Found problems: 15925
2024 Bangladesh Mathematical Olympiad, P7
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that\[f\left(\Big \lceil \frac{f(m)}{n} \Big \rceil\right)=\Big \lceil \frac{m}{f(n)} \Big \rceil\]for all $m,n \in \mathbb{N}$.
[i]Proposed by Md. Ashraful Islam Fahim[/i]
2024 Tuymaada Olympiad, 7
Given are two polynomial $f$ and $g$ of degree $100$ with real coefficients. For each positive integer $n$ there is an integer $k$ such that
\[\frac{f(k)}{g(k)}=\frac{n+1}{n}.\]
Prove that $f$ and $g$ have a common non-constant factor.
2019 Belarus Team Selection Test, 7.3
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that
$$
(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
$$
[i](B. Serankou, M. Karpuk)[/i]
2016 Rioplatense Mathematical Olympiad, Level 3, 2
Determine all positive integers $n$ for which there are positive real numbers $x,y$ and $z$ such that $\sqrt x +\sqrt y +\sqrt z=1$ and $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$ is an integer.
2017 Harvard-MIT Mathematics Tournament, 6
A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^2}$ for $n=1, 2, \dots, 2016$. Find $\lfloor 2017P(2017)\rfloor$.
Revenge EL(S)MO 2024, 5
Inscribe three mutually tangent pink disks of radii $450$, $450$, and $720$ in an uncolored circle $\Omega$ of radius $1200$. In one move, Elmo selects an uncolored region inside $\Omega$ and draws in it the largest possible pink disk. Can Elmo ever draw a disk with a radius that is a perfect square of a rational?
Proposed by [i]Ritwin Narra[/i]
2023 SAFEST Olympiad, 6
Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$
[i]Proposed by Navid Safaei, Iran[/i]
1999 IMO Shortlist, 6
Prove that for every real number $M$ there exists an infinite arithmetic progression such that:
- each term is a positive integer and the common difference is not divisible by 10
- the sum of the digits of each term (in decimal representation) exceeds $M$.
2016 HMNT, 10
Determine the largest integer $n$ such that there exist monic quadratic polynomials $p_1(x)$, $p_2(x)$, $p_3(x)$ with integer coefficients so that for all integers $ i \in [1, n]$ there exists some $j \in [1, 3]$ and $m \in Z$ such that $p_j (m) = i$.
1989 IMO Longlists, 11
Given the equation \[ y^4 \plus{} 4y^2x \minus{} 11y^2 \plus{} 4xy \minus{} 8y \plus{} 8x^2 \minus{} 40x \plus{} 52 \equal{} 0,\] find all real solutions.
2024 Balkan MO, 4
Let $\mathbb{R}^+ = (0, \infty)$ be the set of all positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0) = 0$ which satisfy the equality $f(f(x) + P(y)) = f(x - y) + 2y$ for all real numbers $x > y > 0$.
[i]Proposed by Sardor Gafforov, Uzbekistan[/i]
2005 Bulgaria National Olympiad, 3
Let $M=(0,1)\cap \mathbb Q$. Determine, with proof, whether there exists a subset $A\subset M$ with the property that every number in $M$ can be uniquely written as the sum of finitely many distinct elements of $A$.
2019 PUMaC Algebra B, 8
A [i]weak binary representation[/i] of a nonnegative integer $n$ is a representation $n=a_0+2\cdot a_1+2^2\cdot a_2+\dots$ such that $a_i\in\{0,1,2,3,4,5\}$. Determine the number of such representations for $513$.
2021 Nigerian Senior MO Round 2, 4
let $x_1$, $x_2$ .... $x_6$ be non-negative reals such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5$ + $x_2x_4x_6$ $\geq$ $\frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of $x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2$. Find $p+q$
2022 Czech-Austrian-Polish-Slovak Match, 2
Find all functions $f: \mathbb{R^{+}} \rightarrow \mathbb {R^{+}}$ such that $f(f(x)+\frac{y+1}{f(y)})=\frac{1}{f(y)}+x+1$ for all $x, y>0$.
[i]Proposed by Dominik Burek, Poland[/i]
1996 Canada National Olympiad, 2
Find all real solutions to the following system of equations. Carefully justify your answer.
\[ \left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y \\ \\ \displaystyle\frac{4y^2}{1+4y^2} = z \\ \\ \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right. \]
2023 JBMO Shortlist, A4
Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove that
$$\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2$$
2020 USA EGMO Team Selection Test, 3
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]
and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?
[i]Carl Schildkraut and Milan Haiman[/i]
1962 All-Soviet Union Olympiad, 4
Prove that there are no integers $a, b, c, d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$ and $2$ at $x=62$.
2022 Balkan MO Shortlist, A5
Find all functions $f: (0, \infty) \to (0, \infty)$ such that
\begin{align*}
f(y(f(x))^3 + x) = x^3f(y) + f(x)
\end{align*}
for all $x, y>0$.
[i]Proposed by Jason Prodromidis, Greece[/i]
2007 Thailand Mathematical Olympiad, 9
Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?
1999 All-Russian Olympiad Regional Round, 8.4
There are $40$ identical gas cylinders, gas pressure values in which we are unknown and may be evil. It is allowed to connect any cylinders with each other in an amount not exceeding a given natural number $k$, and then separate them; while the pressure gas in the connected cylinders is set equal to the arithmetic average of the pressures in them before the connection. At what minimum $k$ is there a way to equalize the pressures in all $40$ cylinders, regardless of initial pressure distribution in the cylinders?
2013 Princeton University Math Competition, 16
Is $\cos 1^\circ$ rational? Prove.
2000 Harvard-MIT Mathematics Tournament, 10
How many times per day do at least two of the three hands on a clock coincide?
2023 CMIMC Algebra/NT, 10
For a given $n$, consider the points $(x,y)\in \mathbb{N}^2$ such that $x\leq y\leq n$. An ant starts from $(0,1)$ and, every move, it goes from $(a,b)$ to point $(c,d)$ if $bc-ad=1$ and $d$ is maximized over all such points. Let $g_n$ be the number of moves made by the ant until no more moves can be made. Find $g_{2023} - g_{2022}$.
[i]Proposed by David Tang[/i]