geometry problems from Formula of Unity / The Third Millennium Olympiad (Russian) [aslo know as Formulo de Intereco - FdI ]
(final rounds with aops links)
(final rounds with aops links)
collected inside aops
2013 - 2021 final / 2nd round
A square is divided into five rectangles (see the picture). Prove that if the areas of the four "corner" rectangles are equal then the "central" one is a square.
In a hexagon $ABCDEF$, opposite sides are parallel, moreover, $AB=DE$. Prove that $BC=EF$ and $CD=FA$.
The points $X$ and $Y$ are taken on the sides $AB$ and $BC$ of a triangle $ABC$. $AY$ and $CX$ intersect each other at the point $Z$. It is known that $AY = YC$ and $AB = CZ$. Prove that the points $B, Y,Z,X$ lie on a circumference.
Consider a circle and three equal chords passing through one point. Prove that each chord is a diameter.
Does there exist a tetrahedron with height of $60$ cm, based perimeter $62$ cm and the height of every lateral face (drawn to a side of the base) $61$ cm?
Suppose $ABCD$ is a regular tetrahedron with edge of the length $1$. Through an interior point of face $ABC$ three planes are drawn parallel to faces $ABD, ACD$ and $BCD$ respectively. These planes split the tetrahedron in several parts. Find the total sum of the lengths of the edges of the part that contains vertex $D$.
Given a triangle $ABC$ with points $M$ and $N$ on the sides $AB$ and $AC$ respectively, such that $AM = AN$. Segments $CM$ and $BN$ intersect in $O$, so that $BO = CO$. Prove that the triangle $ABC$ is isosceles.
In a convex quadrilateral $ABCD$, there is a point $E$ inside the triangle $ADC$, such that $\angle BAE = \angle BEA = 80^o$, $\angle CAD =\angle CDA = 80^o$, $\angle EAD = \angle EDA = 50^o$. Prove that $\vartriangle BEC$ is equilateral.
Inside a triangle $\vartriangle ABC$, a point $D$ is given such that $BD + AC < BC$. Prove that $\angle DAC + \angle ADB > 180^o$.
Formula of Unity / Third Millennium 2018 final 9.4 (All Soviet Union 1968)
In a convex octagon, all the angles are equal, and the length of each side is a rational number. Prove that the octagon has a center of symmetry.
Let $ABC$ be a triangle in which all angles are less than $120^o$ and $AB \ne AC$. Consider a point $T$ inside $\vartriangle ABC$ such that $\angle BTC = \angle CTA = \angle ATB = 120^o$. Let the line $BT$ intersect the side $AC$ in $E$, and the line $CT$ intersect the side $AB$ in $F$. Prove that the lines $EF$ and $BC$ have a common point $M$, and $MB : MC = TB : TC$.
Can you come up with a closed spatial broken line made of $5$ segments such that all the segments are of the same length, and any two connected segments are perpendicular to each other?
A room has the shape of a triangle $\vartriangle ABC$ ($\angle B = 60^o$, $\angle C = 45^o$, $AB = 5 m$). A bee that was sitting in the corner $A$, starts flying in a straight line, in a random direction, turning $60$ degrees whenever it hits a wall. (See the picture.) Is it possible that after some time the bee will have flown more than
a) $9.9$ meters? [grade 7]
b) $12$ meters? [grade 11]
The square is cut into four parts of equal area, as it is shown at the picture. Find the ratio $BE : EC$.
(A. R. Arab)
(A. R. Arab)
(P. Mulenko)
In a parallelogramm $ABCD$ ($AB \ne BC$), two heights $BH$ and $BK$ are dropped from the obtuse angle $B$. The points $H$ and $K$ lie on sides and do not coincide with vertices. The triangle $BHK$ is isosceles. Find all possible values of angle $BAD$.
(L. Koreshkova)
A rectangle of size $2021\times 4300$ is given. There is a billiard ball at some point inside it. It is launched along a straight line forming an angle $45^o$ with the sides of the rectangle. After reaching a side, the ball is reflected also at angle $45^o$; if the ball enters a corner, it leaves it along the same line along which it entered. (An example of the beginning of the ball’s path is shown on the picture.)
a) Is it true for any point that if you launch a ball from it according to these rules, it will return there again?
b) Suppose that, starting in a point A, the ball returns to it again. What maximal amount of hits can the ball make before it returns to A first time?
