Olympiad problems

2022 Moscow Mathematical Olympiad, 6

Tags: combinatorics

The Sultan gathered $300$ court sages and offered them a test. There are caps of $25$ different colors, known in advance to the sages. The Sultan said that one of these caps will be put on each of the sages, and if for each color write the number of caps worn, then all numbers will be different. Every sage can see the caps of the other sages, but not own cap. Then all the sages will simultaneously announce the supposed color of their cap. Can sages advance agree to act in such a way that at least $150$ of them are guaranteed to name a color right?

2010 ELMO Shortlist, 1

Tags: inequalities , function , number theory proposed , number theory

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]

2016 Postal Coaching, 5

Tags: geometry , incircle , 3D geometry , prism

Two triangles $ABC$ and $DEF$ have the same incircle. If a circle passes through $A,B,C,D,E$ prove that it also passes through $F$.

1987 Traian Lălescu, 2.1

Tags: abstract algebra , Ring Theory , extension rings , superrings , algebra , polynomial

Any polynom, with coefficients in a given division ring, that is irreducible over it, is also irreducible over a given extension skew ring of it that's finite. Prove that the ring and its extension coincide.

2022 Kyiv City MO Round 1, Problem 2

Tags: algebra , inequalities

For any reals $x, y$, show the following inequality: $$\sqrt{(x+4)^2 + (y+2)^2} + \sqrt{(x-5)^2 + (y+4)^2} \le \sqrt{(x-2)^2 + (y-6)^2} + \sqrt{(x-5)^2 + (y-6)^2} + 20$$ [i](Proposed by Bogdan Rublov)[/i]

2004 Oral Moscow Geometry Olympiad, 3

Tags: equal angles , geometry , Locus

Given a square $ABCD$. Find the locus of points $M$ such that $\angle AMB = \angle CMD$.

1962 AMC 12/AHSME, 8

Tags:

Given the set of $ n$ numbers; $ n > 1$, of which one is $ 1 \minus{} \frac {1}{n}$ and all the others are $ 1.$ The arithmetic mean of the $ n$ numbers is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ n \minus{} \frac {1}{n} \qquad \textbf{(C)}\ n \minus{} \frac {1}{n^2} \qquad \textbf{(D)}\ 1 \minus{} \frac {1}{n^2} \qquad \textbf{(E)}\ 1 \minus{} \frac {1}{n} \minus{} \frac {1}{n^2}$

2017 Korea Winter Program Practice Test, 1

Tags: number theory

Find all prime number $p$ such that the number of positive integer pair $(x,y)$ satisfy the following is not $29$. [list] [*]$1\le x,y\le 29$ [*]$29\mid y^2-x^p-26$ [/list]

2011 Serbia JBMO TST, 4

Tags: inequalities

If a, b, c are positive real numbers with $ a+b+c=1 $. Find the minimum value of $ \sqrt{a}+\sqrt{b}+\sqrt{c}+\frac{1}{\sqrt{abc}} $

2018 PUMaC Team Round, 7

Tags: PuMAC , Team Round

Let triangle $\triangle{MNP}$ have side lengths $MN=13$, $NP=89$, and $PM=100$. Define points $S$, $R$, and $B$ as the midpoints of $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ respectively. A line $\ell$ cuts lines $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ at points $I$, $J$, and $A$ respectively. Find the minimum value of $(SI+RJ+BA)^2.$

2002 Moldova Team Selection Test, 2

Tags: combinatorics

Let $A$ be a set containing $4k$ consecutive positive integers, where $k \geq 1$ is an integer. Find the smallest $k$ for which the set A can be partitioned into two subsets having the same number of elements, the same sum of elements, the same sum of the squares of elements, and the same sum of the cubes of elements.

2012 HMNT, 1

Tags: number theory

Find the number of integers between $1$ and $200$ inclusive whose distinct prime divisors sum to $16$. (For example, the sum of the distinct prime divisors of $12$ is $2 + 3 = 5$.) In this section, the word [i]divisor [/i]is used to refer to a [i]positive divisor[/i] of an integer.

