Olympiad problems

2015 Moldova Team Selection Test, 1

Let $c\in \Big(0,\dfrac{\pi}{2}\Big) , a = \Big(\dfrac{1}{sin(c)}\Big)^{\dfrac{1}{cos^2 (c)}}, b = \Big(\dfrac{1}{cos(c)}\Big)^{\dfrac{1}{sin^2 (c)}}$. \\Prove that at least one of $a,b$ is bigger than $\sqrt[11]{2015}$.

2008 AMC 10, 12

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Postman Pete has a pedometer to count his steps. The pedometer records up to $ 99999$ steps, then flips over to $ 00000$ on the next step. Pete plans to determine his mileage for a year. On January $ 1$ Pete sets the pedometer to $ 00000$. During the year, the pedometer flips from $ 99999$ to $ 00000$ forty-four times. On December $ 31$ the pedometer reads $ 50000$. Pete takes $ 1800$ steps per mile. Which of the following is closest to the number of miles Pete walked during the year? $ \textbf{(A)}\ 2500 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3500 \qquad \textbf{(D)}\ 4000 \qquad \textbf{(E)}\ 4500$

2012 Abels Math Contest (Norwegian MO) Final, 1a

Tags: number theory , algebra

Berit has $11$ twenty kroner coins, $14$ ten kroner coins, and $12$ five kroner coins. An exchange machine can exchange three ten kroner coins into one twenty kroner coin and two five kroner coins, and the reverse. It can also exchange two twenty kroner coins into three ten kroner coins and two five kroner coins, and the reverse. (i) Can Berit get the same number of twenty kroner and ten kroner coins, but no five kroner coins? (ii) Can she get the same number each of twenty kroner, ten kroner, and five kroner coins?

1992 Hungary-Israel Binational, 3

Tags: number theory , logarithm

We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. We call a nonnegative integer $r$-Fibonacci number if it is a sum of $r$ (not necessarily distinct) Fibonacci numbers. Show that there inﬁnitely many positive integers that are not $r$-Fibonacci numbers for any $r, 1 \leq r\leq 5.$

2016 PUMaC Team, 6

Tags: number theory

Compute the sum of all positive integers less than $100$ that do not have consecutive $1$s in their binary representation.

2012 Greece Team Selection Test, 1

Tags: number theory

Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.

2019 PUMaC Individual Finals A, B, A2

Tags: number theory

Prove that for every positive integer $m$, every prime $p$ and every positive integer $j \le p^{m-1}$, $p^m$ divides $${p^m \choose p^j }- {p^{m-1} \choose j}$$

2010 Kosovo National Mathematical Olympiad, 1

Tags: function , algebra

If the real function $f(x)=\cos x+\sum_{i=1}^{n}\cos(a_ix)$ is periodic, prove that $a_i,i\in\{1,2,...,n\}$, are rational numbers.

2013 Princeton University Math Competition, 5

Tags: geometry , 3d geometry , sphere , conic

Suppose you have a sphere tangent to the $xy$-plane with its center having positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$.

2009 Baltic Way, 13

Tags: geometry

The point $H$ is the orthocentre of a triangle $ABC$, and the segments $AD,BE,CF$ are its altitudes. The points $I_1,I_2,I_3$ are the incentres of the triangles $EHF,FHD,DHE$ respectively. Prove that the lines $AI_1,BI_2,CI_3$ intersect at a single point.

2012 Hanoi Open Mathematics Competitions, 13

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A cube with sides of length 3cm is painted red and then cut into 3 x 3 x 3 = 27 cubes with sides of length 1cm. If a denotes the number of small cubes (of 1cm x 1cm x 1cm) that are not painted at all, b the number painted on one sides, c the number painted on two sides, and d the number painted on three sides, determine the value a-b-c+d.

2014 Indonesia MO Shortlist, G2

Tags: circumcircle , geometry

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

1965 Spain Mathematical Olympiad, 3

Tags: length , geometry

A disk in a record turntable makes $100$ revolutions per minute and it plays during $24$ minutes and $30$ seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from $29$cm to $11.5$cm. Compute the length of the recorded line.

