Olympiad problems

1968 Miklós Schweitzer, 3

Let $ K$ be a compact topological group, and let $ F$ be a set of continuous functions defined on $ K$ that has cardinality greater that continuum. Prove that there exist $ x_0 \in K$ and $ f \not\equal{}g \in F$ such that \[ f(x_0)\equal{}g(x_0)\equal{}\max_{x\in K}f(x)\equal{}\max_{x \in K}g(x).\] [i]I. Juhasz[/i]

2010 AMC 8, 22

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The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8 $

2023 Romanian Master of Mathematics Shortlist, N2

Tags: number theory , decimal representation , algebra , polynomial

For every non-negative integer $k$ let $S(k)$ denote the sum of decimal digits of $k$. Let $P(x)$ and $Q(x)$ be polynomials with non-negative integer coecients such that $S(P(n)) = S(Q(n))$ for all non-negative integers $n$. Prove that there exists an integer $t$ such that $P(x) - 10^tQ(x)$ is a constant polynomial.

2021 Thailand Mathematical Olympiad, 4

Tags: pigeonhole principle , combinatorics

Kan Krao Park is a circular park that has $21$ entrances and a straight line walkway joining each pair of two entrances. No three walkways meet at a single point. Some walkways are paved with bricks, while others are paved with asphalt. At each intersection of two walkways, excluding the entrances, is planted lotus if the two walkways are paved with the same material, and is planted waterlily if the two walkways are paved with different materials. Each walkway is decorated with lights if and only if the same type of plant is placed at least $45$ different points along that walkway. Prove that there are at least $11$ walkways decorated with lights and paved with the same material.

2010 National Olympiad First Round, 2

Tags: matrix , linear algebra

How many ordered pairs of positive integers $(x,y)$ are there such that $y^2-x^2=2y+7x+4$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{Infinitely many} $

2018 Baltic Way, 18

Tags: number theory

Let $n \ge 3$ be an integer such that $4n+1$ is a prime number. Prove that $4n+1$ divides $n^{2n}-1$.

2009 Hanoi Open Mathematics Competitions, 10

Tags: altitude , geometry

Let $ABC$ be an acute-angled triangle with $AB =4$ and $CD$ be the altitude through $C$ with $CD = 3$. Find the distance between the midpoints of $AD$ and $BC$

1991 Federal Competition For Advanced Students, 3

Tags: number theory , quadratic , induction , modular arithmetic

Find the number of squares in the sequence given by $ a_0\equal{}91$ and $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$ for $ n \ge 0.$

2023 Stanford Mathematics Tournament, 5

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Suppose $\alpha,\beta,\gamma\in\{-2,3\}$ are chosen such that \[M=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy\] is finite and positive (note: $\mathbb{R}_{\ge0}$ is the set of nonnegative real numbers). What is the sum of the possible values of $M$?

2023 Estonia Team Selection Test, 2

Tags: algebra , real coefficients , polynomial

Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.

2012 Canadian Mathematical Olympiad Qualification Repechage, 5

Tags: number theory

Given a positive integer $n$, let $d(n)$ be the largest positive divisor of $n$ less than $n$. For example, $d(8) = 4$ and $d(13) = 1$. A sequence of positive integers $a_1, a_2,\dots$ satisfies \[a_{i+1} = a_i +d(a_i),\] for all positive integers $i$. Prove that regardless of the choice of $a_1$, there are inﬁnitely many terms in the sequence divisible by $3^{2011}$.

2013 IMAC Arhimede, 6

Tags: sums and products , product , sum , inequalities , algebra

Let $p$ be an odd positive integer. Find all values of the natural numbers $n\ge 2$ for which holds $$\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0$$ where $x_1,x_2,..,x_n$ are any real numbers.

2013 Stanford Mathematics Tournament, 18

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Caroline wants to plant 10 trees in her orchard. Planting $n$ apple trees requires $n^2$ square meters, planting $n$ apricot trees requires $5n$ square meters, and planting $n$ plum trees requires $n^3$ square meters. If she is committed to growing only apple, apricot, and plum trees, what is the least amount of space in square meters, that her garden will take up?

2022 Argentina National Olympiad Level 2, 1

Tags: algebra

Find all real numbers $x$ such that exactly one of the four numbers $x-\sqrt 2$, $x-\dfrac{1}{x}$, $x+\dfrac{1}{x}$ and $x^2+2\sqrt{2}$ is [b]not[/b] an integer.

2013 India PRMO, 6

Tags: number theory , sum of digits

Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?

2021 Junior Balkаn Mathematical Olympiad, 2

Tags: number theory , set , maximum

For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$. Find the largest possible value of $T_A$.

2018 MIG, 6

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How many more hours are in $10$ years than seconds in $1$ day? $\textbf{(A) }1000\qquad\textbf{(B) }1100\qquad\textbf{(C) }1150\qquad\textbf{(D) }1200\qquad\textbf{(E) }1300$

2006 Stanford Mathematics Tournament, 24

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The number 555,555,555,555 factors into eight distinct prime factors, each with a multiplicity of 1. What are the three largest prime factors of 555,555,555,555?

2011 Baltic Way, 6

Tags: inequalities , combinatorics , number theory , analytic geometry , probability , relatively prime

Let $n$ be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates $(x,y),0\le x,y\le n$, is at least $\frac{n^2}{4}$.

2005 iTest, 21

Tags: geometry , tangent

Two circles have a common internal tangent of length $17$ and a common external tangent of length $25$. Find the product of the radii of the two circles.

MOAA Individual Speed General Rounds, 2021.10

Tags: speed

Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2017 Yasinsky Geometry Olympiad, 6

Tags: geometry , median , angle bisector , ratio

In the triangle $ABC$ , the angle bisector $AD$ divides the side $BC$ into the ratio $BD: DC = 2: 1$. In what ratio, does the median $CE$ divide this bisector?

1989 All Soviet Union Mathematical Olympiad, 493

Tags: combinatorial geometry , combinatorics

One bird lives in each of $n$ bird-nests in a forest. The birds change nests, so that after the change there is again one bird in each nest. Also for any birds $A, B, C, D$ (not necessarily distinct), if the distance $AB < CD$ before the change, then $AB > CD$ after the change. Find all possible values of $n$.

2011 Indonesia TST, 1

Tags: system of equations , algebra

Find all $4$-tuple of real numbers $(x, y, z, w)$ that satisfy the following system of equations: $$x^2 + y^2 + z^2 + w^2 = 4$$ $$\frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2 }+\frac{1}{w^2} = 5 -\frac{1}{(xyzw)^2}$$

2013 Taiwan TST Round 1, 2

Tags: number theory

If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.