Olympiad problems

2005 Germany Team Selection Test, 3

Tags: number theory

Let $a$, $b$, $c$, $d$ and $n$ be positive integers such that $7\cdot 4^n = a^2+b^2+c^2+d^2$. Prove that the numbers $a$, $b$, $c$, $d$ are all $\geq 2^{n-1}$.

2006 All-Russian Olympiad Regional Round, 8.5

Tags: number theory , even

The product $a_1 \cdot a_2 \cdot ... \cdot a_{100}$ is written on the board , where $a_1$, $a_2$, $ ... $, $a_{100}$, are natural numbers. Let's consider $99$ expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that the values of exactly $32$ of these expressions are even. What is the largest number of even numbers among $a_1$, $a_2$, $ ... $, $a_{100}$ could it be?

2015 India Regional MathematicaI Olympiad, 1

Tags: geometry , geometric transformation , reflection , angle bisector

Let ABC be a triangle. Let B' and C' denote the reflection of B and C in the internal angle bisector of angle A. Show that the triangles ABC and AB'C' have the same incenter.

2003 AMC 10, 12

Tags: geometry , rectangle , probability , analytic geometry

A point $ (x,y)$ is randomly picked from inside the rectangle with vertices $ (0,0)$, $ (4,0)$, $ (4,1)$, and $ (0,1)$. What is the probability that $ x<y$? $ \textbf{(A)}\ \frac{1}{8} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{3}{8} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{3}{4}$

2020 New Zealand MO, 4

Tags: number theory , prime number , prime

Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers.

2011 VJIMC, Problem 1

Tags: linear algebra , matrix

Let $n>k$ and let $A_1,\ldots,A_k$ be real $n\times n$ matrices of rank $n-1$. Prove that $$A_1\cdots A_k\ne0.$$

2017 Polish MO Finals, 4

Tags: set theory , combinatorics

Prove that the set of positive integers $\mathbb Z^+$ can be represented as a sum of five pairwise distinct subsets with the following property: each $5$-tuple of numbers of form $(n,2n,3n,4n,5n)$, where $n\in\mathbb Z^+$, contains exactly one number from each of these five subsets.

2022 Stars of Mathematics, 4

Tags: composite number , number theory , fermat primes

Let $a{}$ be an even positive integer which is not a power of two. Prove that at least one of $2^{2^n}+1$ and $a^{2^n}+1$ is composite, for infinitely many positive integers $n$. [i]Bojan Bašić[/i]

2016 CMIMC, 10

Tags: team

Let $\mathcal{P}$ be the unique parabola in the $xy$-plane which is tangent to the $x$-axis at $(5,0)$ and to the $y$-axis at $(0,12)$. We say a line $\ell$ is $\mathcal{P}$-friendly if the $x$-axis, $y$-axis, and $\mathcal{P}$ divide $\ell$ into three segments, each of which has equal length. If the sum of the slopes of all $\mathcal{P}$-friendly lines can be written in the form $-\tfrac mn$ for $m$ and $n$ positive relatively prime integers, find $m+n$.

1960 Putnam, A1

Tags: integer , equation

Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation $$\frac{xy}{x+y}=n?$$

2023 Princeton University Math Competition, A4 / B6

Tags: number theory

What is the smallest possible sum of six distinct positive integers for which the sum of any five of them is prime?

2006 Bulgaria Team Selection Test, 2

Tags: polynomial , algebra

Find all couples of polynomials $(P,Q)$ with real coefficients, such that for infinitely many $x\in\mathbb R$ the condition \[ \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}\] Holds. [i] Nikolai Nikolov, Oleg Mushkarov[/i]

2005 Cono Sur Olympiad, 2

Tags: number theory

We say that a number of 20 digits is [i]special[/i] if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.

2022 Dutch IMO TST, 2

Tags: common tangent , geometry , concurrency

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

1963 AMC 12/AHSME, 10

Tags: geometry , circumcircle , perpendicular bisector , pythagorean theorem

Point $P$ is taken interior to a square with side-length $a$ and such that is it equally distant from two consecutive vertices and from the side opposite these vertices. If $d$ represents the common distance, then $d$ equals: $\textbf{(A)}\ \dfrac{3a}{5} \qquad \textbf{(B)}\ \dfrac{5a}{8} \qquad \textbf{(C)}\ \dfrac{3a}{8} \qquad \textbf{(D)}\ \dfrac{a\sqrt{2}}{2} \qquad \textbf{(E)}\ \dfrac{a}{2}$

2017 BMT Spring, 10

Tags: algebra

Dene $H_n =\sum^n_{k=1} \frac{1}{k}$ . Evaluate $\sum^{2017}_{n=1} {2017 \choose n} H_n(-1)^n$.

