Found problems: 876
2022 Miklós Schweitzer, 5
Is it possible to select a non-degenerate segment from each line of the plane such that any two selected segments are disjoint?
2016 IMC, 1
Let $(x_1,x_2,\ldots)$ be a sequence of positive real numbers satisfying ${\displaystyle \sum_{n=1}^{\infty}\frac{x_n}{2n-1}=1}$. Prove that $$ \displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{k} \frac{x_n}{k^2} \le2. $$
(Proposed by Gerhard J. Woeginger, The Netherlands)
1986 Miklós Schweitzer, 10
Let $X_1, X_2$ be independent, identically distributed random variables such that $X_i\geq 0$ for all $i$. Let $\mathrm EX_i=m$, $\mathrm{Var} (X_i)=\sigma ^2<\infty$. Show that, for all $0<\alpha\leq 1$
$$\lim_{n\to\infty} n\,\mathrm{Var} \left( \left[ \frac{X_1+\ldots +X_n}{n}\right] ^\alpha\right)=\frac{\alpha ^ 2 \sigma ^ 2}{m^{2(1-\alpha)}}$$
[Gy. Michaletzki]
2002 Putnam, 2
Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.
2007 IMC, 3
Let $ C$ be a nonempty closed bounded subset of the real line and $ f: C\to C$ be a nondecreasing continuous function. Show that there exists a point $ p\in C$ such that $ f(p) \equal{} p$.
(A set is closed if its complement is a union of open intervals. A function $ g$ is nondecreasing if $ g(x)\le g(y)$ for all $ x\le y$.)
2003 Miklós Schweitzer, 2
Let $p$ be a prime and let $M$ be an $n\times m$ matrix with integer entries such that $Mv\not\equiv 0\pmod{p}$ for any column vector $v\neq 0$ whose entries are $0$ are $1$. Show that there exists a row vector $x$ with integer entries such that no entry of $xM$ is $0\pmod{p}$.
(translated by L. Erdős)
2017 Korea USCM, 3
Sequence $\{a_n\}$ defined by recurrence relation $a_{n+1} = 1+\frac{n^2}{a_n}$. Given $a_1>1$, find the value of $\lim\limits_{n\to\infty} \frac{a_n}{n}$ with proof.
1971 Putnam, A4
Show that for $0 <\epsilon <1$ the expression $(x+y)^n(x^2-(2-\epsilon)xy+y^2)$ is a polynomial with positive coefficients for $n$ sufficiently large and integral. For $\epsilon =.002$ find the smallest admissible value of $n$.
2010 Miklós Schweitzer, 3
Let $ A_i,i=1,2,\dots,t$ be distinct subsets of the base set $\{1,2,\dots,n\}$ complying to the following condition
$$ \displaystyle A_ {i} \cap A_ {k} \subseteq A_ {j}$$for any $1 \leq i <j <k \leq t.$ Find the maximum value of $t.$
Thanks @dgrozev
ICMC 3, 5
A particle moves from the point \(P\) to the point \(Q\) in the Cartesian plane. When it passes through any point \((x,y)\), the particle has an instantaneous speed of \(\sqrt{x + y}\). Compute the minimum time required for the particle to move:
(i) from \(P_1=(-1,0)\) to \(Q_1=(1,0)\), and
(ii) from \(P_2=(0,1)\) to \(Q_2=(1,1)\).
[i]proposed by the ICMC Problem Committee[/i]
2016 IMC, 2
Today, Ivan the Confessor prefers continuous functions $f:[0,1]\to\mathbb{R}$ satisfying $f(x)+f(y)\geq |x-y|$ for all pairs $x,y\in [0,1]$. Find the minimum of $\int_0^1 f$ over all preferred functions.
