Found problems: 876
2021 Romania National Olympiad, 1
Find all continuous functions $f:\left[0,1\right]\rightarrow[0,\infty)$ such that:
$\int_{0}^{1}f\left(x\right)dx\cdotp\int_{0}^{1}f^{2}\left(x\right)dx\cdotp...\cdotp\int_{0}^{1}f^{2020}\left(x\right)dx=\left(\int_{0}^{1}f^{2021}\left(x\right)dx\right)^{1010}$
2012 Putnam, 1
Let $d_1,d_2,\dots,d_{12}$ be real numbers in the open interval $(1,12).$ Show that there exist distinct indices $i,j,k$ such that $d_i,d_j,d_k$ are the side lengths of an acute triangle.
MIPT student olimpiad autumn 2022, 4
In $R^n$ space is given a finite set of points $X$. It is known that for any
subset $Y \subseteq X$ of at most $n+1$ points, there is a unit ball $B_Y$ containing $Y$ and not containing the origin. Prove that there is a unit a ball $B_X$ containing $X$ and not containing the origin.
1999 Putnam, 3
Consider the power series expansion \[\dfrac{1}{1-2x-x^2}=\sum_{n=0}^\infty a_nx^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2+a_{n+1}^2=a_m.\]
2022 VTRMC, 6
Let $f : \mathbb{R} \to \mathbb{R}$ be a function whose second derivative is continuous. Suppose that $f$ and $f''$ are bounded. Show that $f'$ is also bounded.
ICMC 5, 5
A [i]tanned vector[/i] is a nonzero vector in $\mathbb R^3$ with integer entries. Prove that any tanned vector of length at most $2021$ is perpendicular to a tanned vector of length at most $100$.
[i]Proposed by Ethan Tan[/i]
ICMC 5, 2
Find all integers $n$ for which there exists a table with $n$ rows, $2022$ columns, and integer entries, such that subtracting any two rows entry-wise leaves every remainder modulo $2022$.
[i]Proposed by Tony Wang[/i]
2005 IMC, 2
Let $f: \mathbb{R}\to\mathbb{R}$ be a function such that $(f(x))^{n}$ is a polynomial for every integer $n\geq 2$. Is $f$ also a polynomial?
2016 IMC, 3
Let $n$ be a positive integer. Also let $a_1, a_2, \dots, a_n$ and $b_1,b_2,\dots, b_n$ be real numbers such that $a_i+b_i>0$ for $i=1,2,\dots, n$. Prove that
$$\sum_{i=1}^n \frac{a_ib_i-b_i^2}{a_i+b_i}\le\frac{\displaystyle \sum_{i=1}^n a_i\cdot \sum_{i=1}^n b_i - \left( \sum_{i=1}^n b_i\right) ^2}{\displaystyle\sum_{i=1}^n (a_i+b_i)}$$.
(Proposed by Daniel Strzelecki, Nicolaus Copernicus University in Toruń, Poland)
ICMC 3, 1
Alice and Bob play a game on a sphere which is initially marked with a finite number of points. Alice and Bob then take turns making moves, with Alice going first:
- On Alice’s move, she counts the number of marked points on the sphere, \(n\). She then marks another \(n + 1\) points on the sphere.
- On Bob’s move, he chooses one hemisphere and removes all marked points on that hemisphere, including any marked points on the boundary of the hemisphere.
Can Bob always guarantee that after a finite number of moves, the sphere contains no marked points?
(A [i]hemisphere[/i] is the region on a sphere that lies completely on one side of any plane passing through the centre of the sphere.)
[i]proposed by the ICMC Problem Committee[/i]
2012 Miklós Schweitzer, 7
Let $\Gamma$ be a simple curve, lying inside a circle of radius $r$, rectifiable and of length $\ell$. Prove that if $\ell > kr\pi$, then there exists a circle of radius $r$ which intersects $\Gamma$ in at least $k+1$ distinct points.
1966 Putnam, B6
Show that all the solutions of the differential equation $y''+e^xy=0$ remain bounded as $x\to \infty.$
1971 Putnam, B3
Two cars travel around a track at equal and constant speeds, each completing a lap every hour. From a common starting point, the first starts at time $t=0$ and the second at an arbitrary later time $t=T>0.$ Prove that there is a total period of exactly one hour during the motion in which the first has completed twice as many laps as the second.
1960 Miklós Schweitzer, 8
[b]8.[/b] Let $f$ be a bounded real function defined on the unit cube $H$ of the $n$-dimensional space and, for a given $y$, let $A_y$ and $B_y$ denote the parts of the interior of $H$ on which $f>y$ and $f<y$, respectively. Show that $f$ is integrable in the Riemannian sense if and only if for every $y$ almost all points of $A_y$ and $B_y$ are inner points. [b](R. 9)[/b]
2014 Contests, 2
Consider the following sequence
$$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$
Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$.
