Olympiad problems

1998 IMO, 5

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

1970 IMO Shortlist, 12

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.

1987 Bulgaria National Olympiad, Problem 5

Tags: triangle , geometry

Let $E$ be a point on the median $AD$ of a triangle $ABC$, and $F$ be the projection of $E$ onto $BC$. From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$.

2010 Belarus Team Selection Test, 1.2

Tags: plane geometry , triangle , geometry

Points $H$ and $T$ are marked respectively on the sides $BC$ abd $AC$ of triangle $ABC$ so that $AH$ is the altitude and $BT$ is the bisectrix $ABC$. It is known that the gravity center of $ABC$ lies on the line $HT$. a) Find $AC$ if $BC$=a nad $AB$=c. b) Determine all possible values of $\frac{c}{b}$ for all triangles $ABC$ satisfying the given condition.

2005 Slovenia National Olympiad, Problem 3

Tags: geometry , triangle

In an isosceles triangle $ABC$ with $AB = AC$, $D$ is the midpoint of $AC$ and $E$ is the projection of $D$ onto $BC$. Let $F$ be the midpoint of $DE$. Prove that the lines $BF$ and $AE$ are perpendicular if and only if the triangle $ABC$ is equilateral.

2001 IMO, 1

Tags: geometry , circumcenter , triangle , geometric inequality

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

1998 IMO Shortlist, 3

Tags: geometry , incenter , trigonometry , triangle

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

1977 Bulgaria National Olympiad, Problem 6

Tags: triangle , geometry

A Pythagorean triangle is any right-angled triangle for which the lengths of two legs and the length of the hypotenuse are integers. We are observing all Pythagorean triangles in which may be inscribed a quadrangle with sidelength integer number, two of which sides lie on the cathets and one of the vertices of which lies on the hypotenuse of the triangle given. Find the side lengths of the triangle with minimal surface from the observed triangles. [i]St. Doduneko[/i]

2002 Germany Team Selection Test, 2

Tags: geometry , triangle , square

Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.

1961 Czech and Slovak Olympiad III A, 4

Tags: equilateral , triangle , computation , triangle area , max , min , geometry

Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.

2015 ISI Entrance Examination, 7

Tags: circles , triangle , geometry

Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$

1978 IMO Longlists, 41

Tags: geometry , inradius , circumcircle , incenter , triangle

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

2007 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: combinatorial geometry , triangle , segment , combinatorics

Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments. Clarification : the vertices of the original triangle are not considered points of intersection or division.

2009 Postal Coaching, 4

Tags: triangle , geometry , circumcircle , perimeter , geometric inequalities

Let $ABC$ be a triangle, and let $DEF$ be another triangle inscribed in the incircle of $ABC$. If $s$ and $s_1$ denote the semiperimeters of $ABC$ and $DEF$ respectively, prove that $2s_1 \le s$. When does equality hold?

2000 Moldova National Olympiad, Problem 8

Tags: geometry , triangle

Points $D$ and $N$ on the sides $AB$ and $BC$ and points $E,M$ on the side $AC$ of an equilateral triangle $ABC$, respectively, with $E$ between $A$ and $M$, satisfy $AD+AE=CN+CM=BD+BN+EM$. Determine the angle between the lines $DM$ and $EN$.

1961 IMO Shortlist, 4

Tags: ratio , inequalities , geometry , triangle , intersection

Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \] at least one is $\leq 2$ and at least one is $\geq 2$

2021 EGMO, 5

Tags: triangle , geometry , combinatorial geometry

A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane such that [list] [*] no three points in $P$ lie on a line and [*] no two points in $P$ lie on a line through the origin. [/list] A triangle with vertices in $P$ is [i]fat[/i] if $O$ is strictly inside the triangle. Find the maximum number of fat triangles.

1991 IMO, 2

Tags: geometry , geometric inequality , triangle , brocard , angle

Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

2007 IMO, 4

Tags: geometry , circumcircle , incenter , triangle

In triangle $ ABC$ the bisector of angle $ BCA$ intersects the circumcircle again at $ R$, the perpendicular bisector of $ BC$ at $ P$, and the perpendicular bisector of $ AC$ at $ Q$. The midpoint of $ BC$ is $ K$ and the midpoint of $ AC$ is $ L$. Prove that the triangles $ RPK$ and $ RQL$ have the same area. [i]Author: Marek Pechal, Czech Republic[/i]

1967 IMO Shortlist, 1

Tags: geometry , triangle , similar triangles , area of a triangle

$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$

2005 Taiwan TST Round 2, 2

Tags: circumcircle , homothety , triangle , geometry

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

1958 February Putnam, B1

Tags: triangle , geometry

i) Given line segments $A,B,C,D$ with $A$ the longest, construct a quadrilateral with these sides and with $A$ and $B$ parallel, when possible. ii) Given any acute-angled triangle $ABC$ and one altitude $AH$, select any point $D$ on $AH$, then draw $BD$ and extend until it intersects $AC$ in $E$, and draw $CD$ and extend until it intersects $AB$ in $F$. Prove that $\angle AHE = \angle AHF$.

1970 IMO, 3

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.

2007 German National Olympiad, 3

Tags: geometry , similar triangles , midpoint , triangle

We say that two triangles are oriented similarly if they are similar and have the same orientation. Prove that if $ALT, ARM, ORT, $ and $ULM$ are four triangles which are oriented similarly, then $A$ is the midpoint of the line segment $OU.$

2014 Czech-Polish-Slovak Match, 1

Tags: trigonometric equation , polynomial equation , triangle , geometry

Prove that if the positive real numbers $a, b, c$ satisfy the equation \[a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2),\] then there is a triangle $ABC$ with internal angles $\alpha, \beta, \gamma$ such that \[\sin \alpha = a, \qquad \sin \beta = b, \qquad \sin \gamma= c.\]