Olympiad problems

2017 Abels Math Contest (Norwegian MO) Final, 3b

In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up. Every other time, Niels picks a triangle and writes $\times$ in it, and every other time, Henrik picks a triangle where he writes a $o$. If one of the players gets four in a row in some direction (see figure), he wins the game. Determine whether one of the players can force a victory. [img]https://cdn.artofproblemsolving.com/attachments/6/e/5e80f60f110a81a74268fded7fd75a71e07d3a.png[/img]

2011 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: geometry , angle bisector

Let $AD$ and $BE$ be angle bisectors in triangle $ABC$. Let $x$, $y$ and $z$ be distances from point $M$, which lies on segment $DE$, from sides $BC$, $CA$ and $AB$, respectively. Prove that $z=x+y$

2008 ITest, 30

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Find the number of ordered triplets $(a,b,c)$ of positive integers such that $a<b<c$ and $abc=2008$.

1979 IMO Shortlist, 21

Tags: number theory , perfect cube , perfect square , counting , additive number theory

Let $N$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3 + 1\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$

2014 PUMaC Algebra B, 1

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Evaluate $\tfrac1{\sqrt1+\sqrt2}+\tfrac1{\sqrt2+\sqrt3}+\cdots+\tfrac1{\sqrt{1368}+\sqrt{1369}}$.

2023 Bulgaria JBMO TST, 4

Tags: subset , combinatorics , set

Given is a set of $n\ge5$ people and $m$ commissions with $3$ persons in each. Let all the commissions be [i]nice[/i] if there are no two commissions $A$ and $B$, such that $\mid A\cap B\mid=1$. Find the biggest possible $m$ (as a function of $n$).

2005 Grigore Moisil Urziceni, 2

Tags: limit , real analysis

[b]a)[/b] Prove that $ \lim_{x\to\infty } \sqrt{x}\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x}=1. $ [b]b)[/b] Show that $ \lim_{x\to\infty } \left( -\left\lfloor\sqrt{x}\right\rfloor +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) =\frac{-1}{2} $ [b]c)[/b] What about $ \lim_{x\to\infty } \left( -\sqrt{x} +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) ? $

2012 Austria Beginners' Competition, 3

Tags: algebra , inequalities

Let $a$ and $b$ be two positive real numbers with $a \le 2b \le 4a$. Prove that $4ab \le2 (a^2+ b^2) \le 5 ab$.

2021 HMNT, 4

Tags: number theory

Let $n$ be the answer to this problem. We define the digit sum of a date as the sum of its $4$ digits when expressed in mmdd format (e.g. the digit sum of $13$ May is $0+5+1+3 = 9$). Find the number of dates in the year $2021$ with digit sum equal to the positive integer $n$.

2016 ASMT, 8

Tags: geometry , rectangle , angle

In rectangle $ABCD$, point $E$ is chosen on $AB$ and $F$ is the foot of $E$ onto side $CD$ such that the circumcircle of $\vartriangle ABF$ intersects line segments $AD$ and $BC$ at points $G$ and $H$ respectively. Let $S$ be the intersection of $EF$ and $GH$, and $T$ the intersection of lines $EC$ and $DS$. If $\angle SF T = 15^o$ , compute the measure of $\angle CSD$.

1976 All Soviet Union Mathematical Olympiad, 233

Tags: combinatorics , polygon , impossible

Given right $n$-gon wit the point $O$ -- its centre. All the vertices are marked either with $+1$ or $-1$. We may change all the signs in the vertices of regular $k$-gon ($2 \le k \le n$) with the same centre $O$. (By $2$-gon we understand a segment, being halved by $O$.) Prove that in a), b) and c) cases there exists such a set of $(+1)$s and $(-1)$s, that we can never obtain a set of $(+1)$s only. a) $n = 15$, b) $n = 30$, c) $n > 2$, d) Let us denote $K(n)$ the maximal number of $(+1)$ and $(-1)$ sets such, that it is impossible to obtain one set from another. Prove, for example, that $K(200) = 2^{80}$

2020 Latvia Baltic Way TST, 5

Tags: combinatorics

Natural numbers $1,2,...,500$ are written on a blackboard. Two players $A$ and $B$ consecutively make moves, $A$ starts. Each move a player chooses two numbers $n$ and $2n$ and erases them from the blackboard. If a player cannot perform a valid move, he loses. Which player can guarantee a win?

2019 Caucasus Mathematical Olympiad, 6

Tags: geometry

In a triangle $ABC$ with $\angle BAC = 90^{\circ}$ let $BL$ be the bisector, $L\in AC$. Let $D$ be a point symmetrical to $A$ with respect to $BL$. Let $M$ be the circumcenter of $ADC$. Prove that $CM$, $DL$, and $AB$ are concurrent.

2011 Today's Calculation Of Integral, 753

Tags: calculus , integration , limit , calculus computations , logarithm

Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$

2019 CCA Math Bonanza, L4.2

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GM Bisain's IQ is so high that he can move around in $10$ dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates. [i]2019 CCA Math Bonanza Lightning Round #4.2[/i]

2002 National Olympiad First Round, 7

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What is the least number of weighings needed to determine the sum of weights of $13$ watermelons such that exactly two watermelons should be weighed in each weigh? $ \textbf{a)}\ 7 \qquad\textbf{b)}\ 8 \qquad\textbf{c)}\ 9 \qquad\textbf{d)}\ 10 \qquad\textbf{e)}\ 11 $

1998 Flanders Math Olympiad, 2

Tags: geometry , 3d geometry , analytic geometry , vector , trigonometry

Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure. Find the area of intersection of the cube with the plane through the points $a,m,e$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=279[/img]

2016 CCA Math Bonanza, T10

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Plusses and minuses are inserted in the expression \[\pm 1 \pm 2 \pm 3 \dots \pm 2016\] such that when evaluated the result is divisible by 2017. Let there be $N$ ways for this to occur. Compute the remainder when $N$ is divided by 503. [i]2016 CCA Math Bonanza Team #10[/i]

2012 ELMO Shortlist, 9

Tags: analytic geometry , geometry , geometric transformation , combinatorics , number theory

For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$? [i]David Yang.[/i]

2015 Sharygin Geometry Olympiad, 3

Tags: geometry , isosceles , angle

In triangle $ABC$ we have $AB = BC, \angle B = 20^o$. Point $M$ on $AC$ is such that $AM : MC = 1 : 2$, point $H$ is the projection of $C$ to $BM$. Find angle $AHB$. (M. Yevdokimov)

1965 IMO, 3

Tags: geometry , 3d geometry , tetrahedron , prism

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

2010 Malaysia National Olympiad, 6

Tags: diophantine equation , algebra

Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.

2012 Online Math Open Problems, 31

Tags: analytic geometry , geometry , geometric transformation , reflection , trigonometry , congruent triangles

Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$. [i]Author: Alex Zhu[/i]

2024 Princeton University Math Competition, A4 / B6

Tags: algebra

Compute the number of solutions to $1+\cos(\theta)+\cos(2\theta)+\ldots+\cos(2024\theta) = \tfrac{1}{2}$ for $\theta \in [0,2\pi].$

2013 Swedish Mathematical Competition, 3

Tags: diophantine equation , diophantine , number theory

Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$