Found problems: 85335
2008 Spain Mathematical Olympiad, 1
Let $p$ and $q$ be two different prime numbers. Prove that there are two positive integers, $a$ and $b$, such that the arithmetic mean of the divisors of $n=p^aq^b$ is an integer.
2014 Romania Team Selection Test, 2
Let $p$ be an[color=#FF0000] odd [/color]prime number. Determine all pairs of polynomials $f$ and $g$ from $\mathbb{Z}[X]$ such that
\[f(g(X))=\sum_{k=0}^{p-1} X^k = \Phi_p(X).\]
2018 Kazakhstan National Olympiad, 5
Given set $S = \{ xy\left( {x + y} \right)\; |\; x,y \in \mathbb{N}\}$.Let $a$ and $n$ natural numbers such that $a+2^k\in S$ for all $k=1,2,3,...,n$.Find the greatest value of $n$.
2022 Romania Team Selection Test, 4
Can every positive rational number $q$ be written as
$$\frac{a^{2021} + b^{2023}}{c^{2022} + d^{2024}},$$
where $a, b, c, d$ are all positive integers?
[i]Proposed by Dominic Yeo, UK[/i]
2013 Harvard-MIT Mathematics Tournament, 26
Triangle $ABC$ has perimeter $1$. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of $\min(AB,BC,CA)$.
2022 JBMO Shortlist, C2
Let $n \ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of the numbers written in its extremities and by $p_s$ the number of all other drawn segments which intersect $s$ in its interior.
Find the greatest $n$ for which Alex can write the numbers on the circle such that $p_s \le |d_s|$, for each drawn segment $s$.
Ukraine Correspondence MO - geometry, 2020.8
Let $ABC$ be an acute triangle, $D$ be the midpoint of $BC$. Bisectors of angles $ADB$ and $ADC$ intersect the circles circumscribed around the triangles $ADB$ and $ADC$ at points $E$ and $F$, respectively. Prove that $EF\perp AD$.
2018 PUMaC Live Round, Estimation 1
A $2$-by-$2018$ grid is completely covered by non-overlapping L-tiles (see diagram below) and $1$-by-$1$ squares. If the L-tiles can be rotated and flipped, there are a total of $M$ such tilings.
[asy]
size(1cm);
draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle);
draw((0,1)--(1,1)--(1,0));
[/asy]
What is $\ln M?$
Give your answer as an integer or decimal. If your answer is $A$ and the correct answer is $C$, then your score will be $\max\{\lfloor7.5-\tfrac{|A-C|^{1.5}}{20}\rfloor,0\}.$
1979 Spain Mathematical Olympiad, 7
Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.
2020 Iran MO (3rd Round), 4
We call a polynomial $P(x)$ intresting if there are $1398$ distinct positive integers $n_1,...,n_{1398}$ such that
$$P(x)=\sum_{}{x^{n_i}}+1$$
Does there exist infinitly many polynomials $P_1(x),P_2(x),...$ such that for each distinct $i,j$ the polynomial $P_i(x)P_j(x)$ is interesting.
2022 Moscow Mathematical Olympiad, 4
The starship is in a half-space at a distance $a$ from its boundary. The crew knows about it, but has no idea in which direction to move in order to reach the boundary plane. The starship can fly in space along any trajectory, measuring the length of the path traveled, and has a sensor that sends a signal when
the border has been reached. Can a starship be guaranteed to reach the border with a path no longer than $14a$?
2025 Kosovo EGMO Team Selection Test, P4
Let $a,b$ be positive real numbers such that $a^3+b^3=2(a^2+b^2)$.
Prove the following inequality:
$$ \sqrt{a^3+1} + \sqrt{b^3+1} \leq a+b+2. $$
When is equality achieved?
2025 Israel National Olympiad (Gillis), P5
$2024$ otters live in the river. Some are friends with each other. Is it possible that, for any collection of $1012$ otters, there is exactly one additional otter that is friends with all $1012$ otters?
2017 Mathematical Talent Reward Programme, SAQ: P 6
Let us consider an infinite grid plane as shown below. We start with 4 points $A$, $B$, $C$, $D$, that form a square.
We perform the following operation: We pick two points $X$ and $Y$ from the currant points. $X$ is reflected about $Y$ to get $X'$. We remove $X$ and add $X'$ to get a new set of 4 points and treat it as our currant points.
