Olympiad problems

1976 Polish MO Finals, 6

An increasing function $f : N \to R$ satisfies $$f(kl) = f(k)+ f(l)\,\,\, for \,\,\, all \,\,\, k,l \in N.$$ Show that there is a real number $p > 1$ such that $f(n) =\ log_pn$ for all $n$.

2013 Mexico National Olympiad, 4

Tags: geometry , 3d geometry , analytic geometry , prism , combinatorics

A $n \times n \times n$ cube is constructed using $1 \times 1 \times 1$ cubes, some of them black and others white, such that in each $n \times 1 \times 1$, $1 \times n \times 1$, and $1 \times 1 \times n$ subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0). Show it is possible to replace half of the black cubes with white cubes such that each $n \times 1 \times 1$, $1 \times n \times 1$ and $1 \times 1 \times n$ subprism contains exactly one black cube.

2005 USA Team Selection Test, 1

Tags: geometry , rectangle , combinatorics

Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that (a) each element of $T$ is an $m$-element subset of $S_{m}$; (b) each pair of elements of $T$ shares at most one common element; and (c) each element of $S_{m}$ is contained in exactly two elements of $T$. Determine the maximum possible value of $m$ in terms of $n$.

1993 Turkey Team Selection Test, 1

Tags: modular arithmetic , arithmetic sequence , number theory

Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.

2008 National Chemistry Olympiad, 5

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Which element is the major component in solar cells? $ \textbf{(A)}\hspace{.05in}\ce{As}\qquad\textbf{(B)}\hspace{.05in}\ce{Ge} \qquad\textbf{(C)}\hspace{.05in}\ce{P}\qquad\textbf{(D)}\hspace{.05in}\ce{Si} \qquad $

MathLinks Contest 7th, 7.3

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Let $ n$ be a positive integer, and let $ M \equal{} \{1,2,\ldots, 2n\}$. Find the minimal positive integer $ m$, such that no matter how we choose the subsets $ A_i \subset M$, $ 1\leq i\leq m$, with the properties: (1) $ |A_i\minus{}A_j|\geq 1$, for all $ i\neq j$, (2) $ \bigcup_{i\equal{}1}^m A_i \equal{} M$, we can always find two subsets $ A_k$ and $ A_l$ such that $ A_k \cup A_l \equal{} M$ (here $ |X|$ represents the number of elements in the set $ X$.)

LMT Team Rounds 2010-20, B8

Tags: geometry

In rectangle $ABCD$, $AB = 3$ and $BC = 4$. If the feet of the perpendiculars from $B$ and $D$ to $AC$ are $X$ and $Y$ , the length of $X Y$ can be expressed in the form m/n , where m and n are relatively prime positive integers. Find $m +n$.

2020 LMT Fall, B26

Tags: geometry

Aidan owns a plot of land that is in the shape of a triangle with side lengths $5$,$10$, and $5\sqrt3$ feet. Aidan wants to plant radishes such that there are no two radishes that are less than $1$ foot apart. Determine the maximum number of radishes Aidan can plant

PEN I Problems, 7

Tags: floor function , inequalities

Prove that for all positive integers $n$, \[\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}\rfloor =\lfloor \sqrt[3]{8n+3}\rfloor.\]

2020 Online Math Open Problems, 1

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A circle with radius $r$ has area $505$. Compute the area of a circle with diameter $2r$. [i]Proposed by Luke Robitaille & Yannick Yao[/i]

2007 India IMO Training Camp, 1

Tags: ratio , geometry , projective geometry , homothety

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

2019 Iranian Geometry Olympiad, 4

Tags: geometry

Quadrilateral $ABCD$ is given such that $$\angle DAC = \angle CAB = 60^\circ,$$ and $$AB = BD - AC.$$ Lines $AB$ and $CD$ intersect each other at point $E$. Prove that \[ \angle ADB = 2\angle BEC. \] [i]Proposed by Iman Maghsoudi[/i]

1999 French Mathematical Olympiad, Problem 4

Tags: combinatorics , game

On a table are given $1999$ red candies and $6661$ yellow candies. The candies are indistinguishable due to the same packing. A gourmet applies the following procedure as long as it is possible: (i) He picks any of the remaining candies, notes its color, eats it and goes to (ii). (ii) He picks any of the remaining candies, and notes its color: if it is the same as the color of the last eaten candy, eats it and goes to (ii); otherwise returns it upon repacking and goes to (i). Prove that all the candies will be eaten and find the probability that the last eaten candy will be red.

