Olympiad problems

Novosibirsk Oral Geo Oly VII, 2021.1

Tags: geometry , rectangle

Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.

2021 AMC 12/AHSME Fall, 17

Tags:

How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions? $\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$

2023 Sharygin Geometry Olympiad, 10.1

Tags: geometry

Let $M$ be the midpoint of cathetus $AB$ of triangle $ABC$ with right angle $A$. Point $D$ lies on the median $AN$ of triangle $AMC$ in such a way that the angles $ACD$ and $BCM$ are equal. Prove that the angle $DBC$ is also equal to these angles.

2019 USAMO, 6

Tags: polynomial , weird problem , the saddest

Find all polynomials $P$ with real coefficients such that $$\frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x)$$ holds for all nonzero real numbers $x,y,z$ satisfying $2xyz=x+y+z$. [i]Proposed by Titu Andreescu and Gabriel Dospinescu[/i]

2007 Italy TST, 2

Tags: modular arithmetic , combinatorics

In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$. (a) Find the minimum of cyclical 3-subset (depending on $n$); (b) Find the maximum of cyclical 3-subset (depending on $n$).

2014 China Northern MO, 7

Tags: modular arithmetic , number theory

Prove that there exist infinitely many positive integers $n$ such that $3^n+2$ and $5^n+2$ are all composite numbers.

KoMaL A Problems 2021/2022, A. 826

Tags: combinatorics

An antelope is a chess piece which moves similarly to the knight: two cells $(x_1,y_1)$ and $(x_2,y_2)$ are joined by an antelope move if and only if \[ \{|x_1-x_2|,|y_1-y_2|\}=\{3,4\}.\] The numbers from $1$ to $10^{12}$ are placed in the cells of a $10^6\times 10^6$ grid. Let $D$ be the set of all absolute differences of the form $|a-b|$, where $a$ and $b$ are joined by an antelope move in the arrangement. How many arrangements are there such that $D$ contains exactly four elements? Proposed by [i]Nikolai Beluhov[/i], Bulgaria

2023 Sinapore MO Open, P1

Tags: geometry , homothety

In a scalene triangle $ABC$ with centroid $G$ and circumcircle $\omega$ centred at $O$, the extension of $AG$ meets $\omega$ at $M$; lines $AB$ and $CM$ intersect at $P$; and lines $AC$ and $BM$ intersect at $Q$. Suppose the circumcentre $S$ of the triangle $APQ$ lies on $\omega$ and $A, O, S$ are collinear. Prove that $\angle AGO = 90^{o}$.

2012 National Olympiad First Round, 27

Tags: trigonometry

What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$? $ \textbf{(A)}\ \sqrt3 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ \sqrt 2 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \text{None}$

1956 Miklós Schweitzer, 2

Tags: complex number

[b]2.[/b] Find the minimum of $max ( |1+z|, |1+z^{2}|)$ if $z$ runs over all complex numbers. [b](F. 2)[/b]

1988 Poland - Second Round, 2

Tags: algebra , inequalities

Given real numbers $ x_i $, $ y_i $ ($ i = 1, 2, \ldots, n $) such that $$ \qquad x_1 \geq x_2 \geq \ldots \geq x_n \geq 0, \ \ y_1 > y_2 > \ldots > y_n \geq 0,$$ and $$ \prod_{i=1}^k x_i \geq \prod_{i=1}^k y_i, \ \ \text{ for } \ \ k=1,2,\ldots, n.$$ Prove that $$ \sum_{i=1}^n x_i > \sum_{i=1}^n y_i.$$

1999 AMC 8, 22

Tags:

In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? $ \text{(A)}\ \frac{3}{8}\qquad\text{(B)}\ \frac{1}{2}\qquad\text{(C)}\ \frac{3}{4}\qquad\text{(D)}\ 2\frac{2}{3}\qquad\text{(E)}\ 3\frac{1}{3} $

2007 AMC 12/AHSME, 18

Tags: algebra , polynomial , geometry , 3d geometry , quadratic

The polynomial $ f(x) \equal{} x^{4} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ has real coefficients, and $ f(2i) \equal{} f(2 \plus{} i) \equal{} 0.$ What is $ a \plus{} b \plus{} c \plus{} d?$ $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 16$

