Found problems: 876
2014 Contests, 2
There are $n$ cards numbered and stacked in increasing order from up to down (i.e. the card in the top is the number 1, the second is the 2, and so on...). With this deck, the next steps are followed:
-the first card (from the top) is put in the bottom of the deck.
-the second card (from the top) is taken away of the deck.
-the third card (from the top) is put in the bottom of the deck.
-the fourth card (from the top) is taken away of the deck.
- ...
The proccess goes on always the same way: the card in the top is put at the end of the deck and the next is taken away of the deck, until just one card is left. Determine which is that card.
1985 Miklós Schweitzer, 3
[b]3.[/b] Let $k$ and $K$ be concentric circles on the plane, and let $k$ be contained inside $K$. Assume that $k$ is covered by a finite system of convex angular domains with vertices on $K$. Prove that the sum of the angles of the domains is not less than the angle under which $k$ can be seen from a point of $K$. ([b]G.38[/b])
[Zs.. Páles]
1985 Miklós Schweitzer, 12
Let $(\Omega, \mathcal A, P)$ be a probability space, and let $(X_n, \mathcal F_n)$ be an adapted sequence in $(\Omega, \mathcal A, P)$ (that is, for the $\sigma$-algebras $\mathcal F_n$, we have $\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A$, and for all $n$, $X_n$ is an $\mathcal F_n$-measurable and integrable random variable). Assume that
$$\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\,\, (n=2, 3, \ldots )$$
Prove that $\mathrm{sup}_n \mathrm{E}|X_n|<\infty$ implies that $X_n$ converges with probability one as $n\to\infty$. [I. Fazekas]
2022 IMC, 5
We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either
red or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments.
A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour.
Suppose that there are $2022$ blue monochromatic triangles. How many red monochromatic triangles
are there?
1994 Putnam, 4
Let $A$ and $B$ be $2\times 2$ matrices with integer entries such that $A, A+B, A+2B, A+3B,$ and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.
1957 Miklós Schweitzer, 6
[b]6.[/b] Let $f(x)$ be an arbitrary function, differentiable infinitely many times. Then the $n$th derivative of $f(e^{x})$ has the form
$\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$ ($n=0,1,2,\dots$).
From the coefficients $a_{kn}$ compose the sequence of polynomials
$P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}$ ($n=0,1,2,\dots$)
and find a closed form for the function
$F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$
[b](S. 22)[/b]
1955 Miklós Schweitzer, 2
[b]2.[/b] Let $f_{1}(x), \dots , f_{n}(x)$ be Lebesgue integrable functions on $[0,1]$, with $\int_{0}^{1}f_{1}(x) dx= 0$ $ (i=1,\dots ,n)$. Show that, for every $\alpha \in (0,1)$, there existis a subset $E$ of $[0,1]$ with measure $\alpha$, such that $\int_{E}f_{i}(x)dx=0$. [b](R. 17)[/b]
1954 Miklós Schweitzer, 5
[b]5.[/b] Let $\xi _{1},\xi _{2},\dots ,\xi _{n},... $ be independent random variables of uniform distribution in $(0,1)$. Show that the distribution of the random variable
$\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)$
tends to a limit distribution for $n \to \infty $. [b](P. 6)[/b]
2002 Miklós Schweitzer, 8
Prove that there exists an absolute constant $c$ such that any set $H$ of $n$ points of the plane in general position can be coloured with $c\log n$ colours in such a way that any disk of the plane containing at least one point of $H$ intersects some colour class of $H$ in exactly one point.
1956 Miklós Schweitzer, 7
[b]7.[/b] Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that, with some positive number $C$,
$\sum_{k=1}^{n}k\mid a_k \mid<n C$ ($n=1,2, \dots $)
Putting $s_n= a_0 +a_1+\dots+a_n$, suppose that
$\lim_{n \to \infty }(\frac{s_{0}+s_{1}+\dots+s_n}{n+1})= s$
exists. Prove that
$\lim_{n \to \infty }(\frac{s_{0}^2+s_{1}^2+\dots+s_n^2}{n+1})= s^2$
[b](S. 7)[/b]
2021 Alibaba Global Math Competition, 5
Suppose that $A$ is a finite subset of $\mathbb{R}^d$ such that
(a) every three distinct points in $A$ contain two points that are exactly at unit distance apart, and
(b) the Euclidean norm of every point $v$ in $A$ satisfies
\[\sqrt{\frac{1}{2}-\frac{1}{2\vert A\vert}} \le \|v\| \le \sqrt{\frac{1}{2}+\frac{1}{2\vert A\vert}}.\]
Prove that the cardinality of $A$ is at most $2d+4$.