(O. Pyaive)
Points $A, B, C, D$, are chosen on a plane so that $AB = BC = CD$, $BD = DA = AC$. Find the angles of the quadrilateral with vertices in these points.
(A. Tesler )
In a triangle $ABC$, $O_1$ is the incenter, and $O_2$ is the center of the tangent circle for the side $BC$ and the extensions of the other sides. A point $D$ is chosen on the arc $BO_2$ of the circumcircle of $\vartriangle O_1O_2B$, so that $\angle BO_2D$ is a half of $\angle BAC$. $M$ is the midpoint of the arc $BC$ of the circumcircle of $\vartriangle ABC$. Prove that the points $D, M, C$ lie on one straight line.
(O. Pyaive)
There is an equilateral triangle on the plane. There are three circles with centers in its vertices. Each circle radius is less than the triangle’s height. Points on the plane are colored in such a way: if a point is inside exactly one circle, it is colored yellow; if a point is inside exactly two circles, it is colored green; and if it is inside all three circles, it is colored blue. It turned out that the yellow area is equal to $1000$, the green area is equal to $100$, and the blue area is equal to $1$.
a) Find the area of the triangle. (P. Mulenko)
b) Find out what is greater: the length of the side of the triangle, or sum of lengths of green segments located on the sides of the triangle. (P. Mulenko, A. Tesler )
2013 - 2021 1st round
In an acute triangle $ABC$ the angle $C=45^o$ , $AA_1$ and $BB_1$ are the heights. Prove that $A_1B_1 = \sqrt{\frac{A_1B^2 + A_1C^2}{2}}$.
Given rectangle $ABCD$ and a point $K$ on the ray $DC$ such that $DK = BD$. Let $M$ be a midpoint of segment $BK$. Prove that $AM$ is the bisector of the angle $BAC$.
Let $BK$ be a bisector of triangle $ABC$. Given that $AB = AC$ and $BC =AK + BK$, find the angles of the triangle.
Formula of Unity / Third Millennium 2015 first round 8.10 (also 2016 1st 8.6 9.4 10.4 11.4)
Angles $B$ and $C$ of a triangle $ABC$ are equal $30^o$ and $105^o$ and $P$ is the midpoint of $BC$. What is the value of angle $BAP$?
On the side $AB$ of triangle $ABC$ a point $D$ is marked so that $\angle ACD =\angle ABC$. Let $S$ be the circumcenter of $\vartriangle BCD$, and $P$ be the midpoint of $BD$. Prove that the points A$, C, S,$ and $P$ belong to the same circle.
In triangles $ABC$ and $A_1B_1C_1$, $\sin A =\ cos A_1, \sin B = \cos B_1, \sin C =\cos C_1$. Find all possible values of the largest of these six angles.
A point $H$ inside a triangle $ABC$ is such that $\angle HAB = \angle HCB$ and $\angle HBC = \angle HAC$. Prove that $H$ is the orthocenter of $\vartriangle ABC$
On each side of a given square mark a point so that the quadrilateral with vertices at these points had minimal perimeter.
Formula of Unity / Third Millennium 2016 first round 8.6 9.4 10.4 11.4 (also 2015 1st 8.10)
Angles $B$ and $C$ of a triangle $ABC$ are equal $30^o$ and $105^o$ and $P$ is the midpoint of $BC$. What is the value of angle $BAP$?
The inscribed circle of a triangle $\vartriangle ABC$ is tangent to $AB, BC,AC$ at point $C_1, A_1,B_1$ respectively. Prove the inequality: $$\frac{AC}{AB_1}+\frac{CB}{CA_1}+\frac{BA}{BC_1}> 4$$
Formula of Unity / Third Millennium 2016 first round 11.6
Give an example of $4$ positive numbers that cannot be radii of $4$ pairwise tangent spheres.
Let $E$ be the intersection point of the diagonals of a parallelogram $ABCD$. The bisectors of angles $DAE$ and $EBC$ intersect at F. Find the measure of $\angle AFB$ if $ECFD$ is a parallelogram.
