MIPT Metropolitan 2014-21 (Russia) 26p (-15)

geometry problems from Metropolitan Olympiad of Moscow Institute of Physics and Technology (MIPT) (Russia) with aops links in the names

Столичная олимпиада МФТИ

collected inside aops here

it is a also qualifying round for Psytech Olympiad

2014 , 2016 - 2021  (-2015)

2014 MIPT Metropolitan Olympiad 8.5

The line passing through the midpoint $M$ of the hypotenuse $BC$ of the right triangle $ABC$ parallel to line $AB$, intersects the extension of the bisector $BL$ of angle $ABC$ beyond point $L$ at point $P$. Find $\angle ACP$ if $\angle ABC=\alpha$.

2014 MIPT Metropolitan Olympiad 9.5

Let $BC$ be the longest side of the triangle $ABC$. On the side $AB$ point $K$ is selected, on the side $AC$ point $L$ and on the side $BC$ points $M$ and $N$ such that $AK=AL$, $BK=BN$, $CL=CM$. Prove that the points $K,L,M,N$ lie on the same circle.

2014 MIPT Metropolitan Olympiad 10.6

The square $ABCD$ is inscribed in the circle $\omega$. Let $E$ be the midpoint of side $AD$. Line $CE$ intersects the circle for second time at point $F$. Segments $FB$ and $AD$ intersect at point $H$. Find $HD$ if $AH =5$.

2014 MIPT Metropolitan Olympiad 11.6

In the triangular pyramid $ABCD$, four heights are drawn , perpendiculars from vertices to opposite faces. Let's call the height of the pyramid long if it is not shorter than each of the three heights of the triangle, which is by the face to which this height is drawn (for example, the height from the vertex $B$ - long if it is not shorter than each of the heights of the $ACD$ triangle). What the greatest number of long heights can have a pyramid $ABCD$?.

2015 missing

2016 MIPT Metropolitan Olympiad 9.2

In a right-angled triangle $ABC$, points $E$ and $F$ are selected on the hypotenuse $AB$, and on the legs $AC$ and $BC$ are points $P$ and $Q$ respectively so that $AE = P E = QF = BF$. Let $AP = a$, $P C = b$, $CQ = c$. Find $QB$.

2016 MIPT Metropolitan Olympiad 10.2

The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect the circumscribed circle around this triangle at points $Q$ and $P$, respectively. It is known that lines$AP$ and $CQ$ are parallel. Find the angle $ABC$.

2016 MIPT Metropolitan Olympiad 11.4

Points $M$ and $N$ divide the edge $SA$ of the $SABC$ pyramid into three equal parts ($SM = MN = NA$). What is the largest volume a pyramid can have if the lengths of the segments $SA, BN$ and $CM$ are equal to $a, b$ and $c$, respectively?

2017 MIPT Metropolitan Olympiad 9.3

The circle passing through vertices $A$ and $C$ of triangle $ABC$, intersects sides $AB$ and $CB$ at points $E$ and $F$. Lines passing through points $E$ and $F$ parallel to the sides of the triangle, intersects $AC$ at points $M$ and $N$, respectively. Prove that points $E, F, M, N$ lie on one circle if it is known that the segments $EM$ and $FN$ intersect.

2017 MIPT Metropolitan Olympiad 10.2

In a triangle, the centers of the inscribed and circumscribed circles are symmetric with respect to one of its sides. Find the angles of the triangle.

2017 MIPT Metropolitan Olympiad 11.4

Point $H$ is chosen inside an acute-angled triangle $ABC$. It is known that the product of the distance from point $H$ to any of the vertices of triangle $ABC$ with the distance from point $H$ to the opposite side of triangle $ABC$ to this vertex, is the same for each of the vertices. Prove that $H$ is the intersection point of the altitudes of the triangle $ABC$.

2018 MIPT Metropolitan Olympiad 8.2

On sides $BC, CD$ and $DA$ of square $ABCD$, points $L, M$ and $N$ are selected, respectively, so that $AN = CL =DM$. Find the angle $MLN$.

2018 MIPT Metropolitan Olympiad 9.2

From point $N$ on side $BC$ of triangle $ABC$, perpendicular $NP$ is drawn on side $AB$. It is known that the circle circumscribed around the triangle $BNP$ is tangent to the line $AN$, and the circle circumscribed around the triangle $ANC$ is tangent to the line $AB$. Find the largest angle of triangle $ABC$.

2018 MIPT Metropolitan Olympiad 10.2

A circle inscribed in a triangle divides its median into three segments of equal length. Prove that one side of the triangle is $2$ times the other.

