Olympiad problems

2022 Benelux, 4

Tags: number theory

A subset $A$ of the natural numbers $\mathbb{N} = \{0, 1, 2,\dots\}$ is called [i]good[/i] if every integer $n>0$ has at most one prime divisor $p$ such that $n-p\in A$. (a) Show that the set $S = \{0, 1, 4, 9,\dots\}$ of perfect squares is good. (b) Find an infinite good set disjoint from $S$. (Two sets are [i]disjoint[/i] if they have no common elements.)

2019 USAJMO, 1

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There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear. A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.

2011 Cono Sur Olympiad, 2

Tags: invariant , combinatorics

The numbers $1$ through $4^{n}$ are written on a board. In each step, Pedro erases two numbers $a$ and $b$ from the board, and writes instead the number $\frac{ab}{\sqrt{2a^2+2b^2}}$. Pedro repeats this procedure until only one number remains. Prove that this number is less than $\frac{1}{n}$, no matter what numbers Pedro chose in each step.

1974 AMC 12/AHSME, 27

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If $ f(x)\equal{}3x\plus{}2$ for all real $ x$, then the statement: "$ |f(x)\plus{}4| < a$ whenever $ |x\plus{}2| < b$ and $ a>0$ and $ b>0$" is true when $ \textbf{(A)}\ b \le a/3 \qquad \textbf{(B)}\ b > a/3 \qquad \textbf{(C)}\ a \le b/3 \qquad \textbf{(D)}\ a > b/3 \\ \qquad \textbf{(E)}\ \text{The statement is never true.}$

1989 IMO Longlists, 98

Tags: linear algebra , matrix , algebra

Let $ A$ be an $ n \times n$ matrix whose elements are non-negative real numbers. Assume that $ A$ is a non-singular matrix and all elements of $ A^{\minus{}1}$ are non-negative real numbers. Prove that every row and every column of $ A$ has exactly one non-zero element.

the 11th XMO, 4

Tags: combinatorics

We define a beehive of order $n$ as follows: a beehive of order 1 is one hexagon To construct a beehive of order $n$, take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$ Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected. Let $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$.

2019 Estonia Team Selection Test, 4

Tags: algebra , polynomial

Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.

PEN D Problems, 21

Tags: congruence , euler , modular arithmetic

Determine the last three digits of \[2003^{2002^{2001}}.\]

2002 All-Russian Olympiad, 4

Tags: combinatorics , extremal combinatorics

A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$. Heracle defeats a hydra by cutting it into two parts which are no joined. Find the minimum $N$ for which Heracle can defeat any hydra with $100$ necks by no more than $N$ hits.

2013 Saudi Arabia GMO TST, 2

Tags: polynomial , algebra , irreducible

Let $f(X) = a_nX^n + a_{n-1}X^{n-1} + ...+ a_1X + p$ be a polynomial of integer coefficients where $p$ is a prime number. Assume that $p >\sum_{i=1}^n |a_i|$. Prove that $f(X)$ is irreducible.

2021 Romania National Olympiad, 2

Tags: superior algebra , rings

Determine all non-trivial finite rings with am unit element in which the sum of all elements is invertible. [i]Mihai Opincariu[/i]

1988 Dutch Mathematical Olympiad, 4

Tags: geometry , geometric inequality , geometric inequalities , square

Given is an isosceles triangle $ABC$ with $AB = 2$ and $AC = BC = 3$. We consider squares where $A, B$ and $C$ lie on the sides of the square (so not on the extension of such a side). Determine the maximum and minimum value of the area of such a square. Justify the answer.

2019 Azerbaijan Senior NMO, 1

Tags: algebra , square root

Solve the following equation $$\sqrt{\frac{x^2}3-ax+a^2}+\sqrt{\frac{x^2}3-bx+b^2}=\sqrt{a^2-ab+b^2}$$ where $a;b\in\mathbb{R^+}$

2022 Bulgaria EGMO TST, 1

Tags: algebra , closure , binary operation , addition

The finite set $M$ of real numbers is such that among any three of its elements there are two whose sum is in $M$. What is the maximum possible cardinality of $M$? [hide=Remark about the other problems] Problem 2 is UK National Round 2022 P2, Problem 3 is UK National Round 2022 P4, Problem 4 is Balkan MO 2021 Shortlist N2 (the one with Bertrand), Problem 5 is IMO Shortlist 2021 A1 and Problem 6 is USAMO 2002/1. Hence neither of these will be posted here. [/hide]

2020 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt

Alice writes $1001$ letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\{a, b, c\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice can perform a sequence of moves which results in one letter remaining on the blackboard? [i]Proposed by Daniel Zhu.[/i]

2001 National High School Mathematics League, 9

Tags: 3d geometry , geometry

The length of edge of cube $ABCD-A_1B_1C_1D_1$ is $1$, then the distance between lines $A_1C_1$ and $BD_1$ is________.

2009 USAMTS Problems, 3

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Prove that if $a$ and $b$ are positive integers such that $a^2 + b^2$ is a multiple of $7^{2009}$, then $ab$ is a multiple of $7^{2010}$.

1969 All Soviet Union Mathematical Olympiad, 116

Tags: algebra

There is a wolf in the centre of a square field, and four dogs in the corners. The wolf can easily kill one dog, but two dogs can kill the wolf. The wolf can run all over the field, and the dogs -- along the fence (border) only. Prove that if the dog's speed is $1.5$ times more than the wolf's, than the dogs can prevent the wolf escaping.

2010 Malaysia National Olympiad, 5

Tags: number theory

Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$.

2021 AMC 10 Spring, 11

Tags:

For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$? $\textbf{(A) } 3 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

2013 Dutch BxMO/EGMO TST, 4

Tags: functional equation , determine all functions , algebra

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]

2025 Ukraine National Mathematical Olympiad, 9.1

Tags: algebra , system of equations

Solve the system of equations in reals: \[ \begin{cases} y = x^2 + 2x \\ z = y^2 + 2y \\ x = z^2 + 2z \end{cases} \] [i]Proposed by Mykhailo Shtandenko[/i]

2012 Kosovo Team Selection Test, 2

Tags: number theory

Find all three digit numbers, for which the sum of squares of each digit is $90$ .

2006 Italy TST, 3

Tags: function , induction , algebra , functional equation

Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$, \[f(m - n + f(n)) = f(m) + f(n).\]

2021 Taiwan TST Round 3, G

Tags: geometry , circumcircle

Let $ABC$ be a triangle with $AB<AC$, and let $I_a$ be its $A$-excenter. Let $D$ be the projection of $I_a$ to $BC$. Let $X$ be the intersection of $AI_a$ and $BC$, and let $Y,Z$ be the points on $AC,AB$, respectively, such that $X,Y,Z$ are on a line perpendicular to $AI_a$. Let the circumcircle of $AYZ$ intersect $AI_a$ again at $U$. Suppose that the tangent of the circumcircle of $ABC$ at $A$ intersects $BC$ at $T$, and the segment $TU$ intersects the circumcircle of $ABC$ at $V$. Show that $\angle BAV=\angle DAC$. [i]Proposed by usjl.[/i]