Olympiad problems

2023 Sharygin Geometry Olympiad, 5

Tags: geometry , perpendicular

Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.

1961 Miklós Schweitzer, 2

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[b]2.[/b] Show that a ring $R$ has a unit element if and only if any $R$-module $G$ can be written as a direct sum of $RG$ and of the trivial submodule of $G$. (An $R$-module is a linear space with $R$ as its scalar domain. $RG$ denotes the submodule generated by the elements of the form $rg$($r \in R, g \in G$). The trivial submodule of $G$ consists of the elements $g$ of $G$ for which $rg=0$ holds for every $r \in R$.) [b](A. 20)[/b]

2004 Baltic Way, 10

Tags: number theory , modular arithmetic , induction , prime number

Is there an infinite sequence of prime numbers $p_1$, $p_2$, $\ldots$, $p_n$, $p_{n+1}$, $\ldots$ such that $|p_{n+1}-2p_n|=1$ for each $n \in \mathbb{N}$?

2019 Baltic Way, 4

Tags: algebra

Determine all integers $n$ for which there exist an integer $k\geq 2$ and positive integers $x_1,x_2,\hdots,x_k$ so that $$x_1x_2+x_2x_3+\hdots+x_{k-1}x_k=n\text{ and } x_1+x_2+\hdots+x_k=2019.$$

1996 Vietnam National Olympiad, 1

Tags: system of equations , algebra

Solve the system of equations: $ \sqrt {3x}(1 \plus{} \frac {1}{x \plus{} y}) \equal{} 2$ $ \sqrt {7y}(1 \minus{} \frac {1}{x \plus{} y}) \equal{} 4\sqrt {2}$

1979 Poland - Second Round, 1

Tags: geometry , algebra

Given are the points $A$ and $B$ on the edge of a circular pool. The athlete has to get from point $A$ to point $B$ by walking along the edge of the pool or swimming in the pool; he can change the way he moves many times. How should an athlete move to get from point A to B in the shortest time, given that he moves twice as slowly in water as on land?

2007 Macedonia National Olympiad, 5

Tags: combinatorics

Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$

2024 CMIMC Team, 6

Tags: team

Cyclic quadrilateral $ABCD$ has circumradius $3$. Additionally, $AC = 3\sqrt{2}$, $AB/CD = 2/3$, and $AD = BD$. Find $CD$. [i]Proposed by Justin Hsieh[/i]

2006 Baltic Way, 6

Tags: combinatorics

Determine the maximal size of a set of positive integers with the following properties: $1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$. $2.$ No digit occurs more than once in the same integer. $3.$ The digits in each integer are in increasing order. $4.$ Any two integers have at least one digit in common (possibly at different positions). $5.$ There is no digit which appears in all the integers.

2023 IFYM, Sozopol, 3

Tags: combinatorics

Exactly $2^{1012}$ of the subsets of $\{1, 2, \ldots, 2023\}$ are colored red. Is it always true that there exist three distinct red sets $A$, $B$, and $C$ such that every element of $A$ belongs to at least one of $B$ or $C$?

2013 Spain Mathematical Olympiad, 5

Tags: integer sequence , number theory

Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions $i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct). $ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$

1958 November Putnam, B5

Tags: broken line

The lengths of successive segments of a broken line are represented by the successive terms of the harmonic progression $1, 1\slash 2, 1\slash 3, \ldots.$ Each segment makes with the preceding a given angle $\theta.$ What is the distance and what is the direction of the limiting points (if there is one) from the initial point of the first segment?

2025 AIME, 11

Tags: geometry , regular polygon

Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments.

2004 Thailand Mathematical Olympiad, 13

Tags: remainder , number theory

Compute the remainder when $29^{30 }+ 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.

2012 Brazil Team Selection Test, 3

Tags: equation , sequence , algebra

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

2019 LIMIT Category A, Problem 10

Tags: algebra , equation

Number of solutions of the equation $3^x+4^x=8^x$ in reals is $\textbf{(A)}~0$ $\textbf{(B)}~1$ $\textbf{(C)}~2$ $\textbf{(D)}~\infty$

2020 SIME, 5

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Let $ABCD$ be a rectangle with side lengths $\overline{AB} = \overline{CD} = 6$ and $\overline{BC} = \overline{AD} = 3$. A circle $\omega$ with center $O$ and radius $1$ is drawn inside rectangle $ABCD$ such that $\omega$ is tangent to $\overline{AB}$ and $\overline{AD}$. Suppose $X$ and $Y$ are points on $\omega$ that are not on the perimeter of $ABCD$ such that $BX$ and $DY$ are tangent to $\omega$. If the value of $XY^2$ can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.

2011 Germany Team Selection Test, 2

Tags: geometry , circumcircle , reflection , parallelogram

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

1997 Italy TST, 1

Tags: algebra

Let $x,y,z,t$ be real numbers with $x,y,z$ not all equal such that \[x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.\] Find all possible values of $ t$ such that $xyz+t=0$.

1996 Tournament Of Towns, (499) 1

Tags: 3d geometry , cube , geometry

Does there exist a cube in space such that the perpendiculars dropped from its eight vertices to a given plane are of length $0, 1, 2, 3, 4, 5, 6$ and $7$? (V Proizvolov)

2003 AMC 10, 5

Tags: vieta , quadratic , algebra , polynomial

Let $ d$ and $ e$ denote the solutions of $ 2x^2\plus{}3x\minus{}5\equal{}0$. What is the value of $ (d\minus{}1)(e\minus{}1)$? $ \textbf{(A)}\ \minus{}\frac{5}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

MOAA Team Rounds, 2023.10

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Let $S$ be the set of lattice points $(a,b)$ in the coordinate plane such that $1\le a\le 30$ and $1\le b\le 30$. What is the maximum number of lattice points in $S$ such that no four points form a square of side length 2? [i]Proposed by Harry Kim[/i]

2020 MBMT, 4

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Ken has a six sided die. He rolls the die, and if the result is not even, he rolls the die one more time. Find the probability that he ends up with an even number. [i]Proposed by Gabriel Wu[/i]

2016 NIMO Problems, 1

Tags: arithmetic sequence

Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor +\lfloor 2.4 \rfloor +\lfloor 3.1 \rfloor +\cdots+\lfloor 99 \rfloor$. [i]Proposed by Jack Cornish[/i]

2018 CMIMC Team, 3-1/3-2

Tags: team

Let $\Omega$ be a semicircle with endpoints $A$ and $B$ and diameter 3. Points $X$ and $Y$ are located on the boundary of $\Omega$ such that the distance from $X$ to $AB$ is $\frac{5}{4}$ and the distance from $Y$ to $AB$ is $\frac{1}{4}$. Compute \[(AX+BX)^2 - (AY+BY)^2.\] Let $T = TNYWR$. $T$ people each put a distinct marble into a bag; its contents are mixed randomly and one marble is distributed back to each person. Given that at least one person got their own marble back, what is the probability that everyone else also received their own marble?