Olympiad problems

2018 239 Open Mathematical Olympiad, 8-9.7

Tags: inequalities , algebra

The sequence $a_n$ is defined by the following conditions: $a_1=1$, and for any $n\in \mathbb N$, the number $a_{n+1}$ is obtained from $a_n$ by adding three if $n$ is a member of this sequence, and two if it is not. Prove that $a_n<(1+\sqrt 2)n$ for all $n$. [i]Proposed by Mikhail Ivanov[/i]

2011 Junior Balkan Team Selection Tests - Romania, 4

Tags: system of equations , diophantine equation , diophantine , number theory

Show that there is an infinite number of positive integers $t$ such that none of the equations $$ \begin{cases} x^2 + y^6 = t \\ x^2 + y^6 = t + 1 \\ x^2 - y^6 = t \\ x^2 - y^6 = t + 1 \end{cases}$$ has solutions $(x, y) \in Z \times Z$.

2009 Croatia Team Selection Test, 2

Tags: induction , graph theory , combinatorics

On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.

2001 India IMO Training Camp, 1

Tags: inequalities

Let $x$ , $y$ , $z>0$. Prove that if $xyz\geq xy+yz+zx$, then $xyz \geq 3(x+ y+z)$.

2010 Contests, 3

Tags: inequalities

Positive real $A$ is given. Find maximum value of $M$ for which inequality $ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $ holds for all $x, y>0$

2023 BMT, 1

Tags: geometry

A semicircle of radius $2$ is inscribed inside of a rectangle, as shown in the diagram below. The diameter of the semicircle coincides with the bottom side of the rectangle, and the semicircle is tangent to the rectangle at all points of intersection. Compute the length of the diagonal of the rectangle. [img]https://cdn.artofproblemsolving.com/attachments/c/7/81fcfb759188eae7dcb82fa5d58fb9525d85de.png[/img]

2008 Indonesia TST, 4

Tags: greatest common divisor , number theory

Let $ a $ and $ b $ be natural numbers with property $ gcd(a,b)=1 $ . Find the least natural number $ k $ such that for every natural number $ r \ge k $ , there exist natural numbers $ m,n >1 $ in such a way that the number $ m^a n^b $ has exactly $ r+1 $ positive divisors.

2016 Czech-Polish-Slovak Match, 1

Tags: geometry

Let $P$ be a non-degenerate polygon with $n$ sides, where $n > 4$. Prove that there exist three distinct vertices $A, B, C$ of $P$ with the following property:If $\ell_1,\ell_2,\ell_3$ are the lengths of the three polygonal chains into which $A, B, C$ break the perimeter of $P$, then there is a triangle with side lengths $\ell_1,\ell_2$ and $\ell_3$. [size=150]Remark[/size]: By a non-degenerate polygon we mean a polygon in which every two sides are disjoint, apart from consecutive ones, which share only the common endpoint.(Poland)

2016 Poland - Second Round, 5

Tags: geometry , quadrilateral , cyclic quadrilateral , circles

Quadrilateral $ABCD$ is inscribed in circle. Points $P$ and $Q$ lie respectively on rays $AB^{\rightarrow}$ and $AD^{\rightarrow}$ such that $AP = CD$, $AQ = BC$. Show that middle point of line segment $PQ$ lies on the line $AC$.

1993 Romania Team Selection Test, 1

Tags: complex , sequence , algebra

Consider the sequence $z_n = (1+i)(2+i)...(n+i)$. Prove that the sequence $Im$ $z_n$ contains infinitely many positive and infinitely many negative numbers.

2012 Philippine MO, 5

Tags: combinatorics

There are exactly $120$ Twitter subscribers from National Science High School. Statistics show that each of $10$ given celebrities has at least $85$ followers from National Science High School. Prove that there must be two students such that each of the $10$ celebrities is being followed in Twitter by at least one of these students.

2010 Switzerland - Final Round, 5

Tags: combinatorics

Some sides and diagonals of a regular $ n$-gon form a connected path that visits each vertex exactly once. A [i]parallel pair[/i] of edges is a pair of two different parallel edges of the path. Prove that (a) if $ n$ is even, there is at least one [i]parallel pair[/i]. (b) if $ n$ is odd, there can't be one single [i]parallel pair[/i].

