Olympiad problems

2004 Junior Balkan Team Selection Tests - Romania, 1

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At a chess tourney, each player played with all the other players two matches, one time with the white pieces, and one time with the black pieces. One point was given for a victory, and 0,5 points were given for a tied game. In the end of the tourney all the players had the same number of points. a) Prove that there exist two players with the same number of tied games; b) Prove that there exist two players which have the same number of lost games when playing with the white pieces.

2015 Saudi Arabia Pre-TST, 1.4

We color each unit square of a $8\times 8$ table into green or blue such that there are $a$ green unit squares in each $3 \times 3$ square and $b$ green unit squares in each $2 \times 4$ rectangle. Find all possible values of $(a, b)$. (Le Anh Vinh)

1997 Taiwan National Olympiad, 9

Tags: combinatorics , logarithm

For $n\geq k\geq 3$, let $X=\{1,2,...,n\}$ and let $F_{k}$ a the family of $k$-element subsets of $X$, any two of which have at most $k-2$ elements in common. Show that there exists a subset $M_{k}$ of $X$ with at least $[\log_{2}{n}]+1$ elements containing no subset in $F_{k}$.

2023 Costa Rica - Final Round, 3.1

Tags: functional equation , algebra , nonnegative integers

Let $\mathbb Z^{\geq 0}$ be the set of all non-negative integers. Consider a function $f:\mathbb Z^{\geq 0} \to \mathbb Z^{\geq 0}$ such that $f(0)=1$ and $f(1)=1$, and that for any integer $n \geq 1$, we have \[f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.\] Determine the value of $f(2023)/f(2022)$.

MOAA Team Rounds, 2023.7

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In a cube, let $M$ be the midpoint of one of the segments. Choose two vertices of the cube, $A$ and $B$. What is the number of distinct possible triangles $\triangle AMB$ up to congruency? [i]Proposed by Harry Kim[/i]

2013 AMC 12/AHSME, 20

Tags: trigonometry , geometry , trapezoid , parabola , analytic geometry , graphing lines , conic

For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$? $ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $

2011 Junior Balkan Team Selection Tests - Moldova, 2

Tags: inequalities , algebra

The real numbers $a, b, x$ satisfy the inequalities $| a + x + b | \le 1, | 4a + 2x + b | \le1, | 9a + 6x + 4b | \le 1$. Prove that $| x | \le15$.

1952 AMC 12/AHSME, 27

Tags: ratio , geometry , perimeter

The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is: $ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 1: 3 \qquad\textbf{(C)}\ 1: \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3}: 2 \qquad\textbf{(E)}\ 2: 3$

2012 May Olympiad, 1

Tags: algebra , number theory

Pablo says: “I add $2$ to my birthday and multiply the result by $2$. I add to the number obtained $4$ and multiply the result by $5$. To the new number obtained I add the number of the month of my birthday (for example, if it's June, I add $6$) and I get $342$. " What is Pablo's birthday date? Give all the possibilities

2004 France Team Selection Test, 2

Tags: trigonometry , angle bisector , geometry

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

2020 Brazil Cono Sur TST, 1

Tags: geometry , incenter , circumcircle

Let $D$ and $E$ be points on sides $AB$ and $AC$ of a triangle $ABC$ such that $DB = BC = CE$. The segments $BE$ and $CD$ intersect at point $P$. Prove that the incenter of triangle $ABC$ lies on the circles circumscribed around the triangles $BDP$ and $CEP$.

1994 China National Olympiad, 2

Tags: pigeonhole principle , combinatorics

There are $m$ pieces of candy held in $n$ trays($n,m\ge 4$). An [i]operation[/i] is defined as follow: take out one piece of candy from any two trays respectively, then put them in a third tray. Determine, with proof, if we can move all candies to a single tray by finite [i]operations[/i].

2024 ELMO Shortlist, N5

Tags: number theory

Let $T$ be a finite set of squarefree integers. (a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$. (b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists? [i]Allen Wang[/i]

1994 National High School Mathematics League, 9

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Point Sets $A=\{(x,y)|(x-3)^2+(y-4)^2\leq\left( \frac{5}{2}\right)^2\},B=\{(x,y)|(x-4)^2+(y-5)^2>\left( \frac{5}{2}\right)^2\}$, then the number of integral points in $A\cap B$ is________.

2017 CCA Math Bonanza, I1

Tags: factorial

Find the integer $n$ such that $6!\times7!=n!$. [i]2017 CCA Math Bonanza Individual Round #1[/i]

2005 Kyiv Mathematical Festival, 1

Tags: inequalities

Prove that there exists a positive integer $ n$ such that for every $ x\ge0$ the inequality $ (x\minus{}1)(x^{2005}\minus{}2005x^{n\plus{}1}\plus{}2005x^n\minus{}1)\ge0$ holds.

1999 VJIMC, Problem 1

Tags: limit , real analysis

Find the limit $$\lim_{n\to\infty}\left(\prod_{k=1}^n\frac k{k+n}\right)^{e^{\frac{1999}n}-1}.$$

2022 MMATHS, 1

Tags: algebra

Suppose that $ a + b = 20$, $b + c = 22$, and $c + a = 2022$. Compute $\frac{a-b}{c-a}$ .

2014 Macedonia National Olympiad, 2

Tags: number theory

Solve the following equation in $\mathbb{Z}$: \[3^{2a + 1}b^2 + 1 = 2^c\]

2021 Austrian MO National Competition, 6

Tags: perfect square , number theory

Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers. Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer. (Walther Janous)

1978 IMO Longlists, 15

Tags: number theory

Prove that for every positive integer $n$ coprime to $10$ there exists a multiple of $n$ that does not contain the digit $1$ in its decimal representation.

2023 CUBRMC, 3

Tags: number theory

Find all positive integer pairs $(m, n)$ such that $m- n$ is a positive prime number and $mn$ is a perfect square. Justify your answer.

2022 Caucasus Mathematical Olympiad, 8

Tags: binomial coefficient , algebra

Paul can write polynomial $(x+1)^n$, expand and simplify it, and after that change every coefficient by its reciprocal. For example if $n=3$ Paul gets $(x+1)^3=x^3+3x^2+3x+1$ and then $x^3+\frac13x^2+\frac13x+1$. Prove that Paul can choose $n$ for which the sum of Paul’s polynomial coefficients is less than $2.022$.

1975 Spain Mathematical Olympiad, 4

Tags: algebra , inequalities , sum , product

Prove that if the product of $n$ real and positive numbers is equal to $1$, its sum is greater than or equal to $n$.

2023 LMT Fall, 14

Tags: geometry

In obtuse triangle $ABC$ with $AB = 7$, $BC = 20$, and $C A = 15$, let point $D$ be the foot of the altitude from $C$ to line $AB$. Evaluate $[ACD]+[BCD]$. (Note that $[XY Z]$ means the area of triangle $XY Z$.) [i]Proposed by Jonathan Liu[/i]