Olympiad problems

2018 Belarus Team Selection Test, 1.4

Tags: inequalities , geometry

Let $A_1H_1,A_2H_2,A_3H_3$ be altitudes and $A_1L_1,A_2L_2,A_3L_3$ be bisectors of acute-angles triangle $A_1A_2A_3$. Prove the inequality $S(L_1L_2L_3)\ge S(H_1H_2H_3)$ where $S$ stands for the area of a triangle. [i](B. Bazylev)[/i]

2019 Balkan MO, 3

Tags: geometry

Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that: $1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively. $2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively. Prove that $KL$ and $ST$ intersect on the line $BC$.

2007 Bulgaria National Olympiad, 3

Tags: calculus , abstract algebra , integration , induction , polynomial , geometry , algebra

Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.

2011 Philippine MO, 2

Tags: geometry , reflection , geometric transformation

In triangle $ABC$, let $X$ and $Y$ be the midpoints of $AB$ and $AC$, respectively. On segment $BC$, there is a point $D$, different from its midpoint, such that $\angle{XDY}=\angle{BAC}$. Prove that $AD\perp BC$.

II Soros Olympiad 1995 - 96 (Russia), 9.3

Tags: angle , geometry

Two straight lines are drawn on a plane, intersecting at an angle of $40^o$. A circle with center at point $O$ touches these lines. Let's consider a line, different from the given ones, tangent to the same circle and intersecting the given lines at points $B$ and $C$. What can the angle $\angle BOC$ be equal to?

2021 Harvard-MIT Mathematics Tournament., 9

Tags: incenter , circumcircle , geometry

Let scalene triangle $ABC$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $BC,CA,$ and $AB$ at $D,E,$ and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let line $DP$ intersect $\omega$ again at $Q \ne D$. The line $OI$ intersects the altitude from $A$ to$ BC$ at $T$. Given that $OI \|BC,$ show that $PQ=PT$.

2012 Korea National Olympiad, 2

Tags: ratio , dilation , geometry , reflection , geometric transformation

Let $ w $ be the incircle of triangle $ ABC $. Segments $ BC, CA $ meet with $ w $ at points $ D, E$. A line passing through $ B $ and parallel to $ DE $ meets $ w $ at $ F $ and $ G $. ($ F $ is nearer to $ B $ than $ G $.) Line $ CG $ meets $ w $ at $ H ( \ne G ) $. A line passing through $ G $ and parallel to $ EH $ meets with line $ AC $ at $ I $. Line $ IF $ meets with circle $ w $ at $ J (\ne F ) $. Lines $ CJ $ and $ EG $ meets at $ K $. Let $ l $ be the line passing through $ K $ and parallel to $ JD $. Prove that $ l, IF, ED $ meet at one point.

2005 QEDMO 1st, 8 (Z2)

Tags: 3d geometry , complex number , rotation , geometry

Prove that if $n$ can be written as $n=a^2+ab+b^2$, then also $7n$ can be written that way.

2016 Fall CHMMC, 9

Tags: geometry

In quadrilateral $ABCD$, $AB = DB$ and $AD = BC$. If $\angle ABD = 36^{\circ}$ and $\angle BCD = 54^{\circ}$, find $\angle ADC$ in degrees.

2011 May Olympiad, 3

Tags: right triangle , isosceles , angle , perpendicular bisector , geometry

In a right triangle rectangle $ABC$ such that $AB = AC$, $M$ is the midpoint of $BC$. Let $P$ be a point on the perpendicular bisector of $AC$, lying in the semi-plane determined by $BC$ that does not contain $A$. Lines $CP$ and $AM$ intersect at $Q$. Calculate the angles that form the lines $AP$ and $BQ$.

2012 Chile National Olympiad, 4

Tags: angle , ratio , isosceles triangle , isosceles , geometry

Consider an isosceles triangle $ABC$, where $AB = AC$. $D$ is a point on the $AC$ side and $P$ a point on the segment $BD$ so that the angle $\angle APC = 90^o$ and $ \angle ABP = \angle BCP $. Determine the ratio $AD: DC$.

the 16th XMO, 2

Tags: circumcircle , geometry

In a triangle $ABC$ , let $O$ be the circumcenter , $AO$ meet $BC$ at $K$ , A circle $\Omega$ with the centre $T$ and the center $K$ and the radius $AK$ meet $AC$ again at $T$ , $D$ is a point on the plain satisfies that $BC$ is the bisector of the angle $\angle ABD$ , let the orthocenter of the triangle $ABC$ and $BCD$ be $M$ and $N$ . If $MN//AC$ than $DT$ is tangent to $\Omega$

II Soros Olympiad 1995 - 96 (Russia), 9.10

Tags: geometry , equal circles

Two disjoint circles are inscribed in an angle with vertex $A$, whose measure is equal to $a$. The distance between their centers is $d$. A straight line tangent to both circles and not passing through $A$ intersects the sides of the angle at points $B$ and $C$. Find the radius of the circle circumscribed about triangle $ABC$.

