geometry problems from Mathlinks contests with aops links in the names
(a few with solutions in pdf)
Aops post Collections:
Let $ABC$ be a triangle with its centroid $G$. Let $D$ and $E$ be points on segments $AB$ and $AC$, respectively, such that, $\frac{AB}{AD}+\frac{AC}{AE}=3.$ Prove that the points $D, G$ and $E$ are collinear.
Dorlir Ahmeti, Kosovo
7th edition
Given is an acute triangle $ ABC$ and the points $ A_1,B_1,C_1$, that are the feet of its altitudes from $ A,B,C$ respectively. A circle passes through $ A_1$ and $ B_1$ and touches the smaller arc $ AB$ of the circumcircle of $ ABC$ in point $ C_2$. Points $ A_2$ and $ B_2$ are defined analogously.
Prove that the lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ have a common point, which lies on the Euler line of $ ABC$.
Let $ ABC$ be a given triangle with the incenter $ I$, and denote by $ X$, $ Y$, $ Z$ the intersections of the lines $ AI$, $ BI$, $ CI$ with the sides $ BC$, $ CA$, and $ AB$, respectively. Consider $ \mathcal{K}_{a}$ the circle tangent simultanously to the sidelines $ AB$, $ AC$, and internally to the circumcircle $ \mathcal{C}(O)$ of $ ABC$, and let $ A^{\prime}$ be the tangency point of $ \mathcal{K}_{a}$ with $ \mathcal{C}$. Similarly, define $ B^{\prime}$, and $ C^{\prime}$.
Prove that the circumcircles of triangles $ AXA^{\prime}$, $ BYB^{\prime}$, and $ CZC^{\prime}$ all pass through two distinct points.
Let $ A,B,C,D,E$ be five distinct points, such that no three of them lie on the same line. Prove that
$ AB+BC+CA + DE < AD + AE + BD+BE + CD+CE .$
Let $ A^{\prime}$ be an arbitrary point on the side $ BC$ of a triangle $ ABC$. Denote by $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ the circles simultanously tangent to $ AA^{\prime}$, $ A^{\prime}B$, $ \Gamma$ and $ AA^{\prime}$, $ A^{\prime}C$, $ \Gamma$, respectively, where $ \Gamma$ is the circumcircle of $ ABC$. Prove that $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ are congruent if and only if $ AA^{\prime}$ passes through the Nagel point of triangle $ ABC$.
(If $ M,N,P$ are the points of tangency of the excircles of the triangle $ ABC$ with the sides of the triangle $ BC$, $ CA$ and $ AB$ respectively, then the Nagel point of the triangle is the intersection point of the lines $ AM$, $ BN$ and $ CP$.)
Let $ \Omega$ be the circumcircle of triangle $ ABC$. Let $ D$ be the point at which the incircle of $ ABC$ touches its side $ BC$. Let $ M$ be the point on $ \Omega$ such that the line $ AM$ is parallel to $ BC$. Also, let $ P$ be the point at which the circle tangent to the segments $ AB$ and $ AC$ and to the circle $ \Omega$ touches $ \Omega$. Prove that the points $ P$, $ D$, $ M$ are collinear.
Prove that the set of all the points with both coordinates begin rational numbers can be written as a
reunion of two disjoint sets $ A$ and $ B$ such that any line that that is parallel with $ Ox$, and
respectively $ Oy$ intersects $ A$, and respectively $ B$ in a finite number of points.
3rd MathLinks Contest 4.3
sources: Mathlinks Contest Aops Forum, [Old] Mathlinks Contest Collections
6th edition
Let $ABCD$ be a rectangle of center $O$ in the plane $\alpha$, and let $V \notin\alpha$ be a point in
space such that $V O \perp \alpha$. Let $A' \in (V A)$, $B'\in (V B)$, $C'\in (V C)$, $D'\in (V D)$
be four points, and let $M$ and $N$ be the midpoints of the segments $A'C'$ and $B'D'$.
Prove that $MN \parallel \alpha$ if and only if $V , A', B', C', D'$ all lie on a sphere.
6th MathLinks Contest 3.2
Let $ABCD$ be a convex quadrilateral, and the points $A_1 \in (CD)$,
$A_2 \in (BC)$, $C_1 \in (AB)$, $C_2 \in (AD)$. Let $M, N$ be the intersection points between the
lines $AA_2, CC_1$ and $AA_1, CC_2$ respectively. Prove that if three of the quadrilaterals $ABCD$,
$A_2BC_1M$, $AMCN$, $A_1NC_2D$ are circumscriptive (i.e. there exists an incircle tangent to all
the sides of the quadrilateral) then the forth quadrilateral is also circumscriptive.