(Proposed by Tomas Barta, Charles University, Prague)
2014 Putnam, 5
In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible $n\times n$ matrices with entries in the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p,$ where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are:
(1) A player cannot choose an element that has been chosen by either player on any previous turn.
(2) A player can only choose an element that commutes with all previously chosen elements.
(3) A player who cannot choose an element on his/her turn loses the game.
Patniss takes the first turn. Which player has a winning strategy?
2018 IMC, 8
Let $\Omega =\{ (x,y,z)\in \mathbb{Z}^3:y+1\geqslant x\geqslant y\geqslant z\geqslant 0\}$. A frog moves along the points of $\Omega$ by jumps of length $1$. For every positive integer $n$, determine the number of paths the frog can take to reach $(n,n,n)$ starting from $(0,0,0)$ in exactly $3n$ jumps.
[i]Proposed by Fedor Petrov and Anatoly Vershik, St. Petersburg State University[/i]
2004 Miklós Schweitzer, 9
Let $F$ be a smooth (i.e. $C^{\infty}$) closed surface. Call a continuous map $f\colon F\rightarrow \mathbb{R}^2$ an [i]almost-immersion[/i] if there exists a smooth closed embedded curve $\gamma$ (possibly disconnected) in $F$ such that $f$ is smooth and of maximal rank (i.e., rank 2) on $F\backslash \gamma$ and each point $p\in\gamma$ admits local coordinate charts $(x,y)$ and $(u,v)$ about $p$ and $f(p)$, respectively, such taht the coordinates of $p$ and $f(p)$ are zero and the map $f$ is given by $(x,y)\rightarrow (u,v), u=|x|, v=y$.
Determine the genera of those smooth, closed, connected, orientable surfaces $F$ that admit an almost-immersion in the plane with the curve $\gamma$ having a given positive number $n$ of connected components.
2024 SEEMOUS, P4
Let $n\in\mathbb{N}$, $n\geq 2$. Find all values of $k\in\mathbb{N}$, $k\geq 1$, for which the following statement holds: $$\text{"If }A\in\mathcal{M}_n(\mathbb{C})\text{ is such that }A^kA^*=A\text{, then }A=A^*\text{."}$$ (here, $A^*$ denotes the conjugate transpose of $A$).
2014 IMC, 2
Let $A=(a_{ij})_{i, j=1}^n$ be a symmetric $n\times n$ matrix with real entries, and let $\lambda _1, \lambda _2, \dots, \lambda _n$ denote its eigenvalues. Show that
$$\sum_{1\le i<j\le n} a_{ii}a_{jj}\ge \sum_{1\le i < j\le n} \lambda _i \lambda _j$$
and determine all matrices for which equality holds.
(Proposed by Matrin Niepel, Comenius University, Bratislava)
2016 IMC, 4
Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties:
(i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements;
(ii) for any two sets $A, B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$.
Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements.
(Proposed by Fedor Petrov, St. Petersburg State University)
2007 IMC, 6
How many nonzero coefficients can a polynomial $ P(x)$ have if its coefficients are integers and $ |P(z)| \le 2$ for any complex number $ z$ of unit length?
2020 Miklós Schweitzer, 9
Let $D\subseteq \mathbb{C}$ be a compact set with at least two elements and consider the space $\Omega=\bigtimes_{i=1}^{\infty} D$ with the product topology. For any sequence $(d_n)_{n=0}^{\infty} \in \Omega$ let $f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n$, and for each point $\zeta \in \mathbb{C}$ with $|\zeta|=1$ we define $S=S(\zeta,(d_n))$ to be the set of complex numbers $w$ for which there exists a sequence $(z_k)$ such that $|z_k|<1$, $z_k \to \zeta$, and $f_{d_n}(z_k) \to w$. Prove that on a residual set of $\Omega$, the set $S$ does not depend on the choice of $\zeta$.
2022 VTRMC, 1
Give all possible representations of $2022$ as a sum of at least two consecutive positive integers and prove that these are the only representations.
2004 Miklós Schweitzer, 2
Write $t(G)$ for the number of complete quadrilaterals in the graph $G$ and $e_G(S)$ for the number of edges spanned by a subset $S$ of vertices of $G$. Let $G_1, G_2$ be two (simple) graphs on a common underlying set $V$ of vertices, $|V|-n$, and assume that $|e_{G_1}(S)-e_{G_2}(S)|<\frac{n^2}{1000}$ holds for any subset $S\subseteq V$. Prove that $|t(G_1)-t(G_2)|\le \frac{n^4}{1000}$.