Olympiad problems

2012 AMC 12/AHSME, 16

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Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible? $ \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 $

2020 LMT Fall, 7

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$2020*N$ is a perfect cube. If $N$ can be expressed as $2^a*5^b*101^c,$ find the least possible value of $a+b+c$ such that $a,b,c$ are all positive integers and not necessarily distinct. [i]Proposed by Ephram Chun[/i]

2023 Taiwan TST Round 1, A

Tags: algebra , polynomial

Given some monic polynomials $P_1, \ldots, P_n$ with real coefficients, for any real number $y$, let $S_y$ be the set of real number $x$ such that $y = P_i(x)$ for some $i = 1, 2, ..., n$. If the sets $S_{y_1}, S_{y_2}$ have the same size for any two real numbers $y_1, y_2$, show that $P_1, \ldots, P_n$ have the same degree. [i] Proposed by usjl[/i]

2018 PUMaC Combinatorics B, 6

Tags: combinatorics , probability , expected value

If $a$ and $b$ are selected uniformly from $\{0,1,\ldots,511\}$ without replacement, the expected number of $1$'s in the binary representation of $a+b$ can be written in simplest from as $\tfrac{m}{n}$. Compute $m+n$.

2003 Tournament Of Towns, 1

Tags: quadratic , algebra

Johnny writes down quadratic equation \[ax^2 + bx + c = 0.\] with positive integer coefficients $a, b, c$. Then Pete changes one, two, or none “$+$” signs to “$-$”. Johnny wins, if both roots of the (changed) equation are integers. Otherwise (if there are no real roots or at least one of them is not an integer), Pete wins. Can Johnny choose the coefficients in such a way that he will always win?

1993 Denmark MO - Mohr Contest, 1

Tags: algebra

Three friends A, B and C have a total of $120$ kroner. First, A gives as much money to B as B already has. Next, B gives as many money to C that C already has. In the end, C gives the same amount of money to A as A now has. After these transactions, A, B and C have equal amounts of money. How many money did each of the three companions have originally?

2018 PUMaC Combinatorics B, 8

Tags: combinatorics

Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1+3^{10}+3^{11}+3^{100}+3^{111}+3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of $$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$ then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?

2005 Tournament of Towns, 3

Tags: geometry

$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$. [i](4 points)[/i]

2011 Today's Calculation Of Integral, 737

Tags: calculus , integration , geometry , inequalities , calculus computations , logarithm

Let $a,\ b$ real numbers such that $a>1,\ b>1.$ Prove the following inequality. \[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]

2007 Switzerland - Final Round, 3

Tags: combinatorics , tiling , tiles

The plane is divided into unit squares. Each box should be be colored in one of $n$ colors , so that if four squares can be covered with an $L$-tetromino, then these squares have four different colors (the $L$-Tetromino may be rotated and be mirrored). Find the smallest value of $n$ for which this is possible.

2023 Purple Comet Problems, 3

Tags: geometry

Mike has two similar pentagons. The first pentagon has a perimeter of $18$ and an area of $8 \frac{7}{16}$ . The second pentagon has a perimeter of $24$. Find the area of the second pentagon.

2022 Harvard-MIT Mathematics Tournament, 3

Tags: algebra

Let $x_1, x_2, . . . , x_{2022}$ be nonzero real numbers. Suppose that $x_k + \frac{1}{x_{k+1}} < 0$ for each $1 \leq k \leq 2022$, where $x_{2023}=x_1$. Compute the maximum possible number of integers $1 \leq n \leq 2022$ such that $x_n > 0$.

2022 Iran MO (3rd Round), 2

Tags: number theory , divisibility , sequence

For two rational numbers $r,s$ we say:$$r\mid s$$whenever there exists $k\in\mathbb{Z}$ such that:$$s=kr$$ ${(a_n)}_{n\in\mathbb{N}}$ is an increasing sequence of pairwise coprime natural numbers and ${(b_n)}_{n\in\mathbb{N}}$ is a sequence of distinct natural numbers. Assume that for all $n\in\mathbb{N}$ we have: $$\sum_{i=1}^{n}\frac{1}{a_i}\mid\sum_{i=1}^{n}\frac{1}{b_i}$$ Prove that [b]for all[/b] $n\in\mathbb{N}$ we have: $a_n=b_n$.