Formula of Unity / Third Millennium 2017 first round 9.5 10.3
Diagonals of the faces of a box are equal to $4, 6$ and $7$ decimeters respectively. Would a ball of diameter $2$ decimeters fit into that box?
On the sides $AB$ and $BC$ of a triangle $ABC$, points $X$ and $Y$ are selected, so that $AX =BY$ . Points $A, X, Y, C$ lie on the same circle. Let $B_1$ be the foot of the bisector of the angle $B$. Prove that the lines $XB_1$ and $YC$ are parallel.
A point $E$ lies on the side $CD$ of a square $ABCD$. The bisectors of the angles $EAB$ and $EAD$ intersect sides $BC$ and $CD$ at points $M$ and $N$ respectively. $F$ is such a point on the ray $AE$ that $AF = AB$. Prove that $F$ lies on the line $MN$.
An isosceles triangle $ABC$ has a right angle $A$. Two equal acute triangles $ABP$ and $ACQ$ ($PB = AQ$) are built on the sides $AB$ and $AC$ outside of $\vartriangle ABC$. The lines $PB$ and $CQ$ intersect at $M$. Prove that:
(a) $PA \perp QC$
(b) $MA \perp PQ$.
A point $O$ is the center of an equilateral triangle $ABC$. A circle that passes through points $A$ and $O$ intersects the sides $AB$ and $AC$ at points $M$ and $N$ respectively. Prove that $AN = BM$.
In a convex quadrilateral diagonals are perpendicular. Can the lengths of its sides be equal to four consecutive integers?
Angle bisectors $AK, BL$, and $CM$ of a triangle $ABC$ intersect in the point $I$ (the points $K, L, M$ lie at the sides of the triangle). Prove that $\frac{IK}{IA} +\frac{IL}{IB} +\frac{IM}{IC }\ge \frac32$ .
$ABC$ and $CDE$ are two isosceles right triangles with the hypotenuse lengths $BC = 7$ and $CE = 14$. $C$ lies on the segment $BE$, and points $A$ and $D$ are at the same side of line $BE$. $O$ is the intersection point of segments $AE$ and $BD$. Find the area of the triangle $ODE$.
(A. R. Arab)
(P. Mulenko)
In $\vartriangle ABC, AB = 6,BC = 4,AC = 8$. A point $M$ is chosen on the side $AC$ so that the incircles of the triangles $ABM$ and $BCM$ have a common point. Find the ratio of the areas of these triangles.
(L.Koreshkova)
The surface of a wooden $1$ m$^3$ cube is painted. From each vertex, a pyramid is cut off, as a result, a polyhedron with $14$ faces is obtained. Each painted face is a rectangle, and each unpainted face is an equilateral triangle (the triangles are not necessarily equal). Find the total area of the painted faces of the polyhedron if it is\sqrt3 times less than the total area of the unpainted ones.
(A. Tesler )
In a triangle $ABC$, a segment $AD$ is a bisector. Points $E$ and $F$ are on the sides $AB$ and $AC$ respectively, and $\angle AEF=\angle ACB$ . Points $I$ and $J$ are the incenters (i. e. intersection points of bisectors) of the triangles $AEF$ and $BDE$ respectively. Find $\angle EID + \angle EJD$.
(Amir Reza Arab)
The midline of a triangle divides it into two parts , a triangle and a trapezoid. This trapezoid is also divided into two parts by its midline. As a result we obtain three parts , one triangle and two trapezoids. The areas of two of these parts are integers. Prove that the area of the third part is also an integer.
(A. Tesler )
$CF$ is an angle bisector of a triangle $ABC$. Point $O$ is chosen on $CF$ such that $FO\cdot FC = FB^2$. E is the intersection point of $BO$ and $AC$. Prove that $FB = FE$.
(O. Pyaive)
An $n$-sided regular polygon is inscribed into a circle of radius $R$. The point $M$ moves along this circle, and for each its position we consider the sum of distances from the point $M$ to the lines containing the sides of the polygon. Find all the positions of point $M$ for this sum to be minimal.
(O. Pyaive)
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