Olympiad problems

2004 Estonia Team Selection Test, 6

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

2014 Math Hour Olympiad, 8-10.1

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Sherlock and Mycroft are playing Battleship on a $4\times4$ grid. Mycroft hides a single $3\times1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser?

2023 ELMO Shortlist, C7

Tags: combinatorics

A [i]discrete hexagon with center $(a,b,c)$ \emph{(where $a$, $b$, $c$ are integers)[/i] and radius $r$ [i](a nonnegative integer)[/i]} is the set of lattice points $(x,y,z)$ such that $x+y+z=a+b+c$ and $\max(|x-a|,|y-b|,|z-c|)\le r$. Let $n$ be a nonnegative integer and $S$ be the set of triples $(x,y,z)$ of nonnegative integers such that $x+y+z=n$. If $S$ is partitioned into discrete hexagons, show that at least $n+1$ hexagons are needed. [i]Proposed by Linus Tang[/i]

2023 LMT Fall, 9

Tags: algebra

Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$. [i]Proposed byMuztaba Syed[/i]

1962 All Russian Mathematical Olympiad, 022

Tags: geometry , isosceles , perpendicular

The $M$ point is the midpoint of the base $[AC]$ of an isosceles triangle $ABC$. $[MH]$ is orthogonal to $[BC]$ side. Point $P$ is the midpoint of the segment $[MH]$. Prove that $[AH]$ is orthogonal to $[BP]$.

1995 Romania Team Selection Test, 2

Tags: number theory

Find all positive integers $ x,y,z,t$ such that $ x,y,z$ are pairwise coprime and $ (x \plus{} y)(y \plus{} z)(z \plus{} x) \equal{} xyzt$.

2007 USA Team Selection Test, 6

Tags: algebra , polynomial , algorithm , modular arithmetic , counting , distinguishability , induction

For a polynomial $ P(x)$ with integer coefficients, $ r(2i \minus{} 1)$ (for $ i \equal{} 1,2,3,\ldots,512$) is the remainder obtained when $ P(2i \minus{} 1)$ is divided by $ 1024$. The sequence \[ (r(1),r(3),\ldots,r(1023)) \] is called the [i]remainder sequence[/i] of $ P(x)$. A remainder sequence is called [i]complete[/i] if it is a permutation of $ (1,3,5,\ldots,1023)$. Prove that there are no more than $ 2^{35}$ different complete remainder sequences.

2006 Thailand Mathematical Olympiad, 3

Tags: algebra , polynomial , divide

Let $P(x), Q(x)$ and $R(x)$ be polynomials satisfying the equation $2xP(x^3) + Q(-x -x^3) = (1 + x + x^2)R(x)$. Show that $x - 1$ divides $P(x) - Q(x)$.

2007 Moldova Team Selection Test, 4

Tags: prime number , number theory

Show that there are infinitely many prime numbers $p$ having the following property: there exists a natural number $n$, not dividing $p-1$, such that $p|n!+1$.

2007 Danube Mathematical Competition, 3

Tags: induction , number theory

For each positive integer $ n$, define $ f(n)$ as the exponent of the $ 2$ in the decomposition in prime factors of the number $ n!$. Prove that the equation $ n\minus{}f(n)\equal{}a$ has infinitely many solutions for any positive integer $ a$.

2003 AIME Problems, 15

Tags: algebra , polynomial , function

Let \[P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). \] Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2}=a_{k}+b_{k}i$ for $k=1,2,\ldots,r,$ where $i=\sqrt{-1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let \[\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p}, \] where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

2001 India IMO Training Camp, 2

Tags: modular arithmetic , number theory

Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]

2018 BMT Spring, 2

Tags: number theory

Suppose for some positive integers, that $\frac{p+\frac{1}{q}}{q+\frac{1}{p}}= 17$. What is the greatest integer $n$ such that $\frac{p+q}{n}$ is always an integer?

2010 Today's Calculation Of Integral, 641

Tags: calculus , integration , calculus computations , logarithm

Evaluate \[\int_{e^e}^{e^{e^{e}}}\left\{\ln (\ln (\ln x))+\frac{1}{(\ln x)\ln (\ln x)}\right\}dx.\] Own

1993 Balkan MO, 2

Tags: calculus , integration , ceiling function , floor function , inequalities , combinatorics

A positive integer given in decimal representation $\overline{ a_na_{n-1} \ldots a_1a_0 }$ is called [i]monotone[/i] if $a_n\leq a_{n-1} \leq \cdots \leq a_0$. Determine the number of monotone positive integers with at most 1993 digits.

LMT Team Rounds 2010-20, 2020.S16

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For non-negative integer $n$, the function $f$ is given by \[f(x)=\begin{cases} \frac{x}{2} & \text{if $n$ is even} \\ x-1 & \text{if $n$ is odd.} \end{cases} \] Furthermore, let $h(n)$ be the smallest $k$ for which $f^k(n)=0$. Compute \[\sum_{n=1}^{1024} h(n).\]

2017 QEDMO 15th, 8

Tags: median , geometry , triangle inequality

Let $ABC$ be a triangle of area $1$ with medians $s_a, s_b,s_c$. Show that there is a triangle whose sides are the same length as $s_a, s_b$, and $s_c$, and determine its area.

2014 Junior Regional Olympiad - FBH, 4

Tags: number theory , prime number

Find all prime numbers $p$ and $q$ such that $$(2p-q)^2=17p-10q$$

2024 Caucasus Mathematical Olympiad, 3

Tags: number theory

Let $n$ be a $d$-digit (i.e., having $d$ digits in its decimal representation) positive integer not divisible by $10$. Writing all the digits of $n$ in reverse order, we obtain the number $n'$. Determine if it is possible that the decimal representation of the product $n\cdot n'$ consists of digits $8$ only, if (a) $d = 9998$; (b) $d = 9999?$

2021 AMC 12/AHSME Fall, 16

Tags: number theory , greatest common divisor

Let $a, b,$ and $c$ be positive integers such that $a+b+c=23$ and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^{2}+b^{2}+c^{2}$? $\textbf{(A)} ~259\qquad\textbf{(B)} ~438\qquad\textbf{(C)} ~516\qquad\textbf{(D)} ~625\qquad\textbf{(E)} ~687$ Proposed by [b]djmathman[/b]

1961 AMC 12/AHSME, 11

Tags: geometry , perimeter

Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of triangle $APR$ is ${{ \textbf{(A)}\ 42\qquad\textbf{(B)}\ 40.5 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 39\frac{7}{8} }\qquad\textbf{(E)}\ \text{not determined by the given information} } $

2014 ASDAN Math Tournament, 18

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A two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers.

2009 National Olympiad First Round, 32

Tags: algorithm , abstract algebra , group theory

There are $ n$ sets having $ 4$ elements each. The difference set of any two of the sets is equal to one of the $ n$ sets. $ n$ can be at most ? (A difference set of $A$ and $B$ is $ (A\setminus B)\cup(B\setminus A) $) $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ \text{None}$

MIPT student olimpiad autumn 2024, 3

Tags: function , real analysis

$\exists ? f: R\to R$ continuos function that: $\forall x_0\in R \lim\limits_{x \to x_0} \frac{|f(x)-f(x_0)|}{|x-x_0|}=+\infty$

2020 Austrian Junior Regional Competition, 2

Tags: combinatorics , digit

How many positive five-digit integers are there that have the product of their five digits equal to $900$? (Karl Czakler)