2014 BMT Spring, 1

Tags: geometry

Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?

2023 Czech-Polish-Slovak Junior Match, 5

Tags: algebra

Bartek patiently performs operations on fractions. In each move, he adds its inverse to the current result, obtaining a new result. Bartek starts with the number $1$: after the first move, he receives the result 2, after the second move, the result is $\frac{5}{2}$, after the third move $\frac{29}{10}$, etc. After $300$ moves, Bartek receives the result $x$. Determine the largest integer not greater than $x$.

2004 Irish Math Olympiad, 3

Tags: vector , geometry

$AB$ is a chord of length $6$ of a circle centred at $O$ and of radius $5$. Let $PQRS$ denote the square inscribed in the sector $OAB$ such that $P$ is on the radius $OA$, $S$ is on the radius $OB$ and $Q$ and $R$ are points on the arc of the circle between $A$ and $B$. Find the area of $PQRS$.

2009 Brazil National Olympiad, 3

Tags: chip-firing , analytic geometry , search , floor function , combinatorics

There are $ 2009$ pebbles in some points $ (x,y)$ with both coordinates integer. A operation consists in choosing a point $ (a,b)$ with four or more pebbles, removing four pebbles from $ (a,b)$ and putting one pebble in each of the points \[ (a,b\minus{}1),\ (a,b\plus{}1),\ (a\minus{}1,b),\ (a\plus{}1,b)\] Show that after a finite number of operations each point will necessarily have at most three pebbles. Prove that the final configuration doesn't depend on the order of the operations.

2013 ELMO Shortlist, 10

Tags: circumcircle , ratio , symmetry , projective geometry , trigonometry , geometry

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

2012 National Olympiad First Round, 25

Tags: geometry , trigonometry , circumcircle

The midpoint $M$ of $[AC]$ of a triangle $\triangle ABC$ is between $C$ and the feet $H$ of the altitude from $B$. If $m(\widehat{ABH}) = m(\widehat{MBC})$, $m(\widehat{ACB}) = 15^{\circ}$, and $|HM|=2\sqrt{3}$, then $|AC|=?$ $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 5 \sqrt 2 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ \frac{16}{\sqrt3} \qquad \textbf{(E)}\ 10$

1993 Moldova Team Selection Test, 3

Tags: function , parameterization

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function defined as the maximum of a finite number of functions $g:\mathbb{R}\rightarrow\mathbb{R}$ of the form $g(x)=C\cdot10^{-|x-d|}$ (with different values of parameters $d{}$ and $C>0$). For real numbers $a<b$ we have $f(a)=f(b)$. Prove that on the segment $[a;b]$ the sum of legnths of segments on which $f$ is increasing is equal to the sum of legnths of segments on which $f$ is decreasing.

2015 Saint Petersburg Mathematical Olympiad, 6

Tags: combinatorics , graph theory , coloring

There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for which this is always possible. (D. Karpov)

1973 IMO Shortlist, 10

Tags: sequence , algebra , inequality , construction , geometric sequence

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

2007 Bosnia Herzegovina Team Selection Test, 1

Tags: angle bisector , geometry

Let $ABC$ be a triangle such that length of internal angle bisector from $B$ is equal to $s$. Also, length of external angle bisector from $B$ is equal to $s_1$. Find area of triangle $ABC$ if $\frac{AB}{BC} = k$

2014 China Northern MO, 3

Tags: diophantine equation , number theory

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

2023 Bulgaria EGMO TST, 6

Tags: ceva theorem , geometry , incircle , mixtilinear , concurrency , radical axis , brianchon

Let $ABC$ be a triangle with incircle $\gamma$. The circle through $A$ and $B$ tangent to $\gamma$ touches it at $C_2$ and the common tangent at $C_2$ intersects $AB$ at $C_1$. Define the points $A_1$, $B_1$, $A_2$, $B_2$ analogously. Prove that: a) the points $A_1$, $B_1$, $C_1$ are collinear; b) the lines $AA_2$, $BB_2$, $CC_2$ are concurrent.