6th MathLinks Contest 5.3
Let $ABC$ be a triangle, and let $ABB_2A_3$, $BCC_3B_1$ and $CAA_1C_2$ be squares constructed
outside the triangle. Denote with $S$ the area of the triangle $ABC$ and with s the area of the triangle
formed by the intersection of the lines $A_1B_1$, $B_2C_2$ and $C_3A_3$.
Prove that $s \le (4 - 2\sqrt3)S$.
6th MathLinks Contest 6.3
Let $C_1, C_2$ and $C_3$ be three circles, of radii $2, 4$ and $6$ respectively. It is known that each
of them are tangent exteriorly with the other two circles. Let $\Omega_1$ and $\Omega_2$ be two more
circles, each of them tangent to all of the $3$ circles above, of radius $\omega_1$ and $\omega_2$
respectively. Prove that $\omega_1 + \omega_2 = 2\omega_1\omega_2$.
6th MathLinks Contest 7.2
Let $ABCD$ be a cyclic quadrilateral. Let $M, N$ be the midpoints of the diagonals $AC$ and $BD$
and let $P$ be the midpoint of $MN$. Let $A',B',C',D'$ be the intersections of the rays $AP$, $BP$,
$CP$ and $DP$ respectively with the circumcircle of the quadrilateral $ABCD$.
Find, with proof, the value of the sum
\[ \sigma = \frac{ AP}{PA'} + \frac{BP}{PB'} + \frac{CP}{PC'} + \frac{DP}{PD'} . \]
A lattice point in the Carthesian plane is a point with both coordinates integers. A rectangle minor
(respectively a square minor) is a set of lattice points lying inside or on the boundaries of a rectangle
(respectively square) with vertices lattice points. We assign to each lattice point a real number, such
that the sum of all the numbers in any square minor is less than $1$ in absolute value. Prove that the
sum of all the numbers in any rectangle minor is less than $4$ in absolute value.
5th edition
Let $ABC$ be a triangle and let $A' \in BC$, $B' \in CA$ and $C' \in AB$ be three collinear points.
a) Prove that each pair of circles of diameters $AA'$, $BB'$ and $CC'$ has the same radical axis;
b) Prove that the circumcenter of the triangle formed by the intersections of the lines $AA' , BB'$ and
$CC'$ lies on the common radical axis found above.
Suppose that $\{D_n\}_{n\ge 1}$ is an finite sequence of disks in the plane whose total area is less than
$1$. Prove that it is possible to rearrange the disks so that they are disjoint from each other and all
contained inside a disk of area $4$.
Let $ABC$ be an acute angled triangle. Let $M$ be the midpoint of $BC$, and let $BE$ and $CF$ be
the altitudes of the triangle. Let $D \ne M$ be a point on the circumcircle of the triangle $EFM$ such
that $DE = DF$. Prove that $AD \perp BC$.
Given is a unit cube in space. Find the maximal integer $n$ such that there are $n$ points, satisfying the following conditions:
(a) All points lie on the surface of the cube;
(b) No face contains all these points;
(c) The $n$ points are the vertices of a polygon.
Let $ABC$ be a triangle and let $C$ be a circle that intersects the sides $BC, CA$ and $AB$ in the
points $A_1, A_2, B_1, B_2$ and $C_1, C_2$ respectively. Prove that if $AA_1, BB_1$ and $CC_1$
are concurrent lines then $AA_2, BB_2$ and $CC_2$ are also concurrent lines.
Given is a square of sides $3\sqrt7 \times 3\sqrt7$. Find the minimal positive integer $n$ such that no
matter how we put $n$ unit disks inside the given square, without overlapping, there exists a line that
intersects $4$ disks.
4th edition
Let $\Omega_1(O_1, r_1)$ and $\Omega_2(O_2, r_2)$ be two circles that intersect in two points $X, Y$ . Let $A, C$ be the points in which the line $O_1O_2$ cuts the circle $\Omega_1$, and let $B$ be the point in which the circle $\Omega_2$ itnersect the interior of the segment $AC$, and let $M$ be the intersection of the lines $AX$ and $BY$ .
Prove that $M$ is the midpoint of the segment $AX$ if and only if $O_1O_2 =\frac12 (r_1 + r_2)$.
Prove that the six sides of any tetrahedron can be the sides of a convex hexagon.
