Olympiad problems

2009 Indonesia TST, 1

Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

2019 JBMO Shortlist, G7

Tags: geometry

Let $ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$. Let $K$ be the midpoint of $BC$, and let $AKLM$ be a parallelogram with centre $C$. Let $T$ be the intersection of the line $AC$ and the perpendicular bisector of $BM$. Let $\omega_1$ be the circle with centre $C$ and radius $CA$ and let $\omega_2$ be the circle with centre $T$ and radius $TB$. Prove that one of the points of intersection of $\omega_1$ and $\omega_2$ is on the line $LM$. [i]Proposed by Greece[/i]

2010 Mathcenter Contest, 6

Tags: algebra , functional equation

Find all $a\in\mathbb{N}$ such that exists a bijective function $g :\mathbb{N} \to \mathbb{N}$ and a function $f:\mathbb{N}\to\mathbb{N}$, such that for all $x\in\mathbb{N}$, $$f(f(f(...f(x)))...)=g(x)+a$$ where $f$ appears $2009$ times. [i](tatari/nightmare)[/i]

1994 All-Russian Olympiad Regional Round, 11.2

Tags: combinatorics

It was noted that during one day in a town, each person made at most one phone call. Prove that the people in the town can be divided into three groups such that no two persons in the same group talked by phone that day.

2016 Ecuador Juniors, 1

Tags: number theory , digit

A natural number of five digits is called [i]Ecuadorian [/i]if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$, but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$. Find how many Ecuadorian numbers exist.

2023 Malaysian Squad Selection Test, 2

Tags: geometry

Let $ABC$ be a triangle with orthocenter $H$. Let $\ell_b, \ell_c$ be the reflection of lines $AB$ and $AC$ about $AH$ respectively. Suppose $\ell_b$ intersect $CH$ at $P$, and $\ell_c$ intersect $BH$ at $Q$. Prove that $AH, PQ, BC$ are concurrent. [i]Proposed by Ivan Chan Kai Chin[/i]

1993 Kurschak Competition, 1

Tags: inequalities , quadratic , algebra , number theory

Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.

2013 China Team Selection Test, 3

Tags: analytic geometry , geometry , combinatorics

A point $(x,y)$ is a [i]lattice point[/i] if $x,y\in\Bbb Z$. Let $E=\{(x,y):x,y\in\Bbb Z\}$. In the coordinate plane, $P$ and $Q$ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let $T=P\cap Q$. Prove that if $T\ne\emptyset$ and $T\cap E=\emptyset$, then $T$ is a non-degenerate convex quadrilateral region.

1980 Vietnam National Olympiad, 3

Tags: inequalities

Let be given an integer $n\ge 2$ and a positive real number $p$. Find the maximum of \[\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},\] where $x_i$ are non-negative real numbers with sum $p$.

Kharkiv City MO Seniors - geometry, 2021.10.5

Tags: geometry , collinear , incircle

The inscribed circle $\Omega$ of triangle $ABC$ touches the sides $AB$ and $AC$ at points $K$ and $ L$, respectively. The line $BL$ intersects the circle $\Omega$ for the second time at the point $M$. The circle $\omega$ passes through the point $M$ and is tangent to the lines $AB$ and $BC$ at the points $P$ and $Q$, respectively. Let $N$ be the second intersection point of circles $\omega$ and $\Omega$, which is different from $M$. Prove that if $KM \parallel AC$ then the points $P, N$ and $L$ lie on one line.

2022 BMT, 27

Tags: combinatorics

Submit a positive integer $n$ less than $10^5$. Let the sum of the valid submissions from all teams to this question be $S$. If you submit an invalid answer, you will receive $0$ points. Otherwise, your score will be $ \max \left(0,\lfloor 25 - \frac{|S'-n|}{10} \rfloor \right)$ , where $S'$ is the sum of the squares of the digits of $S$.

