Olympiad problems

1989 Greece National Olympiad, 3

If $a\ge 0$ prove that $a^4+ a^3-10 a^2+9 a+4>0$.

2001 Tournament Of Towns, 3

Tags: combinatorics

Kolya is told that two of his four coins are fake. He knows that all real coins have the same weight, all fake coins have the same weight, and the weight of a real coin is greater than that of a fake coin. Can Kolya decide whether he indeed has exactly two fake coins by using a balance twice?

2014 Postal Coaching, 4

Tags: number theory

Denote by $F_n$ the $n^{\text{th}}$ Fibonacci number $(F_1=F_2=1)$.Prove that if $a,b,c$ are positive integers such that $a| F_b,b|F_c,c|F_a$,then either $5$ divides each of $a,b,c$ or $12$ divides each of $a,b,c$.

2014 Cezar Ivănescu, 3

Tags: function , real analysis

Find the real numbers $ \lambda $ that have the property that there is a nonconstant, continuous function $ u: [0,1]\longrightarrow\mathbb{R} $ satisfying $$ u(x)=\lambda\int_0^1 (x-3y)u(y)dy , $$ for any $ x $ in the interval $ [0,1]. $

Estonia Open Junior - geometry, 2007.1.2

Tags: midpoint , equal area , geometry , parallelogram

The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.

2019 China Team Selection Test, 5

Tags: geometry

Let $M$ be the midpoint of $BC$ of triangle $ABC$. The circle with diameter $BC$, $\omega$, meets $AB,AC$ at $D,E$ respectively. $P$ lies inside $\triangle ABC$ such that $\angle PBA=\angle PAC, \angle PCA=\angle PAB$, and $2PM\cdot DE=BC^2$. Point $X$ lies outside $\omega$ such that $XM\parallel AP$, and $\frac{XB}{XC}=\frac{AB}{AC}$. Prove that $\angle BXC +\angle BAC=90^{\circ}$.

2011 Vietnam National Olympiad, 2

Tags: incenter , angle bisector , geometry

Let $\triangle ABC$ be a triangle such that $\angle C$ and $\angle B$ are acute. Let $D$ be a variable point on $BC$ such that $D\neq B, C$ and $AD$ is not perpendicular to $BC.$ Let $d$ be the line passing through $D$ and perpendicular to $BC.$ Assume $d \cap AB= E, d \cap AC =F.$ If $M, N, P$ are the incentres of $\triangle AEF, \triangle BDE,\triangle CDF.$ Prove that $A, M, N, P$ are concyclic if and only if $d$ passes through the incentre of $\triangle ABC.$

2010 Contests, 525

Tags: calculus , integration , geometry , vector , inequalities , quadratic , trigonometry

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

2008 Germany Team Selection Test, 2

Tags: geometry , rectangle , combinatorics

Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?

2008 Romania Team Selection Test, 2

Tags: modular arithmetic , number theory

Let $ m, n \geq 1$ be two coprime integers and let also $ s$ an arbitrary integer. Determine the number of subsets $ A$ of $ \{1, 2, ..., m \plus{} n \minus{} 1\}$ such that $ |A| \equal{} m$ and $ \sum_{x \in A} x \equiv s \pmod{n}$.

1997 Abels Math Contest (Norwegian MO), 2b

Tags: isosceles , product , independent , geometry , circles

Let $A,B,C$ be different points on a circle such that $AB = AC$. Point $E$ lies on the segment $BC$, and $D \ne A$ is the intersection point of the circle and line $AE$. Show that the product $AE \cdot AD$ is independent of the choice of $E$.

2012 Today's Calculation Of Integral, 828

Tags: calculus , integration , function , trigonometry , calculus computations

Find a function $f(x)$, which is differentiable and $f'(x) $ is continuous, such that $\int_0^x f(t)\cos (x-t)\ dt=xe^{2x}.$

2024 ELMO Shortlist, N4

Tags: number theory

Find all pairs $(a,b)$ of positive integers such that $a^2\mid b^3+1$ and $b^2\mid a^3+1$. [i]Linus Tang[/i]

1990 Tournament Of Towns, (248) 2

Tags: square , equal area , ratio , geometry

If a square is intersected by another square equal to it but rotated by $45^o$ around its centre, each side is divided into three parts in a certain ratio $a : b : a$ (which one can compute). Make the following construction for an arbitrary convex quadrilateral: divide each of its sides into three parts in this same ratio $a : b : a$, and draw a line through the two division points neighbouring each vertex. Prove that the new quadrilateral bounded by the four drawn lines has the same area as the original one. (A. Savin, Moscow)