Let $ABC$ be a triangle, and let $C$ be its circumcircle. Let $T$ be the circle tangent to $AB, AC$ and $C$ internally in the points $F, E$ and $D$ respectively. Let $P, Q$ be the intersection points between the line $EF$ and the lines $DB$ and $DC$ respectively. Prove that if $DP = DQ$ then the triangle $ABC$ is isosceles.
We say that two triangles $T_1$ and $T_2$ are contained one in each other, and we write $T_1 \subset T_2$, if and only if all the points of the triangle $T_1$ lie on the sides or in the interior of the triangle $T_2$.
Let $\Delta$ be a triangle of area $S$, and let $\Delta_1 \subset \Delta$ be the largest equilateral triangle with this property, and let $\Delta \subset \Delta_2$ be the smallest equilateral triangle with this property (in terms of areas). Let $S_1, S_2$ be the areas of $\Delta_1, \Delta_2$ respectively.
Prove that $S_1S_2 = S^2$.
Bonus question: : Does this statement hold for quadrilaterals (and squares)?
Let $ABCD$ be a convex quadrilateral, and let $K$ be a point on side$ AB$ such that $\angle KDA = \angle BCD$. Let $L$ be a point on the diagonal $AC$ such that $KL \parallel BC$. Prove that $\angle KDB = \angle LDC$.
Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.
Let $\Omega$ be the incircle of a triangle $ABC$. Suppose that there exists a circle passing through $B$ and $C$ and tangent to $\Omega$ in $A'$. Suppose the similar points $B'$, $C'$ exist. Prove that the lines $AA', BB'$ and $CC'$ are concurrent.
3rd edition
Let $P$ be the set of points in the Euclidian plane, and let $L$ be the set of lines in the same plane.
Does there exist an one-to-one mapping (injective function) $f : L \to P$ such that for each $\ell \in L$
we have $f(\ell) \in \ell$?
3rd MathLinks Contest 2.2
Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters
$BC, CA, AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all three semicircles
has radius $t$. Prove that $$\frac{s}{2} < t \le \frac{s}{2} + \left( 1 - \frac{\sqrt3}{2} \right)r.$$
Each point in the Euclidean space is colored with one of $n \ge 2$ colors, and each of the $n$ colors is
used. Prove that one can find a triangle such that the color assigned to the orthocenter is different from
all the colors assigned to the vertices of the triangle.
Let $S$ be a nonempty set of points of the plane. We say that $S$ determines the distance $d > 0$ if there are two points $A, B$ in $S$ such that $AB = d$. Assuming that $S$ does not contain $8$ collinear points and that it determines not more than $91$ distances, prove that $S$ has less than $2004$ elements.
An integer point of the usual Euclidean $3$-dimensional space is a point whose three coordinates are
all integers. A set $S$ of integer points is called a covered set if for all points $A, B$ in $S$ each integer
point in the segment $[AB]$ is also in $S$. Determine the maximum number of elements that a covered
set can have if it does not contain $2004$ collinear points.
3rd MathLinks Contest 5.3
We say that a tetrahedron is median if and only if for each vertex the plane that passes through the
midpoints of the edges emerging from the vertex is tangent to the inscribed sphere. Also a tetrahedron is
called regular if all its faces are congruent. Prove that a tetrahedron is regular if and only if it is median.
3rd MathLinks Contest 6.1
For a triangle $ABC$ and a point $M$ inside the triangle we consider the lines $AM, BM,CM$ which
intersect the sides $BC, CA, AB$ in $A_1, B_1, C_1$ respectively. Take $A', B', C'$ to be the
intersection points between the lines $AA_1, BB_1, CC_1$ and $B_1C_1, C_1A_1, A_1B_1$
respectively.
a) Prove that the lines $BC', CB'$ and $AA'$ intersect in a point $A_2$;
b) Define similarly points $B_2, C_2$. Find the loci of $M$ such that the triangle $A_1B_1C_1$ is
similar with the triangle $A_2B_2C_2$.
2nd edition
Given are on a line three points $A, B, C$ such that $AB = 1$ and $BC = x$. Consider the circles
$\Omega_a, \Omega_b$ and $\Omega_c$ which are tangent to the given line at the points $A, B, C$
respectively, and such that $\Omega_b$ is tangent externally with both $\Omega_a$ and $\Omega_c$
in points $M, N$ respectively. Find all values of the radius of the circle $\Omega_b$ for which the
triangle $BMN$ is isosceles
2nd MathLinks Contest 2.3
Prove that if two triangles are inscribed in the same circle, then their incircles are not strictly contained
one into each other.
