Found problems: 876
2005 IMC, 6
6. If $ p,q$ are rationals, $r=p+\sqrt{7}q$, then prove there exists a matrix
$\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in M_{2}(Z)- ( \pm I_{2})$
for which $\frac{ar+b}{cr+d}=r$ and $det(A)=1$
2015 Miklos Schweitzer, 5
Let $f(x) = x^n+x^{n-1}+\dots+x+1$ for an integer $n\ge 1.$ For which $n$ are there polynomials $g, h$ with real coefficients and degrees smaller than $n$ such that $f(x) = g(h(x)).$
2001 Putnam, 3
For any positive integer $n$, let $ \left< n \right> $ denote the closest integer to $ \sqrt {n} $. Evaluate: \[ \displaystyle\sum_{n=1}^{\infty} \dfrac {2^{\left< n \right>} + 2^{- \left< n \right>}}{2^n} \]
2021 Alibaba Global Math Competition, 3
Given positive integers $k \ge 2$ and $m$ sufficiently large. Let $\mathcal{F}_m$ be the infinite family of all the (not necessarily square) binary matrices which contain exactly $m$ 1's. Denote by $f(m)$ the maximum integer $L$ such that for every matrix $A \in \mathcal{F}_m$, there always exists a binary matrix $B$ of the same dimension such that (1) $B$ has at least $L$ 1-entries; (2) every entry of $B$ is less or equal to the corresponding entry of $A$; (3) $B$ does not contain any $k \times k$ all-1 submatrix. Show the equality
\[\lim_{m \to \infty} \frac{\ln f(m)}{\ln m}=\frac{k}{k+1}.\]
ICMC 4, 4
Does there exist a set $\mathcal{R}$ of positive rational numbers such that every positive rational number is the sum of the elements of a unique finite subset of $\mathcal{R}$?
[i]Proposed by Tony Wang[/i]
2000 Putnam, 5
Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$.
2004 Putnam, B6
Let $A$ be a nonempty set of positive integers, and let $N(x)$ denote the number of elements of $A$ not exceeding $x$. Let $B$ denote the set of positive integers $b$ that can be written in the form $b=a-a^{\prime}$ with $a\in A$ and $a^{\prime}\in A$. Let $b_1<b_2<\cdots$ be the members of $B$, listed in increasing order. Show that if the sequence $b_{i+1}-b_i$ is unbounded, then $\lim_{x\to \infty}\frac{N(x)}{x}=0$.
2013 IMC, 2
Let $\displaystyle{p,q}$ be relatively prime positive integers. Prove that
\[\displaystyle{ \sum_{k=0}^{pq-1} (-1)^{\left\lfloor \frac{k}{p}\right\rfloor + \left\lfloor \frac{k}{q}\right\rfloor} = \begin{cases} 0 & \textnormal{ if } pq \textnormal{ is even}\\ 1 & \textnormal{if } pq \textnormal{ odd}\end{cases}}\]
[i]Proposed by Alexander Bolbot, State University, Novosibirsk.[/i]
2012 IMC, 1
Consider a polynomial
\[f(x)=x^{2012}+a_{2011}x^{2011}+\dots+a_1x+a_0.\]
Albert Einstein and Homer Simpson are playing the following game. In turn, they choose one of the coefficients $a_0,a_1,\dots,a_{2011}$ and assign a real value to it. Albert has the first move. Once a value is assigned to a coefficient, it cannot be changed any more. The game ends after all the coefficients have been assigned values.
Homer's goal is to make $f(x)$ divisible by a fixed polynomial $m(x)$ and Albert's goal is to prevent this.
(a) Which of the players has a winning strategy if $m(x)=x-2012$?
(b) Which of the players has a winning strategy if $m(x)=x^2+1$?