1996 Bundeswettbewerb Mathematik, 1

Tags: combinatorial geometry , combinatorics , square , geometry

Can a square of side length $5$ be covered by three squares of side length $4$?

2017 NIMO Problems, 3

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In rectangle $ABCD$ with center $O$, $AB=10$ and $BC=8$. Circle $\gamma$ has center $O$ and lies tangent to $\overline{AB}$ and $\overline{CD}$. Points $M$ and $N$ are chosen on $\overline{AD}$ and $\overline{BC}$, respectively; segment $MN$ intersects $\gamma$ at two distinct points $P$ and $Q$, with $P$ between $M$ and $Q$. If $MP : PQ : QN = 3 : 5 : 2$, then the length $MN$ can be expressed in the form $\sqrt{a} - \sqrt{b}$, where $a$, $b$ are positive integers. Find $100a + b$. [i]Proposed by Michael Tang[/i]

2023 Thailand TST, 1

Tags: algebra , sequence

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

2010 ELMO Shortlist, 1

Tags: function , number theory , inequalities

For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have \[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \] and determine when equality holds. [i]Wenyu Cao.[/i]

2014 All-Russian Olympiad, 1

Tags: prime number , number theory

Call a natural number $n$ [i]good[/i] if for any natural divisor $a$ of $n$, we have that $a+1$ is also divisor of $n+1$. Find all good natural numbers. [i]S. Berlov[/i]

1998 Yugoslav Team Selection Test, Problem 1

Tags: game , combinatorics

From a deck of playing cards, four [i]threes[/i], four [i]fours[/i] and four [i]fives[/i] are selected and put down on a table with the main side up. Players $A$ and $B$ alternately take the cards one by one and put them on the pile. Player $A$ begins. A player after whose move the sum of values of the cards on the pile is (a) greater than 34; (b) greater than 37; loses the game. Which player has a winning strategy?

1990 National High School Mathematics League, 8

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Point $A(2,0)$. $P(\sin(2t-\frac{\pi}{3}),\cos(2t-\frac{\pi}{3}))$ is a moving point. When $t$ changes from $\frac{\pi}{12}$ to $\frac{\pi}{4}$, area swept by segment $AP$ is________.

2014 Contests, 2

Tags: inequalities

Let $a,b\in\mathbb{R}_+$ such that $a+b=1$. Find the minimum value of the following expression: \[E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.\]

2016 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , perfect square

Prove that it is not possible that numbers $(n+1)\cdot 2^n$ and $(n+3)\cdot 2^{n+2}$ are perfect squares, where $n$ is positive integer.

2001 Switzerland Team Selection Test, 7

Tags: geometry , circumcenter , perpendicular

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circle $S$ through $A,B$, and $O$ intersects $AC$ and $BC$ again at points $P$ and $Q$ respectively. Prove that $CO \perp PQ$.

2007 Tournament Of Towns, 2

Tags: calculus , integration

Initially, the number $1$ and a non-integral number $x$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write $x^2$ on the blackboard in a finite number of moves?

2011 Today's Calculation Of Integral, 731

Tags: calculus , integration , parabola , geometry , calculus computations , conic

Let $C$ be the point of intersection of the tangent lines $l,\ m$ at $A(a,\ a^2),\ B(b,\ b^2)\ (a<b)$ on the parabola $y=x^2$ respectively. When $C$ moves on the parabola $y=\frac 12 x^2-x-2$, find the minimum area bounded by 2 lines $l,\ m$ and the parabola $y=x^2$.