2000 Miklós Schweitzer, 6
Suppose the real line is decomposed into two uncountable Borel sets. Prove that a suitable translated copy of the first set intersects the second in an uncountable set.
2012 IMC, 5
Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients.
[i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]
2005 IMC, 3
What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$, we have $ \text{trace}\left(AB\right) \equal{} 0$ ?
2016 Miklós Schweitzer, 10
Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?
1995 Putnam, 1
Let $S$ be a set of real numbers which is closed under multiplication (that is $a,b\in S\implies ab\in S$). Let $T,U\subset S$ such that $T\cap U=\emptyset, T\cup U=S$. Given that for any three elements $a,b,c$ in $T$, not necessarily distinct, we have $abc\in T$ and also if $a,b,c\in U$, not necessarily distinct then $abc\in U$. Show at least one of $T$ and $U$ is closed under multiplication.
2022 IMC, 4
Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of
$\{1, 2, \ldots , n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever
$1\leq a<b<c<d \leq n$, the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$.
1985 Miklós Schweitzer, 9
Let $D=\{ z\in \mathbb C\colon |z|<1\}$ and $D=\{ w\in \mathbb C \colon |w|=1\}$. Prove that if for a function $f\colon D\times B\rightarrow\mathbb C$ the equality
$$f\left( \frac{az+b}{\overline{b}z+\overline{a}}, \frac{aw+b}{\overline{b}w+\overline a} \right)=f(z,w)+f\left(\frac{b}{\overline a}, \frac{aw+b}{\overline b w+\overline a} \right)$$
holds for all $z\in D, w\in B$ and $a, b\in \mathbb C,|a|^2=|b|^2+1$, then there is a function $L\colon (0, \infty )\rightarrow \mathbb C$ satisfying
$$L(pq)=L(p)+L(q)\,\,\,\text{for all}\,\,\, p,q > 0$$
such that $f$ can be represented as
$$f(z,w)=L\left( \frac{1-|z|^2}{|w-z|^2}\right)\,\,\,\text{for all}\,\,\, z\in D, w\in B$$.
[Gy. Maksa]
2008 IMC, 3
Let $ n$ be a positive integer. Prove that $ 2^{n\minus{}1}$ divides
\[ \sum_{0\leq k < n/2} \binom{n}{2k\plus{}1}5^k.\]
1956 Miklós Schweitzer, 9
[b]9.[/b] Show that if the trigonometric polynomial $f(\theta)= \sum_{v=1}^{n} a_v \cos v\theta$ monotonically decreases over the closed interval $[0,\pi]$, then the trigonometric polynomial $g(\theta)=\sum_{v=1}^{n}a_v \sin v\theta$ is non negative in the same interval. [b](S. 26)[/b]
2016 VJIMC, 4
Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying
$$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$
for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e.
$$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$
2015 Miklos Schweitzer, 10
Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable,strictly convex function.Let $H$ be a Hilbert space and $A,B$ be bounded,self adjoint linear operators on $H$.Prove that,if $f(A)-f(B)=f'(B)(A-B)$ then $A=B$.
2006 Putnam, A3
Let $1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots$ be a sequence defined by $x_{k}=k$ for $k=1,2\dots,2006$ and $x_{k+1}=x_{k}+x_{k-2005}$ for $k\ge 2006.$ Show that the sequence has 2005 consecutive terms each divisible by 2006.
2024 Brazil Undergrad MO, 3
Consider a game on an \( n \times n \) board, where each square starts with exactly one stone. A move consists of choosing $5$ consecutive squares in the same row or column of the board and toggling the state of each of those squares (removing the stone from squares with a stone and placing a stone in squares without a stone). For which positive integers \( n \geq 5 \) is it possible to end up with exactly one stone on the board after a finite number of moves?
1960 Miklós Schweitzer, 2
[b]2.[/b] Construct a sequence $(a_n)_{n=1}^{\infty}$ of complex numbers such that, for every $l>0$, the series
$\sum_{n=1}^{\infty} \mid a_n \mid ^{l}$
be divergent, but for almost all $\theta$ in $(0,2\pi)$,
$\prod_{n=1}^{\infty} (1+a_n e^{i\theta})$
be convergent. [b](S. 11)[/b]