2019 Moldova Team Selection Test, 4

Tags: geometry

Quadrilateral $ABCD$ is inscribed in circle $\Gamma$ with center $O$. Point $I$ is the incenter of triangle $ABC$, and point $J$ is the incenter of the triangle $ABD$. Line $IJ$ intersects segments $AD, AC, BD, BC$ at points $P, M, N$ and, respectively $Q$. The perpendicular from $M$ to line $AC$ intersects the perpendicular from $N$ to line $BD$ at point $X$. The perpendicular from $P$ to line $AD$ intersects the perpendicular from $Q$ to line $BC$ at point $Y$. Prove that $X, O, Y$ are colinear.

2020 BMT Fall, 24

Tags: algebra , combinatorics

For positive integers $N$ and $m$, where $m \le N$, define $$a_{m,N} =\frac{1}{{N+1 \choose m}} \sum_{i=m-1}^{N-1} \frac{ {i \choose m-1}}{N - i}$$ Compute the smallest positive integer $N$ such that $$\sum^N_{m=1}a_{m,N} >\frac{2020N}{N +1}$$

2008 Princeton University Math Competition, A3/B5

Tags: combinatorics

Evaluate $\sum_{m=0}^{2009}\sum_{n=0}^{m}\binom{2009}{m}\binom{m}{n}$

2011 Princeton University Math Competition, A5 / B8

Tags: number theory

Let $d(n)$ denote the number of divisors of $n$ (including itself). You are given that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\] Find $p(6)$, where $p(x)$ is the unique polynomial with rational coefficients satisfying \[p(\pi) = \sum_{n=1}^{\infty} \frac{d(n)}{n^2}.\]

1966 All Russian Mathematical Olympiad, 079

Tags: combinatorics

For three arbitrary crossroads $A,B,C$ in a certain city there exist a way from $A$ to $B$ not coming through $C$. Prove that for every couple of the crossroads there exist at least two non-intersecting ways connecting them. (there are at least two crossroads in the city)

2002 Dutch Mathematical Olympiad, 1

Tags: geometry , reflection , square

The sides of a $10$ by $10$ square $ABCD$ are reflective on the inside. A beam of light enters the square via the vertex $A$ and heads to the point $P$ on $CD$ with $CP = 3$ and $PD = 7$. In $P$ it naturally reflects on the $CD$ side. The light beam can only leave the square via one of the angular points $A, B, C$ or $D$. What is the distance that the light beam travels within the square before it leaves the square again? By which vertex does that happen?

2008 Vietnam National Olympiad, 2

Tags: trigonometry , geometry

Given a triangle with acute angle $ \angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the opposite ray of $ EC$ such that $ \angle BME \equal{} \angle ECA.$ Denote by $ \theta$ the measure of angle $ \angle BEC.$ Evaluate $ \frac{MC}{AB}$ in terms of $ \theta.$

1998 Austrian-Polish Competition, 6

Tags: parallel , concurrency , circles , geometry

Different points $A,B,C,D,E,F$ lie on circle $k$ in this order. The tangents to $k$ in the points $A$ and $D$ and the lines $BF$ and $CE$ have a common point $P$. Prove that the lines $AD,BC$ and $EF$ also have a common point or are parallel.

2010 IFYM, Sozopol, 7

Tags: geometry , polygon

Let $M$ be a convex polygon. Externally, on its sides are built squares. It is known that the vertices of these squares, that don’t lie on $M$, lie on a circle $k$. Determine $M$ (its type).

2001 All-Russian Olympiad Regional Round, 11.1

Tags: number theory , diophantine , diophantine equation

Find all prime numbers $p$ and $q$ such that $p + q = (p -q)^3.$

1998 VJIMC, Problem 2

Tags: arithmetic sequence , algebra , number theory , palindrome

Decide whether there is a member in the arithmetic sequence $\{a_n\}_{n=1}^\infty$ whose first member is $a_1=1998$ and the common difference $d=131$ which is a palindrome (palindrome is a number such that its decimal expansion is symmetric, e.g., $7$, $33$, $433334$, $2135312$ and so on).