[i]Proposed by Fedor Duzhin, Nanyang Technological University.[/i]
1955 Miklós Schweitzer, 9
[b]9.[/b] Show that to any elliptic paraboloid $\varphi_1$ there may be found an elliptic paraboloid $\varphi_2$ (other than $\varphi_1$) and an affinity $\phi$ which maps $\varphi_1$ onto $\varphi_2$ and has the following property: If $P$ is any point of $\varphi_1$ such that $\phi(P) \neq P$, then the straight line connecting $P$ and $\phi(P)$ is a common tangent of the two paraboloids. [b](G. 18)[/b]
2016 VJIMC, 2
Let $X$ be a set and let $\mathcal{P}(X)$ be the set of all subsets of $X$. Let $\mu: \mathcal{P}(X) \to \mathcal{P}(X)$ be a map with the property that $\mu(A \cup B) = \mu(A) \cup \mu(B)$ whenever $A$ and $B$ are disjoint subsets of $X$. Prove that there exists $F \subset X$ such that $\mu(F) = F$.
1957 Miklós Schweitzer, 1
[b]1.[/b] Let $C_{ij}$ ($i,j=1,2,3$) be the coefficients of a real non-involutive orthogonal transformation. Prove that the function $w= \sum_{i,j=1}^{3} c_{ ij}z_{i}\bar{z_{ j}}$ maps the surface of complex unit sphere $\sum_{i=1}^{3} z_{i}\bar{z_{i}} = 1$ onto a triangle of the w-plane. [b](F. 3)[/b]
2019 Simon Marais Mathematical Competition, B4
A [i]binary string[/i] is a sequence, each of whose terms is $0$ or $1$. A set $\mathcal{B}$ of binary strings is defined inductively according to the following rules.
[list]
[*]The binary string $1$ is in $\mathcal{B}$.[/*]
[*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ odd, then both $s_1,s_2,\dotsc ,s_n,0$ and $0,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*]
[*]If $s_1,s_2,\dotsc ,s_n$ is in $\mathcal{B}$ with $n$ even, then both $s_1,s_2,\dotsc ,s_n,1$ and $1,s_1,s_2,\dotsc ,s_n$ are in $\mathcal{B}$.[/*]
[*]No other binary strings are in $\mathcal{B}$.[/*]
[/list]
For each positive integer $n$, let $b_n$ be the number of binary strings in $\mathcal{B}$ of length $n$.
[list=a]
[*]Prove that there exist constants $c_1,c_2>0$ and $1.6<\lambda_1,\lambda_2<1.9$ such that $c_1\lambda_1^n<b_n<c_2\lambda_2^n$ for all positive integer $n$.[/*]
[*]Determine $\liminf_{n\to \infty} {\sqrt[n]{b_n}}$ and $\limsup_{n\to \infty} {\sqrt[n]{b_n}}$[/*]
[/list]
[i]Note: The problem is open in the sense that no solution is currently known to part (b).[/i]
2012 Miklós Schweitzer, 11
Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$. For what real numbers $c$ is the following statement true:
$$P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}$$
2018 Miklós Schweitzer, 1
Let $S\subset \mathbb{R}$ be a closed set and $f:\mathbb{R}^{2n}\to \mathbb{R}$ be a continuous function. Define a graph $G$ as follows: Let $x$ be a vertex of $G$ iff $x\in \mathbb{R}^{n}$ and $f(x,x)\not\in S$, then connect the vertices $x$ and $y$ by an edge in $G$ iff $f(x,y)\in S$ or $f(y,x)\in S$. Show that the chromatic number of $G$ is countable.
