Olympiad problems

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

Tags:

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

Tags:

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)

2013 Purple Comet Problems, 14

Tags:

How many ordered triples $(a, b, c)$ of positive integers satisfy $a \le b \le c$ and $a \cdot b\cdot c = 1000$?

2018 Bosnia and Herzegovina EGMO TST, 4

Tags: combinatorics , maximization , extremal combinatorics

It is given positive integer $n$. Let $a_1, a_2,..., a_n$ be positive integers with sum $2S$, $S \in \mathbb{N}$. Positive integer $k$ is called separator if you can pick $k$ different indices $i_1, i_2,...,i_k$ from set $\{1,2,...,n\}$ such that $a_{i_1}+a_{i_2}+...+a_{i_k}=S$. Find, in terms of $n$, maximum number of separators