2014 Online Math Open Problems, 4

Tags: geometry , parallelogram

A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom 52 = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$. [i]Proposed by Yang Liu[/i]

2014 May Olympiad, 4

Tags: right triangle , isosceles triangle , equilateral triangle , angle , geometry

Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$

India EGMO 2024 TST, 1

Tags: euler , circumcircle , geometry

Let $ABC$ be a triangle with circumcentre $O$ and centroid $G$. Let $M$ be the midpoint of $BC$ and $N$ the reflection of $M$ across $O$. Prove that $NO = NA$ if and only if $\angle AOG = 90^{\circ}$. [i]Proposed by Pranjal Srivastava[/i]

1983 All Soviet Union Mathematical Olympiad, 351

Tags: geometry , combinatorial geometry , circles

Three disks touch pairwise from outside in the points $X,Y,Z$. Then the radiuses of the disks were expanded by $2/\sqrt3$ times, and the centres were reserved. Prove that the triangle $XYZ$ is completely covered by the expanded disks.

2016 Indonesia TST, 2

Tags: perimeter , geometric inequality , inequalities , geometry

Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds: \[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]

2023 HMNT, 7

Tags: geometry

Let $ABCD$ be a convex trapezoid such that $\angle BAD = \angle ADC = 90^o$, $AB = 20$, $AD = 21$, and $CD = 28$. Point $P \ne A$ is chosen on segment $AC$ such that $\angle BPD = 90^o$. Compute $AP$.

2010 Brazil Team Selection Test, 1

Tags: perpendicular , geometry

Let $ABC$ be an acute triangle and $D$ a point on the side $AB$. The circumcircle of triangle $BCD$ cuts the side $AC$ again at $E$ .The circumcircle of triangle $ACD$ cuts the side $BC$ again at $F$. If $O$ is the circumcenter of the triangle $CEF$. Prove that $OD$ is perpendicular to $AB$.

2020 HMIC, 3

Tags: geometry , 3d geometry , tetrahedron

Let $P_1P_2P_3P_4$ be a tetrahedron in $\mathbb{R}^3$ and let $O$ be a point equidistant from each of its vertices. Suppose there exists a point $H$ such that for each $i$, the line $P_iH$ is perpendicular to the plane through the other three vertices. Line $P_1H$ intersects the plane through $P_2, P_3, P_4$ at $A$, and contains a point $B\neq P_1$ such that $OP_1=OB$. Show that $HB=3HA$. [i]Michael Ren[/i]

2024 Sharygin Geometry Olympiad, 24

Tags: pyramid , 3d geometry , geometry

Let $SABC$ be a pyramid with right angles at the vertex $S$. Points $A', B', C'$ lie on the edges $SA, SB, SC$ respectively in such a way that the triangles $ABC$ and $A'B'C'$ are similar. Does this yield that the planes $ABC$ and $A'B'C'$ are parallel?

2007 ITest, 7

Tags: geometry

An equilateral triangle with side length $1$ has the same area as a square with side length $s$. Find $s$. $\textbf{(A) }\dfrac{\sqrt[4]3}2\hspace{14em}\textbf{(B) }\dfrac{\sqrt[4]3}{\sqrt2}\hspace{14em}\textbf{(C) }1$ $\textbf{(D) }\dfrac34\hspace{14.9em}\textbf{(E) }\dfrac43\hspace{14.9em}\textbf{(F) }\sqrt3$ $\textbf{(G) }\dfrac{\sqrt6}2$

2021 Junior Balkan Team Selection Tests - Romania, P3

Tags: incenter , geometry

The incircle of triangle $ABC$ is tangent to the sides $AB,AC$ and $BC$ at the points $M,N$ and $K$ respectively. The median $AD$ of the triangle $ABC$ intersects $MN$ at the point $L$. Prove that $K,I$ and $L$ are collinear, where $I$ is the incenter of the triangle $ABC$.

Cono Sur Shortlist - geometry, 2009.G4

Tags: altitude , geometry , perpendicular , incenter

Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.