ICMC 3, 6
Let \(\varepsilon < \frac{1}{2}\) be a positive real number and let \(U_{\varepsilon}\) denote the set of real numbers that differ from their nearest integer by at most \(\varepsilon\). Prove that there exists a positive integer \(m\) such that for any real number \(x\), the sets \(\left\{x, 2x, 3x, . . . , mx\right\}\) and \(U_{\varepsilon}\) have at least one element in common.
proposed by the ICMC Problem Committee
2015 Miklos Schweitzer, 11
For $[0,1]\subset E\subset [0,+\infty)$ where $E$ is composed of a finite number of closed interval,we start a two dimensional Brownian motion from the point $x<0$ terminating when we first hit $E$.Let $p(x)$ be the probability of the finishing point being in $[0,1]$.Prove that $p(x)$ is increasing on $[-1,0)$.
2008 IMC, 3
Let $p$ be a polynomial with integer coefficients and let $a_1<a_2<\cdots <a_k$ be integers. Given that $p(a_i)\ne 0\forall\; i=1,2,\cdots, k$.
[list]
(a) Prove $\exists\; a\in \mathbb{Z}$ such that
\[ p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k \]
(b) Does there exist $a\in \mathbb{Z}$ such that
\[ \prod_{i=1}^{k}p(a_i)\mid p(a) \][/list]
1955 Miklós Schweitzer, 5
[b]5.[/b] Show that a ring $R$ is commutative if for every $x \in R$ the element $x^{2}-x$ belongs to the centre of $R$. [b](A. 18)[/b]
2015 IMC, 8
Consider all $26^{26}$ words of length 26 in the Latin
alphabet. Define the $\emph{weight}$ of a word as $1/(k+1)$, where $k$
is the number of letters not used in this word. Prove that the sum
of the weights of all words is $3^{75}$.
Proposed by Fedor Petrov, St. Petersburg State University
2022 VTRMC, 3
Find all positive integers $a, b, c, d,$ and $n$ satisfying $n^a + n^b + n^c = n^d$ and prove that these are the only such solutions.
1977 Putnam, A6
Let $f(x,y)$ be a continuous function on the square $$S=\{(x,y):0\leq x\leq 1, 0\leq y\leq 1\}.$$ For each point $(a,b)$ in the interior of $S$, let $S_{(a,b)}$ be the largest square that is contained in $S$, is centered at $(a,b)$, and has sides parallel to those of $S$. If the double integral $\int \int f(x,y) dx dy$ is zero when taken over each square $S_{(a,b)}$, must $f(x,y)$ be identically zero on $S$?
ICMC 8, 4
Let a chain denote a row of positive integers which continue infinitely in both directions, such that for each number $n$, the $n$ numbers directly to the left of $n$ yield $n$ distinct remainders upon division by $n$.
(a) If a chain has a maximum integer, what are the possible values of that integer?
(b) Does there exist a chain which does not have a maximum integer?
ICMC 2, 1
Observe that, in the usual chessboard colouring of the two-dimensional grid, each square has 4 of its 8 neighbours black and 4 white. Does there exist a way to colour the three-dimensional grid such that each cube has half of its 26 neighbours black and half white? Is this possible in four dimensions?
2021 Alibaba Global Math Competition, 4
Let $(\Omega, \mathcal{A},\mathbb{P})$ be a standard probability space, and $\mathcal{X}$ be the set of all bounded random variables. For $t>0$, defined the mapping $R_t$ by
\[R_t(X)=t\log \mathbb{E}[\exp(X/t)], \quad X \in \mathcal{X},\]
and for $t \in (0,1)$ define the mapping $Q_t$ by
\[Q_t(X)=\inf\{x \in \mathbb{R}: \mathbb{P}(X>x) \le t\}, \quad X \in \mathcal{X}.\]
For two mappings $f,g:\mathcal{X} \to [-\infty,\infty)$, defined the operator $\square$ by
\[f\square g(X)=\inf\{f(Y)+g(X-Y): Y \in \mathcal{X}\}, \quad X \in \mathcal{X}.\]
(a) Show that, for $t,s>0$,
\[R_t \square R_s=R_{t+s}.\]
(b) Show that, for $t,s>0$ with $t+s<1$,
\[Q_t \square Q_s=Q_{t+s}.\]