For example in the figure suppose we choose $A$ and $B$ (we can choose any other pair too). Then reflect $A$ about $B$ to get $A'$. We remove $A$ and add $A'$. Thus $A'$, $B$, $C$, $D$ is our new 4 points. We may again choose $D$ and $A'$ from the currant points. Reflect $D$ about $A'$ to obtain $D'$ and hence $A'$, $B$, $C$, $D'$ are now new set of points. Then similar operation is performed on this new 4 points and so on.
Starting with $A$, $B$, $C$, $D$ can you get a bigger square by some sequence of such operations?
2017 Miklós Schweitzer, 4
Let $K$ be a number field which is neither $\mathbb{Q}$ nor a quadratic imaginary extension of $\mathbb{Q}$. Denote by $\mathcal{L}(K)$ the set of integers $n\ge 3$ for which we can find units $\varepsilon_1,\ldots,\varepsilon_n\in K$ for which
$$\varepsilon_1+\dots+\varepsilon_n=0,$$but $\displaystyle\sum_{i\in I}\varepsilon_i\neq 0$ for any nonempty proper subset $I$ of $\{1,2,\dots,n\}$. Prove that $\mathcal{L}(K)$ is infinite, and that its smallest element can be bounded from above by a function of the degree and discriminant of $K$. Further, show that for infinitely many $K$, $\mathcal{L}(K)$ contains infinitely many even and infinitely many odd elements.
2020 Regional Olympiad of Mexico Center Zone, 3
In an acute triangle $ABC$, an arbitrary point $P$ is chosen on the altitude $AH$. The points $E$ and $F$ are the midpoints of $AC$ and $AB$, respectively. The perpendiculars from $E$ on $CP$ and from $F$ on $BP$ intersect at the point $K$. Show that $KB = KC$.
2021 Philippine MO, 3
Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies
• $f(p) = 1$ for all primes $p$, and
• $f(ab) = af(b) + bf(a)$ for all $ a,b \in \mathbb{Q}^+ $.
For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$?
1988 China Team Selection Test, 3
In triangle $ABC$, $\angle C = 30^{\circ}$, $O$ and $I$ are the circumcenter and incenter respectively, Points $D \in AC$ and $E \in BC$, such that $AD = BE = AB$. Prove that $OI = DE$ and $OI \bot DE$.
2015 IFYM, Sozopol, 7
Determine the greatest natural number $n$, such that for each set $S$ of 2015 different integers there exist 2 subsets of $S$ (possible to be with 1 element and not necessarily non-intersecting) each of which has a sum of its elements divisible by $n$.
2021 Malaysia IMONST 1, 14
Given a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. Each coefficient $a, b, c, d, e$, and$ f$ is equal to either $ 1$ or $-1$. If $p(2) = 11$, what is the value of $p(3)$?
2021 JHMT HS, 5
A function $f$ with domain $A$ and range $B$ is called [i]injective[/i] if every input in $A$ maps to a unique output in $B$ (equivalently, if $x, y \in A$ and $x \neq y$, then $f(x) \neq f(y)$). With $\mathbb{C}$ denoting the set of complex numbers, let $P$ be an injective polynomial with domain and range $\mathbb{C}$. Suppose further that $P(0) = 2021$ and that when $P$ is written in standard form, all coefficients of $P$ are integers. Compute the smallest possible positive integer value of $P(10)/P(1)$.
2010 Flanders Math Olympiad, 3
In a triangle $ABC$, $\angle B= 2\angle A \ne 90^o$ . The inner bisector of $B$ intersects the perpendicular bisector of $[AC]$ at a point $D$. Prove that $AB \parallel CD$.
2016 Costa Rica - Final Round, LR2
There are $2016$ participants in the Olcotournament of chess. It is known that in any set of four participants, there is one of them who faced the other three. Prove there is at least $2013$ participants who faced everyone else.
2017 CMIMC Individual Finals, 1
Cody has an unfair coin that flips heads with probability either $\tfrac13$ or $\tfrac23$, but he doesn't know which one it is. Using this coin, what is the fewest number of independent flips needed to simulate a coin that he knows will flip heads with probability $\tfrac13$?
2006 Indonesia MO, 1
Find all pairs $ (x,y)$ of real numbers which satisfy $ x^3\minus{}y^3\equal{}4(x\minus{}y)$ and $ x^3\plus{}y^3\equal{}2(x\plus{}y)$.