Olympiad problems

2020 LIMIT Category 2, 17

Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$. Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$ (A) $\pi/2$ (B) $\pi/4$ (C) $\pi $ (D) $\pi/3$

1983 IMO Shortlist, 4

Tags: trigonometry , geometry , parallelogram , triangle

On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.

2004 Germany Team Selection Test, 3

Tags: geometry , incenter , circumcircle , triangle

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]

2017 Kyiv Mathematical Festival, 2

Tags: geometry , triangle

A triangle $ABC$ is given. Let $D$ be a point on the extension of the segment $AB$ beyond $A$ such that $AD=BC,$ and $E$ be a point on the extension of the segment $BC$ beyond $B$ such that $BE=AC.$ Prove that the circumcircle of the triangle $DEB$ passes through the incenter of the triangle $ABC.$

1998 Croatia National Olympiad, Problem 3

Tags: geometry , square , triangle , right triangle

Points $E$ and $F$ are chosen on the sides $AB$ and $BC$ respectively of a square $ABCD$ such that $BE=BF$. Let $BN$ be an altitude of the triangle $BCE$. Prove that the triangle $DNF$ is right-angled.

1982 Bulgaria National Olympiad, Problem 6

Tags: triangle , square , centroid , triangle centers , locus , geometry

Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.

1966 IMO Longlists, 19

Tags: excircle , geometry , triangle , construction

Construct a triangle given the radii of the excircles.

1974 Czech and Slovak Olympiad III A, 2

Tags: triangle , min , max , geometry

Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.

2008 Czech and Slovak Olympiad III A, 3

Tags: geometric inequality , triangle , median , inequalities

Find the greatest value of $p$ and the smallest value of $q$ such that for any triangle in the plane, the inequality \[p<\frac{a+m}{b+n}<q\] holds, where $a,b$ are it's two sides and $m,n$ their corresponding medians.

1997 IMO Shortlist, 20

Tags: geometry , circumcircle , cyclic quadrilateral , triangle

A quick solution: Let R be the foot of the perpend. from X to BC. Let's assume Q and R are in the interior of the segms AC and BC (respectively) and P in the ext of AD. P, R, Q are colinear (Simson's thm). PQ tangent to circle XRD iff XRQ=XDR iff Pi-XCA=XDR iff XBA=XDR=XDC=ADB iff XBC+ABC=ADB=DAC+ACB iff XAC+ABC=DAC+ACD iff ABC=ACD=ACB iff AB=AC. It's the same for all the other cases.

2014 Canadian Mathematical Olympiad Qualification, 6

Tags: midpoint , triangle , geometry

Given a triangle $A, B, C, X$ is on side $AB$, $Y$ is on side $AC$, and $P$ and $Q$ are on side $BC$ such that $AX = AY , BX = BP$ and $CY = CQ$. Let $XP$ and $YQ$ intersect at $T$. Prove that $AT$ passes through the midpoint of $PQ$.

1953 Czech and Slovak Olympiad III A, 2

Tags: geometry , triangle , angle

Let $\alpha,\beta,\gamma$ be angles of a triangle. Two of them can be expressed using an auxiliary angle $\varphi$ in a way that $$\alpha=\varphi+\frac\pi4,\quad\beta=\pi-3\varphi.$$ Show that $\alpha>\gamma.$

1991 Bulgaria National Olympiad, Problem 1

Tags: triangle , geometry

Let $M$ be a point on the altitude $CD$ of an acute-angled triangle $ABC$, and $K$ and $L$ the orthogonal projections of $M$ on $AC$ and $BC$. Suppose that the incenter and circumcenter of the triangle lie on the segment $KL$. (a) Prove that $CD=R+r$, where $R$ and $r$ are the circumradius and inradius, respectively. (b) Find the minimum value of the ratio $CM:CD$.

2000 Belarus Team Selection Test, 7.2

Tags: geometry , circumcircle , incenter , angle bisector , triangle

Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.

2011 Indonesia TST, 3

Tags: geometry , trigonometry , maximization , triangle , perpendicular

Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define \[ p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}. \] Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

1970 IMO Longlists, 58

Tags: acute , triangle , point set , counting , combinatorial geometry

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

2020 Adygea Teachers' Geometry Olympiad, 3

Tags: geometry , 3d geometry , tetrahedron , triangle

Is it true that of the four heights of an arbitrary tetrahedron, three can be selected from which a triangle can be made?

2000 IMO Shortlist, 8

Tags: geometry , homothety , circumcircle , triangle , incircle

Let $ AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ ABC$. Its incircle touches the sides $ BC, AC$ and $ AB$ at $ T_1, T_2$ and $ T_3$ respectively. Consider the symmetric images of the lines $ H_1H_2, H_2H_3$ and $ H_3H_1$ with respect to the lines $ T_1T_2, T_2T_3$ and $ T_3T_1$. Prove that these images form a triangle whose vertices lie on the incircle of $ ABC$.

2000 Moldova National Olympiad, Problem 3

Tags: triangle , geometry

The excircle of a triangle $ABC$ corresponding to $A$ touches the side $BC$ at $M$, and the point on the incircle diametrically opposite to its point of tangency with $BC$ is denoted by $N$. Prove that $A,M,$ and $N$ are collinear.

1973 IMO Shortlist, 14

Tags: geometry , triangle inequality , minimization , triangle , optimization

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

2019 India PRMO, 6

Tags: geometry , triangle

Let $ABC$ be a triangle such that $AB=AC$. Suppose the tangent to the circumcircle of ABC at B is perpendicular to AC. Find angle ABC measured in degrees

1962 Putnam, A3

Tags: ratio , triangle , area of a triangle

In a triangle $ABC$, let $A'$ be a point on the segment $BC$, $B'$ be a point on the segment $CA$ and $C'$ a point on the segment $AB$ such that $$ \frac{AB'}{B'C}= \frac{BC'}{C'A} =\frac{CA'}{A'B}=k,$$ where $k$ is a positive constant. Let $\triangle$ be the triangle formed by the interesctions of $AA'$, $BB'$ and $CC'$. Prove that the areas of $\triangle $ and $ABC$ are in the ratio $$\frac{(k-1)^{2}}{k^2 +k+1}.$$

2000 IMO Shortlist, 5

Tags: diophantine equation , number theory , inradius , perimeter , triangle

Prove that there exist infinitely many positive integers $ n$ such that $ p \equal{} nr,$ where $ p$ and $ r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.

1975 Czech and Slovak Olympiad III A, 1

Tags: geometry , acute , triangle , right triangle , area , triangle area

Let $\mathbf T$ be a triangle with $[\mathbf T]=1.$ Show that there is a right triangle $\mathbf R$ such that $[\mathbf R]\le\sqrt3$ and $\mathbf T\subseteq\mathbf R.$ ($[-]$ denotes area of a triangle.)

1997 Brazil Team Selection Test, Problem 5

Tags: geometry , geometric inequality , inequalities , geometric inequalities , triangle , area

Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.