Olympiad problems

2014 China Northern MO, 7

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.

1980 IMO Shortlist, 2

Tags: algebra , recurrence relation , sequence , inequality

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

1940 Putnam, A6

Tags: algebra , polynomial

Let $f(x)$ be a polynomial of degree $n$ such that $f(x)^{p}$ is divisible by $f'(x)^{q}$ for some positive integers $p,q$. Prove that $f(x)$ is divisible by $f'(x)$ and that $f(x)$ has a single root of multiplicity $n$.

1969 IMO Longlists, 13

Tags: modular arithmetic , number theory , divisibility , additive number theory

$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

2018 BMT Spring, 8

Tags: combinatorics

Moor and nine friends are seated around a circular table. Moor starts out holding a bottle, and whoever holds the bottle passes it to the person on his left or right with equal probability until everyone has held the bottle. Compute the expected distance between Moor and the last person to receive the bottle, where distance is the fewest number of times the bottle needs to be passed in order to go back to Moor.

2011 Purple Comet Problems, 2

Tags: geometry

The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$. [asy] size(150); defaultpen(linewidth(0.8)); int i; for(i=5;i>=1;i=i-1) { if (floor(i/2)==i/2) { filldraw(circle(origin,4*i),white); } else { filldraw(circle(origin,4*i),red); } } [/asy]