Olympiad problems

1994 Poland - Second Round, 5

The incircle $\omega$ of a triangle $ABC$ is tangent to the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The angle bisector at $A$ meets $PQ$ at point $S$. Prove $\angle ASC = 90^o$ .

2008 Princeton University Math Competition, 8

Tags: algebra

Suppose that the roots of the quadratic $x^2 + ax + b$ are $\alpha$ and $\beta$. Then $\alpha^3$ and $\beta^3$ are the roots of some quadratic $x^2 + cx + d$. Find $c$ in terms of $a$ and $b$.

2006 Iran MO (3rd Round), 3

Tags: inequalities , trigonometry , geometry , incenter , geometric transformation , reflection , circumcircle

In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]

2022 USAMTS Problems, 2

Tags: function , functional equation , algebra

Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.

2007 Greece JBMO TST, 1

Tags: circumcircle , geometry , angle

Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$. a) Find the angles $\angle B$ and $\angle C$ b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.

2020 MBMT, 6

Tags:

Given that $\sqrt{10} \approx 3.16227766$, find the largest integer $n$ such that $n^2 \leq 10,000,000$. [i]Proposed by Jacob Stavrianos[/i]

2001 IMO Shortlist, 5

Tags: geometry , circumcircle , trigonometry , triangle

Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that \[ \angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB. \] Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum \[ \frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}. \]

2018 AMC 12/AHSME, 22

Tags: algebra , polynomial

Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$? $\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $

2019 Thailand Mathematical Olympiad, 6

Tags: algebra , functional inequality

Determine all function $f:\mathbb{R}\to\mathbb{R}$ such that $xf(y)+yf(x)\leqslant xy$ for all $x,y\in\mathbb{R}$.

2019 Kurschak Competition, 1

Tags: geometry , geometric transformation , reflection , circumcircle

In an acute triangle $\bigtriangleup ABC$, $AB<AC<BC$, and $A_1,B_1,C_1$ are the projections of $A,B,C$ to the corresponding sides. Let the reflection of $B_1$ wrt $CC_1$ be $Q$, and the reflection of $C_1$ wrt $BB_1$ be $P$. Prove that the circumcirle of $A_1PQ$ passes through the midpoint of $BC$.

2009 District Olympiad, 3

Tags: inequalities

[b]a)[/b] For $ a,b\ge 0 $ and $ x,y>0, $ show that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2}\ge \frac{(a+b)^3}{(x+y)^2} . $$ [b]b)[/b] For $ a,b,c\ge 0 $ and $ x,y,z>0 $ under the condition $ a+b+c=x+y+z, $ prove that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2} +\frac{c^3}{z^2} \ge a+b+c. $$

2023 HMNT, 1

Tags: geometry

Let $ABC$ be an equilateral triangle with side length $2$ that is inscribed in a circle $\omega$. A chord of $\omega$ passes through the midpoints of sides $AB$ and $AC$. Compute the length of this chord.

1997 ITAMO, 2

Tags: odd , function , periodic , periodic function , algebra

Let a real function $f$ defined on the real numbers satisfy the following conditions: (i) $f(10+x) = f(10- x)$ (ii) $f(20+x) = - f(20- x)$ for all $x$. Prove that f is odd and periodic.

2000 IMO Shortlist, 5

Tags: geometry , rectangle , combinatorics , intersection , graph theory , double counting

In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is [i]nice[/i] if it has at least one of the vertices of the $n$ rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$. (There can be nonconvex regions as well as regions with more than one boundary curve.)

1996 National High School Mathematics League, 8

Tags: complex number

On the complex plane, non-zero complex numbers $z_1,z_2$ are on the circle with center $\text{i}$, radius of $1$. The real part of $\overline{z_1}\cdot z_2$ is $0$, and $\arg (z_1)=\frac{\pi}{6}$, then $z_2=$________.

2017 IFYM, Sozopol, 8

Tags: polynomial , algebra

Find all polynomials $P\in \mathbb{R}[x]$, for which $P(P(x))=\lfloor P^2 (x)\rfloor$ is true for $\forall x\in \mathbb{Z}$.

2020 USOMO, 4

Tags:

Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs. [i]Proposed by Ankan Bhattacharya[/i]

1989 IMO Longlists, 58

Tags: complex number , geometry

A regular $ n\minus{}$gon $ A_1A_2A_3 \cdots A_k \cdots A_n$ inscribed in a circle of radius $ R$ is given. If $ S$ is a point on the circle, calculate \[ T \equal{} \sum^n_{k\equal{}1} SA^2_k.\]

1995 IMO Shortlist, 1

Tags: algebra , modular arithmetic , number theory

Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.

2014 Postal Coaching, 1

Tags: circumcircle , perpendicular bisector , geometry

Let $ABC$ be a triangle in which $\angle B$ is obtuse.Let $\Gamma$ be its circumcircle and $O$ be the centre of $\Gamma$.Let the tangent to $\Gamma$ at $C$ intersect the line $AB$ in $B_1$.Let $O_1$ be the circumcentre of the circumcircle $\Gamma_1$ of $\triangle AB_1 C$.Take any point $B_2$ on the segment $BB_1$ different from $B,B_1$.Let $C_1$ be the point of contact of the tangent from $B_2$ to $\Gamma$ which is closer to $C$.Let $O_2$ be the circumcentre of $\triangle AB_2 C_1$.Prove that $O,O_2,O_1,C_1,C$ are concyclic if $OO_2\perp AO_1$.

2018 VJIMC, 1

Tags: equation , power , real number

Find all real solutions of the equation \[17^x+2^x=11^x+2^{3x}.\]

1993 Austrian-Polish Competition, 1

Tags: diophantine equation , diophantine , number theory

Solve in positive integers $x,y$ the equation $2^x - 3^y = 7$.

VMEO III 2006 Shortlist, G3

Tags: geometry , 3d geometry , tetrahedron , geometric inequality

The tetrahedron $OABC$ has all angles at vertex $O$ equal to $60^o$. Prove that $$AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2$$

2001 IMO Shortlist, 8

Tags: matrix , combinatorics , ramsey theory , extremal combinatorics

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

2008 Sharygin Geometry Olympiad, 6

Tags: geometry

(B.Frenkin) Consider the triangles such that all their vertices are vertices of a given regular 2008-gon. What triangles are more numerous among them: acute-angled or obtuse-angled?