Found problems: 876
2004 IMC, 1
Let $A$ be a real $4\times 2$ matrix and $B$ be a real $2\times 4$ matrix such that
\[ AB = \left(%
\begin{array}{cccc}
1 & 0 & -1 & 0 \\
0 & 1 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & -1 & 0 & 1 \\
\end{array}%
\right). \]
Find $BA$.
2004 Miklós Schweitzer, 8
Prove that for any $0<\delta <2\pi$ there exists a number $m>1$ such that for any positive integer $n$ and unimodular complex numbers $z_1,\ldots, z_n$ with $z_1^v+\dots+z_n^v=0$ for all integer exponents $1\le v\le m$, any arc of length $\delta$ of the unit circle contains at least one of the numbers $z_1,\ldots, z_n$.
2007 Putnam, 1
Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$
1997 Putnam, 6
For a positive integer $n$ and any real number $c$, define $x_k$ recursively by :
\[ x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1} \]
Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k,\; 1\le k\le n$.
1984 Miklós Schweitzer, 5
[b]5.[/b] Let $a_0 , a_1 , \dots $ be nonnegative real numbers such that
$\sum_{n=0}^{\infty}a_n = \infty$
For arbitrary $ c>0$, let
$n_{j}(c)= \min \left \{ k : c.j \leq \sum_{i=0}^{k} a_i \right \}$, $j= 1,2, \dots $
Prove that if $\sum_{i=0}^{\infty}a_i^2 = \infty$, then there exists a $c>0$ for which $\sum_{j=1}^{\infty} a_{n_j (c)} = \infty$ .([b]S.24[/b])
[P. Erdos, I. Joó, L. Székely]
1995 Iran MO (2nd round), 1
Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another.
(For example, $23=9+8+6.)$
1990 Putnam, B6
Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $ \overline {L} $ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K,t)$ be the band of points whose distance from $\overline{L}$ is at most $ \left( \frac {t}{2} \right) w $, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all $S$? ($K$ runs over all lines in the plane.)
1958 Miklós Schweitzer, 8
[b]8.[/b] Let the function $f(x)$ be periodic with the period $1$, non-negative, concave in the interval $(0,1)$ and continuous at the point $0$. Prove that $f(nx)\leq nf(x)$ for every real $x$ and positive integer $n$. [b](R. 6)[/b]
2014 IMC, 1
Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$.
(Proposed by Stephan Wagner, Stellenbosch University)
2002 Putnam, 3
Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove
that $T_n - n$ is always even.
2003 Putnam, 4
Let $f(z) = az^4+ bz^3+ cz^2+ dz + e = a(z -r_1)(z -r_2)(z -r_3)(z -r_4)$ where $a, b, c, d, e$ are integers, $a \not= 0$. Show that if $r_1 + r_2$ is a rational number, and if $r_1 + r_2 \neq r_3 + r_4$, then $r_1r_2$ is a rational number.
2014 IMC, 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)
2012 Putnam, 2
Let $*$ be a commutative and associative binary operation on a set $S.$ Assume that for every $x$ and $y$ in $S,$ there exists $z$ in $S$ such that $x*z=y.$ (This $z$ may depend on $x$ and $y.$) Show that if $a,b,c$ are in $S$ and $a*c=b*c,$ then $a=b.$
1984 Miklós Schweitzer, 3
[b]3.[/b] Let $a$ and $b$ be positive integers such that when dividing them by any prime $p$, the remainder of $a$ is always less than or equal to the remainder of $b$. Prove that $a=b$. ([b]N.16[/b])
[P. Erdos, P. P. Pálify]
2018 Korea USCM, 4
$n\geq 2$ is a given integer. For two permuations $(\alpha_1,\cdots,\alpha_n)$ and $(\beta_1,\cdots,\beta_n)$ of $1,\cdots,n$, consider $n\times n$ matrix $A= \left(a_{ij} \right)_{1\leq i,j\leq n}$ defined by $a_{ij} = (1+\alpha_i \beta_j )^{n-1}$. Find every possible value of $\det(A)$.