2024 CCA Math Bonanza, I2

Tags: function

Let $S(x) = x+1$ and $V(x) = x^2-1$. Find the sum of the squares of all real solutions to $S(V(S(V(x)))) = 1$. [i]Individual #2[/i]

1976 All Soviet Union Mathematical Olympiad, 230

Tags: equilateral , combinatorial geometry , combinatorics

Let us call "[i]big[/i]" a triangle with all sides longer than $1$. Given a equilateral triangle with all the sides equal to $5$. Prove that: a) You can cut $100$ [i]big [/i] triangles out of given one. b) You can divide the given triangle onto $100$ [i]big [/i] nonintersecting ones fully covering the initial one. c) The same as b), but the triangles either do not have common points, or have one common side, or one common vertex. d) The same as c), but the initial triangle has the side $3$.

2010 Romania Team Selection Test, 5

Tags: floor function , number theory

Let $a$ and $n$ be two positive integer numbers such that the (positive) prime factors of $a$ be all greater than $n$. Prove that $n!$ divides $(a - 1)(a^2 - 1)\cdots (a^{n-1} - 1)$. [i]AMM Magazine[/i]

PEN S Problems, 19

Tags: floor function , calculus , integration , induction

Determine all pairs $(a, b)$ of real numbers such that $a\lfloor bn\rfloor =b\lfloor an\rfloor$ for all positive integer $n$.

2003 Portugal MO, 2

Tags: combinatorics , combinatorial geometry

An architect designed a hexagonal column with $37$ metal tubes of equal thickness. The figure shows the cross-section of this column. Is it possible to build a similar column whose number of tubes ends in $2003$? [img]https://cdn.artofproblemsolving.com/attachments/6/a/eb5714d2324aac8b78042d1f48f03b74ab0d78.png[/img]

Kvant 2021, M2672

Tags: geometry

Let the inscribed circle $\omega$ of the triangle $ABC$ have a center $I{}$ and touch the sides $BC, CA$ and $AB$ at points $D, E$ and $F{}$ respectively. Let $M{}$ and $N{}$ be points on the straight line $EF$ such that $BM \parallel AC$ and $CN \parallel AB$. Let $P{}$ and $Q{}$ be points on the segments $DM{}$ and $DN{}$, respectively, such that $BP \parallel CQ$. Prove that the intersection point of the lines $PF$ and $QE$ lies on $\omega$. [i]Proposed by Don Luu (Vietnam)[/i]

2020 Dutch Mathematical Olympiad, 2

Tags: sequence , recurrence relation , algebra

For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$. (a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$. (b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.

2007 Serbia National Math Olympiad, 1

Tags: incenter , triangle , midpoint , congruent triangles , geometry

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

1982 IMO Longlists, 29

Tags: function , limit , algebra

Let $f : \mathbb R \to \mathbb R$ be a continuous function. Suppose that the restriction of $f$ to the set of irrational numbers is injective. What can we say about $f$? Answer the analogous question if $f$ is restricted to rationals.

2023 Indonesia MO, 1

Tags: geometry

An acute triangle $ABC$ has $BC$ as its longest side. Points $D,E$ respectively lie on $AC,AB$ such that $BA = BD$ and $CA = CE$. The point $A'$ is the reflection of $A$ against line $BC$. Prove that the circumcircles of $ABC$ and $A'DE$ have the same radii.

1994 Tuymaada Olympiad, 5

Tags: inequalities , trigonometric inequality , minimum , trigonometry

Find the smallest natural number $n$ for which $sin \Big(\frac{1}{n+1934}\Big)<\frac{1}{1994}$ .

2016 Kosovo National Mathematical Olympiad, 1

Tags: number theory

Find all triples $(x,y,z)$ of integers such that satisfied: $x^2+y^2+z^2+xy+yz+zx=6$

2002 Argentina National Olympiad, 4

Tags: combinatorics

Initially on the blackboard all the integers from $1$ to $2002$ inclusive are written in one line and in some order, without repetitions. In each step, the first and second numbers of the line are deleted and the absolute value of the subtraction of the two numbers that have just been deleted is written at the beginning of the line; the other numbers are not modified in that step, and there is a new line that has one less number than the previous step. After completing $2001$ steps, only one number remains on the board. Determine all possible values of the number left on the board by varying the order of the $2002$ numbers on the initial line (and performing the $2001$ steps).