2nd MathLinks Contest 3.2
Let $ABC$ be a triangle with altitudes $AD, BE, CF$. Choose the points $A_1, B_1, C_1$ on the lines
$AD, BE, CF$ respectively, such that$$\frac{AA_1}{AD}= \frac{BB_1}{BE}= \frac{CC_1}{CF} = k.$$Find all values of $k$ such that the triangle $A_1B_1C_1$ is similar to the triangle $ABC$ for all triangles $ABC$.
2nd MathLinks Contest 4.2
Given is a finite set of points $M$ and an equilateral triangle $\Delta$ in the plane. It is known that for
any subset $M' \subset M$, which has no more than $9$ points, can be covered by two translations of
the triangle $\Delta$. Prove that the entire set $M$ can be covered by two translations of $\Delta$.
2nd MathLinks Contest 5.1
For which positive integers $n \ge 4$ one can find n points in the plane, no three collinear, such that for
each triangle formed with three of the $n$ points which are on the convex hull, exactly one of the $n - 3$
remaining points belongs to its interior.
2nd MathLinks Contest 6.2
A triangle $ABC$ is located in a cartesian plane $\pi$ and has a perimeter of $3 + 2\sqrt3$. It is known
that the triangle $ABC$ has the property that any triangle in the plane $\pi$, congruent with it, contains
inside or on the boundary at least one lattice point (a point with both coordinates integers). Prove that
the triangle $ABC$ is equilateral.
2nd MathLinks Contest 7.3
A convex polygon $P$ can be partitioned into $27$ parallelograms. Prove that it can also be partitioned
into $21$ parallelograms.
1st edition
In a triangle $\vartriangle ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular
bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect ∠A. Prove that $PQ$ is smaller than
$AB$ if and only if $\angle B$ is obtuse.
1st MathLinks Contest 2.3
Prove that in any acute triangle with sides $a, b, c$ circumscribed in a circle of radius $R$ the following
inequality holds: $$\frac{\sqrt2}{4} <\frac{Rp}{2aR + bc} <\frac{1}{2}$$
where $p$ represents the semi-perimeter of the triangle.
A pack of $2003$ circus flees are deployed by their circus trainer, named Gogu, on a sufficiently large
table, in such a way that they are not all lying on the same line. He now wants to get his Ph.D. in fleas
training, so Gogu has thought the fleas the following trick: we chooses two fleas and tells one of them
to jump over the other one, such that any flea jumps as far as twice the initial distance between him and
the flea over which he is jumping. The Ph.D. circus committee has but only one task to Gogu: he has to
make the his flees gather around on the table such that every flea represents a vertex of a convex polygon
. Can Gogu get his Ph.D., no matter of how the fleas were deployed?
1st MathLinks Contest 4.1Given are $4004$ distinct points, which lie in the interior of a convex polygon of area $1$.
Prove that there exists a convex polygon of area $\frac{1}{2003}$, included in the given polygon,
such that it does not contain any of the given points in its interior
1st MathLinks Contest 4.3
Find the triangle of the least area which can cover any triangle with sides not exceeding $1$.
1st MathLinks Contest 5.1
In a triangle $ABC$, $\angle B = 70^o$, $\angle C = 50^o$. A point $M$ is taken on the side $AB$
such that $\angle MCB = 40^o$ , and a point $N$ is taken on the side $AC$ such that $\angle NBC = 50^o$. Find $\angle NMC$.
1st MathLinks Contest 6.2
Given is a triangle $ABC$ and on its sides the triangles $ABM, BCN$ and $CAP$ are build such that
$\angle AMB = 150^o$, $AM = MB$, $\angle CAP = \angle CBN = 30^o$, $\angle ACP = \angle BCN = 45^o$
. Prove that the triangle $MNP$ is an equilateral triangle.
1st MathLinks Contest 7.2
Consider the circles $\omega$, $\omega_1$, $\omega_2$, where $\omega_1$, $\omega_2$ pass
through the center $O$ of $\omega$. The circle $\omega$ cuts $\omega_1$ at $A, E$ and $\omega_2$
at $C, D$. The circles $\omega_1$ and $\omega_2$ intersect at $O$ and $M$. If A$D$ cuts $CE$ at
$B$ and if $MN \perp BO$, ($N \in BO$) prove that $2MN^2 \le BM \cdot MO$.
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