Formula of Unity / Third Millennium 2013-21 (FdI) 56p

geometry problems from  Formula of Unity  /  The Third Millennium Olympiad  (Russian) [aslo know as Formulo de Intereco - FdI ]
(final rounds with aops links)

                                                                                                             collected inside aops 

round 1 here and round 2 here

2013 - 2021  final / 2nd  round

Formula of Unity / Third Millennium 2013 final 7.1

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.

Formula of Unity / Third Millennium 2013 final 8.1

In a hexagon $ABCDEF$, opposite sides are parallel, moreover, $AB=DE$. Prove that $BC=EF$ and $CD=FA$.

Formula of Unity / Third Millennium 2013 final 10.3 11.3

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.

Formula of Unity / Third Millennium 2014 final 7.3 8.2

Consider a circle and three equal chords passing through one point. Prove that each chord is a diameter.

Formula of Unity / Third Millennium 2014 final 11.4

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? 

Formula of Unity / Third Millennium 2015 final 7.3 8.3

Suppose $ABC$ is an isosceles triangle. Let $O$ be a point of intersection of medians $AA_1$ and $BB_1$. Given that $\angle AOB = 120^o$, find angles of the triangle $ABC$.

Formula of Unity / Third Millennium 2015 final 9.3

Points $M$ and $P$ are chosen on sides of $AB$ and $BC$ respectively of a square $ABCD$, so that $AM = CP$. A circle with the diameter $DP$ intersects the segment $CM$ at the point $K$. Prove that $MK \perp BK$.

Formula of Unity / Third Millennium 2015 final 10.3 11.3

Points $H, K$ and $M$ are marked respectively on the sides $BC, AC$ and $AB$ of a triangle $ABC$. Let $AH$ be an altitude. Prove that $HA$ is the bisector of \KHM if and only if $AH, BK$ and $CM$ intersect at the same point.

Formula of Unity / Third Millennium 2015 final 11.5

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 $D$.

Formula of Unity / Third Millennium 2016 final 8.3 9.2

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.

Formula of Unity / Third Millennium 2016 final 10.4 11.4

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.

Formula of Unity / Third Millennium 2017 final  7.3 8.3

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 2017 final 10.5 11.4

The angle between the diagonals of a trapezoid (a quadrilateral with exactly one pair of parallel sides) is equal to $60^o$. Prove that the sum of lengths of two legs is not less than the length of the greater base.

Formula of Unity / Third Millennium 2018 final 8.3

In a rhombus $ABCD$, points $E$ and $F$ are the centers of the sides $AB$ and $BC$ respectively. $P$ is such a point that $PA = PF, PE = PC$. Prove that $P$ lies on the line $BD$.

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.

Formula of Unity / Third Millennium 2018 final 10.5 11.4

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 $MB : MC = TB : TC$.

Formula of Unity / Third Millennium 2018 final 11.3

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?

Formula of Unity / Third Millennium 2019 final 7.1 11.1

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]

Formula of Unity / Third Millennium 2019 final 8.3 9.2

In a convex pentagon $ABCDE, \angle A = 60^o$, and all other angles are equal. It is known that $AB = 6$,$CD = 4$,$EA = 7$. Find the distance from $A$ to the line $CD$.

Formula of Unity / Third Millennium 2020 final 7.4
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)

Formula of Unity / Third Millennium 2020 final 8.2

The square is cut into five parts of equal area, as it is shown at the picture. Find the ratio $BE : EC$.

(A. R. Arab)

Formula of Unity / Third Millennium 2020 final 9.2 10.2 11.2

A rectangle $ABCD$ is folded along $MN$ so that the points $B$ and $D$ coincide. It turned out that $AD = AC'$. Find the ratio of the rectangle’s sides. 

(P. Mulenko)

Formula of Unity / Third Millennium 2021 final 8.3

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)

Formula of Unity / Third Millennium 2021 final 8.6

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)

Formula of Unity / Third Millennium 2021 final 9.4

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 )

Formula of Unity / Third Millennium 2021 final 10.2 11.2

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)

Formula of Unity / Third Millennium 2021 final  7.5 (a) 11.6 (b)

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

Formula of Unity / Third Millennium 2013 first round  10.4 11.4

In an acute triangle $ABC$ the angle $C=45^o$ , $AA_1$ and $BB_1$ are the heights. Prove that $A_1B_1 = \sqrt{\frac{A_1B^2 + A_1C^2}{2}}$.

Formula of Unity / Third Millennium 2014 first round  8.2 9.2

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 $BAC$.

Formula of Unity / Third Millennium 2014 first round  10.5 11.5

Given a triangle $ABC$ with a height $CH$ and its circumcenter $O$. Let T be a point on $AO$ such that $AO \perp CT$ and let $M$ be an intersection point of $HT$ and $BC$. Find the ratio of lengths of the segments $BM$ and $CM$.

Formula of Unity / Third Millennium 2015 first round 8.2 

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 $BAP$?

Formula of Unity / Third Millennium 2015 first round 9.8 10.8

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.

Formula of Unity / Third Millennium 2015 first round 9.9 10.9 11.9

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.

Formula of Unity / Third Millennium 2015 first round 9.10

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 $\vartriangle ABC$

Formula of Unity / Third Millennium 2015 first round 10.1

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 $BAP$?

Formula of Unity / Third Millennium 2016 first round 10.6
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.

Formula of Unity / Third Millennium 2017 first round 8.4 9.4
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?

Formula of Unity / Third Millennium 2017 first round 10.4 11.4

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.

Formula of Unity / Third Millennium 2018 first round 7.3 8.2

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 $MN$.

Formula of Unity / Third Millennium 2018 first round 9.5

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) $MA \perp PQ$.

Formula of Unity / Third Millennium 2018 first round 10.5 11.4 

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 $AN = BM$.

Formula of Unity / Third Millennium 2019 first round 8.5 9.4

On the sides AB and AD of a square ABCD, two equilateral triangles ALD and ABK are built, such that L is inside the square but K is outside of it. Prove that K lies on the segment DE.

Formula of Unity / Third Millennium 2019 first round 9.6

In a convex quadrilateral diagonals are perpendicular. Can the lengths of its sides be equal to four consecutive integers?

Formula of Unity / Third Millennium 2019 first round 10.6 11.5

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 $\frac{IK}{IA} +\frac{IL}{IB} +\frac{IM}{IC }\ge \frac32$ .

Formula of Unity / Third Millennium 2020 first round 8.5

$ABC$ 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)

Formula of Unity / Third Millennium 2020 first round 9.4

A triangle is inscribed into a circle of diameter $5$. Find all possible values of the perimeter of the triangle if each its side has integer length (and prove that other values are impossible).

 (P. Mulenko)

Formula of Unity / Third Millennium 2020 first round 10.2 11.2

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)

Formula of Unity / Third Millennium 2020 first round 10.4 11.4

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 )

Formula of Unity / Third Millennium 2021 first round  8.3

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)

Formula of Unity / Third Millennium 2021 first round  9.2

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 )

Formula of Unity / Third Millennium 2021 first round  9.4

$CF$ 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)

Formula of Unity / Third Millennium 2021 first round  10.8 11.8

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)

sources: www.formulo.org, matholimp.livejournal.com