each other.

Original formulation: 
Let $ABCD$ be a convex quadrilateral such that $AC = BD.$ Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $AB,BC,CD,DA$ respectively. Show that $O_1O_3$ is perpendicular to $O_2O_4.$

IMO ISL 1992 p7
Two circles $\Omega_{1}$ and $\Omega_{2}$ are externally tangent to each other at a point $I$, and both of these circles are tangent to a third circle $\Omega$ which encloses the two circles $\Omega_{1}$ and $\Omega_{2}$.
The common tangent to the two circles $\Omega_{1}$ and $\Omega_{2}$ at the point $I$ meets the circle $\Omega$ at a point $A$. One common tangent to the circles $\Omega_{1}$ and $\Omega_{2}$ which doesn't pass through $I$ meets the circle $\Omega$ at the points $B$ and $C$ such that the points $A$ and $I$ lie on the same side of the line $BC$.
Prove that the point $I$ is the incenter of triangle $ABC$.

Alternative formulation. 
Two circles touch externally at a point $I$. The two circles lie inside a large circle and both touch it. The chord $BC$ of the large circle touches both smaller circles (not at $I$). The common tangent to the two smaller circles at the point $I$ meets the large circle at a point $A$, where the points $A$ and $I$ are on the same side of the chord $BC$. Show that the point $I$ is the incenter of triangle $ABC$.

IMO ISL 1992 p8
Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions:
(i) its side lengths are $1, 2, 3, \ldots, 1992$ in some order;
(ii) the polygon is circumscribable about a circle.

Alternative formulation
Does there exist a 1992-gon with side lengths $1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.

IMO ISL 1992 p10
Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that $\vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert,$ where $\vert A \vert$ denotes the number of elements in the finite set $A$.

IMO ISL 1992 p11
In a triangle $ABC,$ let $D$ and $E$ be the intersections of the bisectors of $\angle ABC$ and $\angle ACB$ with the sides $AC,AB,$ respectively. Determine the angles $\angle A,\angle B, \angle C$ if $\angle BDE = 24 ^{\circ},$ $\angle CED = 18 ^{\circ}.$

IMO ISL 1992 p20  - 1992 IMO Problem 1 (FRA)

In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.

by Johan Yebbou