2020 LMT Fall, B15

Tags: geometry

Let $\vartriangle AMO$ be an equilateral triangle. Let $U$ and $G$ lie on side $AM$, and let $S$ and $N$ lie on side $AO$ such that $AU =UG = GM$ and $AS = SN = NO$. Find the value of $\frac{[MONG]}{[U S A]}$

1966 AMC 12/AHSME, 17

Tags: ellipse , conic

The number of distinct points common to the curves $x^2+4y^2=1$ and $4x^2+y^2=5$ is: $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4$

2015 Iran Team Selection Test, 5

Tags: number theory , combinatorics

We call a permutation $(a_1, a_2,\cdots , a_n)$ of the set $\{ 1,2,\cdots, n\}$ "good" if for any three natural numbers $i <j <k$, $n\nmid a_i+a_k-2a_j$ find all natural numbers $n\ge 3$ such that there exist a "good" permutation of a set $\{1,2,\cdots, n\}$.

2019 Purple Comet Problems, 18

Tags: system of equations , algebra

Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying $a + b - c - d = 3$ $ab - 3bc + cd - 3da = 4$ $3ab - bc + 3cd - da = 5$ Find $11(a - c)^2 + 17(b -d)^2$.

2022 AMC 12/AHSME, 23

Tags:

Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define \[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\] Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum \[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\] $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

2023 Middle European Mathematical Olympiad, 4

Tags: number theory

Let $n, m$ be positive integers. A set $S$ of positive integers is called $(n, m)$-good, if: (1) $m \in S$; (2) for all $a\in S$, all divisors of $a$ are also in $S$; (3) for all distinct $a, b \in S$, $a^n+b^n \in S$. For which $(n, m)$, the only $(n, m)$-good set is $\mathbb{N}$?

2005 Tournament of Towns, 1

Tags:

A palindrome is a positive integer which reads in the same way in both directions (for example, $1$, $343$ and $2002$ are palindromes, while $2005$ is not). Is it possible to find $2005$ pairs in the form of $(n, n + 110)$ where both numbers are palindromes? [i](3 points)[/i]

2010 Finnish National High School Mathematics Competition, 5

Tags: geometry

Let $S$ be a non-empty subset of a plane. We say that the point $P$ can be seen from $A$ if every point from the line segment $AP$ belongs to $S$. Further, the set $S$ can be seen from $A$ if every point of $S$ can be seen from $A$. Suppose that $S$ can be seen from $A$, $B$ and $C$ where $ABC$ is a triangle. Prove that $S$ can also be seen from any other point of the triangle $ABC$.

1992 Brazil National Olympiad, 1

Tags: calculus , derivative , analytic geometry , graphing lines , slope , algebra

The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$

2010 China National Olympiad, 3

Tags: inequalities , trigonometry , calculus , function , complex number , algebra

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

2021 Iran Team Selection Test, 2

Tags: function , number theory

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have : $$f(n)+1400m^2|n^2+f(f(m))$$

2014 Czech-Polish-Slovak Junior Match, 2

Tags: equation , algebra

Solve the equation $a + b + 4 = 4\sqrt{a\sqrt{b}}$ in real numbers

2023/2024 Tournament of Towns, 1

Tags: polymonial , algebra

For every polynomial of degree 45 with coefficients $1,2,3, \ldots, 46$ (in some order) Tom has listed all its distinct real roots. Then he increased each number in the list by 1 . What is now greater: the amount of positive numbers or the amount of negative numbers? Alexey Glebov

2020 MBMT, 20

Tags: geometry , geometric transformation , rotation

Sam colors each tile in a 4 by 4 grid white or black. A coloring is called [i]rotationally symmetric[/i] if the grid can be rotated 90, 180, or 270 degrees to achieve the same pattern. Two colorings are called [i]rotationally distinct[/i] if neither can be rotated to match the other. How many rotationally distinct ways are there for Sam to color the grid such that the colorings are [i]not[/i] rotationally symmetric? [i]Proposed by Gabriel Wu[/i]