2001 IberoAmerican, 1

Tags: modular arithmetic , number theory unsolved , number theory

We say that a natural number $n$ is [i]charrua[/i] if it satisfy simultaneously the following conditions: - Every digit of $n$ is greater than 1. - Every time that four digits of $n$ are multiplied, it is obtained a divisor of $n$ Show that every natural number $k$ there exists a [i]charrua[/i] number with more than $k$ digits.

2016 ASDAN Math Tournament, 9

Tags: 2016 , Guts Round

An equilateral triangle $\triangle ABC$ with side length $3$ has center $O$. A circle is drawn centered at $O$ with radius $1$. Find the area of the region contained inside both the triangle and circle.

2018 Romanian Master of Mathematics, 2

Tags: algebra , polynomial , RMM , RMM 2018

Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying $$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$

2010 Contests, 3

Tags: analytic geometry , algebra unsolved , algebra

One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false: [b]a)[/b] Every point of the plane is in a line that can be defined by $2$ rational points. [b]b)[/b] Every point of the plane is in a line that can be defined by $2$ irrational points. This maybe is not algebra so sorry if I putted it in the wrong category!

2011 Mathcenter Contest + Longlist, 7

Tags: algebra , system of equations , System

Given $k_1,k_2,...,k_n\in R^+$, find all the naturals $n$ such that $$k_1+k_2+...+k_n=2n-3$$ $$\frac{1}{k_1}+\frac{1}{k_2}+...+\frac{1}{k_n}=3$$ [i](Zhuge Liang)[/i]

2020 USMCA, 7

Tags:

Compute the value of \[\cos \frac{2\pi}{7} + 2\cos \frac{4\pi}{7} + 3\cos \frac{6\pi}{7} + 4\cos \frac{8\pi}{7} + 5\cos \frac{10\pi}{7} + 6\cos \frac{12\pi}{7}.\]

2017 Singapore Junior Math Olympiad, 2

Tags: diophantine , Diophantine equation , number theory , factorial

Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$

2016 ASDAN Math Tournament, 1

Tags: 2016 , team test

Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping?

2018 Mediterranean Mathematics OIympiad, 2

Tags: geometry , circumcircle , Ugly

Let $ABC$ be acute triangle. Let $E$ and $F$ be points on $BC$, such that angles $BAE$ and $FAC$ are equal. Lines $AE$ and $AF$ intersect cirumcircle of $ABC$ at points $M$ and $N$. On rays $AB$ and $AC$ we have points $P$ and $R$, such that angle $PEA$ is equal to angle $B$ and angle $AER$ is equal to angle $C$. Let $L$ be intersection of $AE$ and $PR$ and $D$ be intersection of $BC$ and $LN$. Prove that $$\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.$$

1982 IMO Longlists, 37

Tags: geometry , hexagon , collinearity , Golden Ratio , IMO , IMO 1982

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.

2016 Dutch IMO TST, 3

Tags: geometry , equal segments , isosceles , angle bisector

Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.

1965 IMO Shortlist, 5

Tags: geometry , parallelogram , Locus , Locus problems , IMO , IMO 1965

Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over a) the side $AB$; b) the interior of $\triangle OAB$.

1987 AMC 12/AHSME, 11

Tags: AMC

Let $c$ be a constant. The simultaneous equations \begin{align*} x-y = &\ 2 \\ cx+y = &\ 3 \\ \end{align*} have a solution $(x, y)$ inside Quadrant I if and only if $ \textbf{(A)}\ c=-1 \qquad\textbf{(B)}\ c>-1 \qquad\textbf{(C)}\ c<\frac{3}{2} \qquad\textbf{(D)}\ 0<c<\frac{3}{2}\\ \qquad\textbf{(E)}\ -1<c<\frac{3}{2} $