2010 Ukraine Team Selection Test, 2

Tags: geometry , cyclic quadrilateral , collinear , incircle , excircle

Let $ABCD$ be a quadrilateral inscribled in a circle with the center $O, P$ be the point of intersection of the diagonals $AC$ and $BD$, $BC\nparallel AD$. Rays $AB$ and $DC$ intersect at the point $E$. The circle with center $I$ inscribed in the triangle $EBC$ touches $BC$ at point $T_1$. The $E$-excircle with center $J$ in the triangle $EAD$ touches the side $AD$ at the point T$_2$. Line $IT_1$ and $JT_2$ intersect at $Q$. Prove that the points $O, P$, and $Q$ lie on a straight line.

1975 Polish MO Finals, 6

Tags: algebra

On the interval $[0,1]$ are given functions $S(x) = 1 - x$ and $T(x) = x/2$. Does there exist a function of the form $f = g_1\circ g_2\circ ... \circ g_n$, where $n \in N$ and each $g_k$ is either $S(x)$ or $T(x)$, such that $$f\left(\frac12\right)=\frac{1975}{2^{1975}} \, ?$$

2017 Hanoi Open Mathematics Competitions, 1

Tags: algebra , sum , polynomial

Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 4x^2 -3x + 2$. The sum $|x_1| + |x_2| + |x_3|$ is (A): $4$ (B): $6$ (C): $8$ (D): $10$ (E): None of the above.

2014 IMC, 1

Tags: sequence

For a positive integer $x$, denote its $n^{\mathrm{th}}$ decimal digit by $d_n(x)$, i.e. $d_n(x)\in \{ 0,1, \dots, 9\}$ and $x=\sum_{n=1}^{\infty} d_n(x)10^{n-1}$. Suppose that for some sequence $(a_n)_{n=1}^{\infty}$, there are only finitely many zeros in the sequence $(d_n(a_n))_{n=1}^{\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence $(a_n)_{n=1}^{\infty}$. (Proposed by Alexander Bolbot, State University, Novosibirsk)

2024 Brazil Team Selection Test, 1

Tags: combinatorics

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

2005 AMC 12/AHSME, 11

Tags: probability

An envelope contains eight bills: $ 2$ ones, $ 2$ fives, $ 2$ tens, and $ 2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $ \$ 20$ or more? $ \textbf{(A)}\ \frac {1}{4}\qquad \textbf{(B)}\ \frac {2}{7}\qquad \textbf{(C)}\ \frac {3}{7}\qquad \textbf{(D)}\ \frac {1}{2}\qquad \textbf{(E)}\ \frac {2}{3}$

2025 All-Russian Olympiad Regional Round, 9.7

Tags: algebra

Let's call a set of numbers [i]lucky[/i] if it cannot be divided into two nonempty groups so that the product of the sum of the numbers in one group and the sum of the numbers in the other is positive. The teacher wrote several integers on the blackboard. Prove that the children can add another integer to the existing ones so that the resulting set is lucky. [i]A. Kuznetsov[/i]

2007 IMO Shortlist, 6

Tags: algorithm , graph theory , combinatorics , clique number

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a [i]clique[/i] if each two of them are friends. (In particular, any group of fewer than two competitiors is a clique.) The number of members of a clique is called its [i]size[/i]. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room. [i]Author: Vasily Astakhov, Russia[/i]

2019 USMCA, 1

Tags:

Kelvin the Frog and Alex the Kat are playing a game on an initially empty blackboard. Kelvin begins by writing a digit. Then, the players alternate inserting a digit anywhere into the number currently on the blackboard, including possibly a leading zero (e.g. $12$ can become $123$, $142$, $512$, $012$, etc.). Alex wins if the blackboard shows a perfect square at any time, and Kelvin's goal is prevent Alex from winning. Does Alex have a winning strategy?