1999 Austrian-Polish Competition, 6

Tags: algebra , system of equations

Solve in the nonnegative real numbers the system of equations $$\begin{cases} x_n^2 + x_nx_{n-1} + x_{n-1}^4 = 1 \,\,\,\, for \,\,\,\, n = 1,2,..., 1999 \\\ x_0 = x_{1999} \end{cases}$$

1987 Romania Team Selection Test, 4

Tags: algebra , polynomial , vector , inequalities

Let $ P(X) \equal{} a_{n}X^{n} \plus{} a_{n \minus{} 1}X^{n \minus{} 1} \plus{} \ldots \plus{} a_{1}X \plus{} a_{0}$ be a real polynomial of degree $ n$. Suppose $ n$ is an even number and: a) $ a_{0} > 0$, $ a_{n} > 0$; b) $ a_{1}^{2} \plus{} a_{2}^{2} \plus{} \ldots \plus{} a_{n \minus{} 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n \minus{} 1}$. Prove that $ P(x)\geq 0$ for all real values $ x$. [i]Laurentiu Panaitopol[/i]

2012 Poland - Second Round, 1

Tags: algebra

$a,b,c,d\in\mathbb{R}$, solve the system of equations: \[ \begin{cases} a^3+b=c \\ b^3+c=d \\ c^3+d=a \\ d^3+a=b \end{cases} \]

2016 PUMaC Number Theory A, 4

Tags: number theory

Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \le n < b$ such that $n^2$ and $n$ end in the same digit in base $b$.

1981 All Soviet Union Mathematical Olympiad, 317

Tags: combinatorics , tournament

Eighteen soccer teams have played $8$ tours of a one-round tournament. Prove that there is a triple of teams, having not met each other yet.

PEN E Problems, 16

Tags: binomial coefficient , algebra , polynomial

Prove that for any prime $p$ in the interval $\left]n, \frac{4n}{3}\right]$, $p$ divides \[\sum^{n}_{j=0}{{n}\choose{j}}^{4}.\]

2020 Switzerland - Final Round, 6

Tags: combinatorics

Let $n \ge 2$ be an integer. Consider the following game: Initially, $k$ stones are distributed among the $n^2$ squares of an $n\times n$ chessboard. A move consists of choosing a square containing at least as many stones as the number of its adjacent squares (two squares are adjacent if they share a common edge) and moving one stone from this square to each of its adjacent squares. Determine all positive integers $k$ such that: (a) There is an initial configuration with $k$ stones such that no move is possible. (b) There is an initial configuration with $k$ stones such that an infinite sequence of moves is possible.

MathLinks Contest 5th, 2.1

Tags: functional equation

For what positive integers $k$ there exists a function $f : N \to N$ such that for all $n \in N$ we have $\underbrace{\hbox{f(f(... f(n)....))}}_{\hbox{k times}} = f(n) + 2$ ?

2011 Greece National Olympiad, 3

Tags: inequalities

Let $a,b,c$ be positive real numbers with sum $6$. Find the maximum value of \[S = \sqrt[3]{{{a^2} + 2bc}} + \sqrt[3]{{{b^2} + 2ca}} + \sqrt[3]{{{c^2} + 2ab}}.\]

2020 LMT Spring, 29

Tags:

Let $\mathcal{F}$ be the set of polynomials $f(x)$ with integer coefficients for which there exists an integer root of the equation $f(x)=1$. For all $k>1$, let $m_k$ be the smallest integer greater than one for which there exists $f(x)\in \mathcal{F}$ such that $f(x)=m_k$ has exactly $k$ distinct integer roots. If the value of $\sqrt{m_{2021}-m_{2020}}$ can be written as $m\sqrt{n}$ for positive integers $m,n$ where $n$ is squarefree, compute the largest integer value of $k$ such that $2^k$ divides $\frac{m}{n}$.

1999 Argentina National Olympiad, 4

Tags: combinatorics , geometry , combinatorial geometry

Coins of diameter $1$ have been placed on a square of side $11$, without overlapping or protruding from the square. Can there be $126$ coins? and $127$? and $128$?

2024 Caucasus Mathematical Olympiad, 5

Tags: algebra

Let $a, b, c$ be reals and consider three lines $y=ax+b, y=bx+c, y=cx+a$. Two of these lines meet at a point with $x$-coordinate $1$. Show that the third one passes through a point with two integer coordinates.