1976 Putnam, 1
Evaluate $$lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} ([\frac{2n}{k}] -2[\frac{n}{k}])$$ and express your answer in the form $\log a-b,$ with $a$ and $b$ positive integers.
Here $[x]$ is defined to be the integer such that $[x] \leq x <[x]+1$ and $\log x$ is the logarithm of $x$ to base $e.$
2000 Miklós Schweitzer, 8
Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a map such that the image of every compact set is compact, and the image of every connected set is connected. Prove that $f$ is continuous.
2002 Miklós Schweitzer, 5
Denote by $\lambda (H)$ the Lebesgue outer measure of $H\subseteq \left[ 0,1\right]$. The horizontal and vertical sections of the set $A\subseteq [0, 1]\times [ 0, 1]$ are denoted by $A^y$ and $A_x$ respectively; that is, $A^y=\{ x\in [ 0, 1] \colon (x, y) \in A\}$ and $A_x=\{ y\in [ 0, 1]\colon (x,y)\in A\}$ for all $x,y\in [0,1]$.
(a) Is there a decomposition $A\cup B$ of the unit square $[0,1]\times [0,1]$ such that $A^y$ is the union of finitely many segments of total length less than $\frac12$ and $\lambda (B_x)\le \frac12$ for all $x, y\in [0,1]$?
(b) Is there a decomposition $A\cup B$ of the unit square $[0,1] \times [0,1]$ such that $A^y$ is the union of finitely many segments of total length not greater than $\frac12$ and $\lambda (B_x)<\frac12$ for all $x,y\in [0,1]$?
2003 Miklós Schweitzer, 10
Let $X$ and $Y$ be independent random variables with "Saint-Petersburg" distribution, i.e. for any $k=1,2,\ldots$ their value is $2^k$ with probability $\frac{1}{2^k}$. Show that $X$ and $Y$ can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables $(X', Y')$ on this space with the property that $X+Y=2X'+Y'I(Y'\le X')$ almost surely (here $I(A)$ denotes the indicator function of the event $A$).
(translated by L. Erdős)
2018 Brazil Undergrad MO, 20
What is the largest number of points that can exist on a plane so that each distance between any two of them is an odd integer?
2004 IMC, 5
Prove that
\[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]
1957 Miklós Schweitzer, 8
[b]8.[/b] Find all integers $a>1$ for which the least (integer) solution $n$ of the congruence $a^{n} \equiv 1 \pmod{p}$ differs from 6 (p is any prime number). [b](N. 9)[/b]
2020 Miklós Schweitzer, 2
Prove that if $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous periodic function and $\alpha \in \mathbb{R}$ is irrational, then the sequence $\{n\alpha+f(n\alpha)\}_{n=1}^{\infty}$ modulo 1 is dense in $[0,1]$.
2003 Putnam, 1
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.
2012 Miklós Schweitzer, 10
Let $K$ be a knot in the $3$-dimensional space (that is a differentiable injection of a circle into $\mathbb{R}^3$, and $D$ be the diagram of the knot (that is the projection of it to a plane so that in exception of the transversal double points it is also an injection of a circle). Let us color the complement of $D$ in black and color the diagram $D$ in a chessboard pattern, black and white so that zones which share an arc in common are coloured differently. Define $\Gamma_B(D)$ the black graph of the diagram, a graph which has vertices in the black areas and every two vertices which correspond to black areas which have a common point have an edge connecting them.
[list=a]
[*]Determine all knots having a diagram $D$ such that $\Gamma_B(D)$ has at most $3$ spanning trees. (Two knots are not considered distinct if they can be moved into each other with a one parameter set of the injection of the circle)[/*]
[*]Prove that for any knot and any diagram $D$, $\Gamma_B(D)$ has an odd number of spanning trees.[/*]
[/list]