(Proposed by Fedor Petrov, St. Petersburg State University)
1984 Miklós Schweitzer, 9
[b]9.[/b] Let $X_0, X_1, \dots $ be independent, indentically distributed, nondegenerate random variables, and let $0<\alpha <1$ be a real number. Assume that the series
$\sum_{k=1}^{\infty} \alpha^{k} X_k$
is convergent with probability one. Prove that the distribution function of the sum is continuous. ([b]P. 23[/b])
[T. F. Móri]
2008 IMC, 1
Find all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that
\[ f(x)-f(y)\in \mathbb{Q}\quad \text{ for all }\quad x-y\in\mathbb{Q} \]
MIPT student olimpiad spring 2022, 4
Let us consider sequences of complex numbers that are infinite in both directions $c=(c_k) , k\in Z$ with finite norm
$||c||= (\sum_{k \in Z} |c_k|^2)^{1/2}$
Let $T_m-$ this is a shift operation sequences on m ($(T_mc)_k=c_{k-m}$)
Prove that:
$\lim_{n \to \infty} \frac{\sum_{i=0}^{n-1} T_ic}{n} =0$
(Adding and multiplying a sequence by a number defined component by component)
2017 Miklós Schweitzer, 9
Let $N$ be a normed linear space with a dense linear subspace $M$. Prove that if $L_1,\ldots,L_m$ are continuous linear functionals on $N$, then for all $x\in N$ there exists a sequence $(y_n)$ in $M$ converging to $x$ satisfying $L_j(y_n)=L_j(x)$ for all $j=1,\ldots,m$ and $n\in \mathbb{N}$.
1999 Putnam, 6
Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.
1984 Putnam, A5
Putnam 1984/A5) Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z\leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral
\[\iiint_{R}xy^{9}z^{8}w^{4}\ dx\ dy\ dz\]
in the form $a!b!c!d!/n!$ where $a,b,c,d$ and $n$ are positive integers.
[hide="A solution"]\[\iiint_{R}xy^{9}z^{8}w^{4}\ dx dy dz = 4\iiint_{R}\int_{0}^{1-x-y-z}xy^{9}z^{8}w^{3}\ dw dx dy dz = 4\iiiint_{Q}xy^{9}z^{8}w^{3}\ dw dx dy dz\]
where $Q=\left\{ (x,y,z,w)\in\mathbb{R}^{4}|\ x,y,z,w\geq 0, x+y+z+w\leq 1\right\}$, which is a Dirichlet integral giving
\[4\iiiint_{Q}x^{1}y^{9}z^{8}w^{3}\ dw dx dy dz = 4\cdot\frac{1!9!8!3!}{(2+10+9+4)!}= \frac{1!9!8!4!}{25!}\][/hide]
1972 Putnam, B6
Let $ n_1<n_2<n_3<\cdots <n_k$ be a set of positive integers. Prove that the polynomial $ 1\plus{}z^{n_1}\plus{}z^{n_2}\plus{}\cdots \plus{}z^{n_k}$ has no roots inside the circle $ |z|<\frac{\sqrt{5}\minus{}1}{2}$.
2018 Brazil Undergrad MO, 23
How many prime numbers $ p $ the number $ p ^ 3-4 p + 9 $ is a perfect square
1976 Putnam, 3
Find all integral solutions of the equation $$|p^r-q^s|=1,$$ where $p$ and $q$ are prime numbers and $r$ and $s$ are positive integers larger than unity. Prove that there are no other solutions.
2013 IMC, 2
Let $\displaystyle{f:{\cal R} \to {\cal R}}$ be a twice differentiable function. Suppose $\displaystyle{f\left( 0 \right) = 0}$. Prove there exists $\displaystyle{\xi \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)}$ such that \[\displaystyle{f''\left( \xi \right) = f\left( \xi \right)\left( {1 + 2{{\tan }^2}\xi } \right)}.\]
[i]Proposed by Karen Keryan, Yerevan State University, Yerevan, Armenia.[/i]
2002 Putnam, 2
Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex. Show that the player who signs first will always win by playing as well as possible.
2006 IMC, 3
Let $A$ be an $n$x$n$ matrix with integer entries and $b_{1},b_{2},...,b_{k}$ be integers satisfying $detA=b_{1}\cdot b_{2}\cdot ...\cdot b_{k}$. Prove that there exist $n$x$n$-matrices $B_{1},B_{2},...,B_{k}$ with integers entries such that $A=B_{1}\cdot B_{2}\cdot ...\cdot B_{k}$ and $detB_{i}=b_{i}$ for all $i=1,...,k$.
2004 VJIMC, Problem 3
Denote by $B(c,r)$ the open disk of center $c$ and radius $r$ in the plane. Decide whether there exists a sequence $\{z_n\}^\infty_{n=1}$ of points in $\mathbb R^2$ such that the open disks $B(z_n,1/n)$ are pairwise disjoint and the sequence $\{z_n\}^\infty_{n=1}$ is convergent.
2013 IMC, 1
Let $\displaystyle{z}$ be a complex number with $\displaystyle{\left| {z + 1} \right| > 2}$. Prove that $\displaystyle{\left| {{z^3} + 1} \right| > 1}$.
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]