2018 MIPT Metropolitan Olympiad 11.1

Find all triangles with angles $\alpha, \beta, \gamma$ such that the equality $\sin  \alpha= \cos \beta=  \cos \gamma$ holds.

2018 MIPT Metropolitan Olympiad 11.2

Two disjoint circles $\Gamma_1$ and $\Gamma_2$ are given with centers $O_1$ and $O_2$, respectively. Their common internal tangents intersect at point $O$. From point $O$ is drawn the perpendicular $OH$ on a common external tangent line $AB$ to these circles (points $A$ and $B$ are points of tangency with circles $\Gamma_1$ and $\Gamma_2$, respectively). Prove that rays $HO_1$ and $HO_2$ form equal angles with line $AB$.

2019 MIPT Metropolitan Olympiad 9.42

Altitudes $AD$ and $CE$ are drawn in an acute-angled triangle $ABC$. Points $M$ and $N$ are feet of perpendiculars drawn on the line $DE$ from points $A$ and $C$, respectively. What is the greatest length can have a segment $DN$, if $ME = 10$?

2019 MIPT Metropolitan Olympiad 10.22

In triangle $ABC$ the median $AM$, angle bisector $CL$ are drawn, point $D$ lies on the side $AC$. Find $AB$ if it is known that the midpoint of the bisector $CL$ is the midpoint of the segment $DM$, $AC = 10$ and $\cos \angle BAC =\frac{1}{4}$.

2019 MIPT Metropolitan Olympiad 11.42

A point $K$ is chosen on the median $AM$ of triangle ABC so that $\angle BAC + \angle BKC = 180^ o$. What is the maximum value that the difference $AB\cdot CK - AC \cdot BK$ can take if $AC = 3$?

2020 MIPT Metropolitan Olympiad 8.41

The altitude $BH$ is drawn in an acute-angled triangle$ABC$. It is known that the circle circumscribed around the triangle $ABH$ intersects the side $BC$ in its midpoint $M$. It is known that $\angle BAC=50^o$. Find the angle $BHM$.

2020 MIPT Metropolitan Olympiad 9.41

Let $I$ and $O$ be the centers of a circle inscribed in a non-isosceles acute-angled triangle $ABC$ and circumscribed around it, respectively,. Lines $AI$ and $CI$ intersect the circle circumscribed around the triangle at points $D$ and $E$, respectively. It is known that the points $E,I,O,D$ lie on one circle. Find the angle $ABC$.

2020 MIPT Metropolitan Olympiad 10.41

A point $D$ is selected at the base $AC$ of the triangle $ABC$. It is known that the distance between the centers of circles circumscribed around triangles $ABD$ and $CBD$ is twice the length of $AC$. Find the angle between the lines $AC$ and $DB$.

2020 MIPT Metropolitan Olympiad 11.42

For a convex quadrilateral $ABCD$, the following property holds: the projections of opposite sides onto one diagonal have equal lengths, and the projections of opposite sides onto the second diagonal have equal lengths (the projections lie on the diagonals). What is the smallest value that $\angle BCD$ can take if$\angle ADC=100^o$?

2021 MIPT Metropolitan Olympiad 8.21

Angle bisector $BAD$ of parallelogram $ABCD$ intersects side $BC$ at its midpoint , at point $M$. Find the angle $AMD$.

2021 MIPT Metropolitan Olympiad 9.31

A point $M$ is taken on side $BC$ of triangle $ABC$ so that $BM = AC$. Point $H$ is the foot of the perpendicular drawn from vertex $B$ on the segment $AM$. It is known that $BH = CM$ and $\angle MAC = 30^o$. Find angle $ACB$.

2021 MIPT Metropolitan Olympiad 10.31

Let $I$ and $O$ be, respectively, the centers of a circle inscribed in a non-isosceles acute-angled triangle $ABC$ and circumscribed around it. Direct lines $AI$ and $CI$ intersect the circumscribed circle of the triangle at points $D$ and $E$ respectively. It is known that the points $E, I, O, D$ lie on one circle. Find the angle $ADC$.

2021 MIPT Metropolitan Olympiad 11.31

Given a parallelogram $ABCD$ and a circle $S$. It is known that the circle $S$ passes through vertices $A, B$ and $D$ of the parallelogram, and also intersects its sides $BC$ and $DC$ at points $E$ and F respectively. Let $P$ and $Q$ be the points of intersection of lines $AE$ and $AF$ with diagonal $BD$. Find length $PQ$ if $AP + AQ = 10$ and $BD = 7$?

sources:

https://olimpiada.ru/activity/395/tasks/

https://olimpiada.ru/activity/132