1980 IMO, 8

Tags: calculus , integration , number theory

Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).

2021 Purple Comet Problems, 7

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Among the $100$ constants $a_1,a_2,a_3,...,a_{100}$, there are $39$ equal to $-1$ and $61$ equal to $+1$. Find the sum of all the products $a_ia_j$, where $1\le i < j \le 100$.

2020 New Zealand MO, 6

Tags: geometry , bisects segment

Let $\vartriangle ABC$ be an acute triangle with $AB > AC$. Let $P$ be the foot of the altitude from $C$ to $AB$ and let $Q$ be the foot of the altitude from $B$ to $AC$. Let $X$ be the intersection of $PQ$ and $BC$. Let the intersection of the circumcircles of triangle $\vartriangle AXC$ and triangle $\vartriangle PQC$ be distinct points: $C$ and $Y$ . Prove that $PY$ bisects $AX$.

2018 EGMO, 2

Tags: number theory , multiplication

Consider the set \[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\] [list=a] [*]Prove that every integer $x \geq 2$ can be written as the product of one or more elements of $A$, which are not necessarily different. [*]For every integer $x \geq 2$ let $f(x)$ denote the minimum integer such that $x$ can be written as the product of $f(x)$ elements of $A$, which are not necessarily different. Prove that there exist infinitely many pairs $(x,y)$ of integers with $x\geq 2$, $y \geq 2$, and \[f(xy)<f(x)+f(y).\] (Pairs $(x_1,y_1)$ and $(x_2,y_2)$ are different if $x_1 \neq x_2$ or $y_1 \neq y_2$). [/list]

1952 AMC 12/AHSME, 11

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If $ y \equal{} f(x) \equal{} \frac {x \plus{} 2}{x \minus{} 1}$, then it is incorrect to say: $ \textbf{(A)}\ x \equal{} \frac {y \plus{} 2}{y \minus{} 1} \qquad\textbf{(B)}\ f(0) \equal{} \minus{} 2 \qquad\textbf{(C)}\ f(1) \equal{} 0 \qquad\textbf{(D)}\ f( \minus{} 2) \equal{} 0$ $ \textbf{(E)}\ f(y) \equal{} x$

2024 Canadian Junior Mathematical Olympiad, 4

Tags: number theory , combinatorics

Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?

1989 National High School Mathematics League, 11

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From $1,2,\cdots,14$, take out three numbers $a_1<a_2<a_3$, satisfying that $a_2-a_1\geq3,a_3-a_2\geq3$. Then the number of different ways of taking out numbers is________.

2005 IberoAmerican, 5

Tags: circumcircle , trigonometry , geometry

Let $O$ be the circumcenter of acutangle triangle $ABC$ and let $A_1$ be some point in the smallest arc $BC$ of the circumcircle of $ABC$. Let $A_2$ and $A_3$ points on sides $AB$ and $AC$, respectively, such that $\angle BA_1A_2 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$. Prove that the line $A_2A_3$ passes through the orthocenter of $ABC$.

2011 Romania National Olympiad, 1

Tags: function , algebra

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ a function having the property that $$ \left| f(x+y)+\sin x+\sin y \right|\le 2, $$ for all real numbers $ x,y. $ [b]a)[/b] Prove that $ \left| f(x) \right|\le 1+\cos x, $ for all real numbers $ x. $ [b]b)[/b] Give an example of what $ f $ may be, if the interval $ \left( -\pi ,\pi \right) $ is included in its [url=https://en.wikipedia.org/wiki/Support_(mathematics)]support.[/url]

2019 Belarus Team Selection Test, 1.2

Tags: inequalities , geometry

Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that $$ AB+CD>\max(AM+DM,BN+CN)? $$ [i](Folklore)[/i]

2018 USAMTS Problems, 4:

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Find, with proof, all ordered pairs of positive integers $(a, b)$ with the following property: there exist positive integers $r$, $s$, and $t$ such that for all $n$ for which both sides are defined, [center]${{n\choose{a}}\choose{b}}=r {{n+s}\choose{t}}$ .[/center]

2012 Switzerland - Final Round, 2

Tags: algebra , functional , functional equation

Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$