Found problems: 876
2005 Putnam, B5
Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that
(a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically)
and that
(b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$
Show that $P=0$ identically.
2007 IMC, 4
Let $ G$ be a finite group. For arbitrary sets $ U, V, W \subset G$, denote by $ N_{UVW}$ the number of triples $ (x, y, z) \in U \times V \times W$ for which $ xyz$ is the unity .
Suppose that $ G$ is partitioned into three sets $ A, B$ and $ C$ (i.e. sets $ A, B, C$ are pairwise disjoint and $ G = A \cup B \cup C$). Prove that $ N_{ABC}= N_{CBA}.$
2009 IMC, 1
Let $\ell$ be a line and $P$ be a point in $\mathbb{R}^3$. Let $S$ be the set of points $X$ such that the distance from $X$ to $\ell$ is greater than or equal to two times the distance from $X$ to $P$. If the distance from $P$ to $\ell$ is $d>0$, find $\text{Volume}(S)$.
1958 Miklós Schweitzer, 9
[b]9.[/b] Show that if $f(z) = 1+a_1 z+a_2z^2+\dots$ for $\mid z \mid\leq 1$ and
$\frac{1}{2\pi}\int_{0}^{2\pi}\mid f(e^{i\phi}) \mid^{2} d\phi < \left (1+\frac{\mid a_1\mid ^2} {4} \right )^2$,
then $f(z)$ has a root in the disc $\mid z \mid \leq 1$.[b](F. 4)[/b]
2013 Putnam, 1
Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.
2023 Olimphíada, 1
Let $n \geq 2023$ be an integer. For each real $x$, we say that $\lfloor x \rceil$ is the closest integer to $x$, and if there are two closest integers then it is the greater of the two. Suppose there is a positive real $a$ such that $$\lfloor an \rceil = n + \bigg\lfloor\frac{n}{a} \bigg\rceil.$$
Show that $|a^2 - a - 1| < \frac{n\varphi+1}{n^2}$.
1979 Putnam, A3
Let $x_1,x_2,x_3, \dots$ be a sequence of nonzero real numbers satisfying $$x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}} \text{ for } n=3,4,5, \dots.$$ Establish necessary and sufficient conditions on $x_1$ and $x_2$ for $x_n$ to be an integer for infinitely many values of $n.$
2008 Miklós Schweitzer, 3
A bipartite graph on the sets $\{ x_1,\ldots, x_n \}$ and $\{ y_1,\ldots, y_n\}$ of vertices (that is the edges are of the form $x_iy_j$) is called tame if it has no $x_iy_jx_ky_l$ path ($i,j,k,l\in\{ 1,\ldots, n\}$) where $j<l$ and $i+j>k+l$. Calculate the infimum of those real numbers $\alpha$ for which there exists a constant $c=c(\alpha)>0$ such that for all tame graphs $e\le cn^{\alpha}$, where $e$ is the number of edges and $n$ is half of the number of vertices.
(translated by Miklós Maróti)
2014 Putnam, 6
Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$
2002 Miklós Schweitzer, 6
Let $K\subseteq \mathbb{R}$ be compact. Prove that the following two statements are equivalent to each other.
(a) For each point $x$ of $K$ we can assign an uncountable set $F_x\subseteq \mathbb{R}$ such that
$$\mathrm{dist}(F_x, F_y)\ge |x-y|$$
holds for all $x,y\in K$;
(b) $K$ is of measure zero.
1960 Miklós Schweitzer, 9
[b]9.[/b] Let $A_1, \dots , A_n$ and $B$ be ideals of an assoticative ring $R$ such that $B$ is contained in the set-union of the ideals $A_i$($i=1, \dots , n$) but not contained in the union of any $n-1$ of the ideals $A_i$. Show that, for some positive integer $k$, $B_k$ is contained in the intersection of the ideals $A_i$. [b](A. 19)[/b]
2011 IMC, 2
An alien race has three genders: male, female and emale. A married triple consists of three persons, one from each gender who all like each other. Any person is allowed to belong to at most one married triple. The feelings are always mutual ( if $x$ likes $y$ then $y$ likes $x$).
The race wants to colonize a planet and sends $n$ males, $n$ females and $n$ emales. Every expedition member likes at least $k$ persons of each of the two other genders. The problem is to create as many married triples so that the colony could grow.
a) Prove that if $n$ is even and $k\geq 1/2$ then there might be no married triple.
b) Prove that if $k \geq 3n/4$ then there can be formed $n$ married triple ( i.e. everybody is in a triple).
1955 Miklós Schweitzer, 3
[b]3.[/b] Let the density function $f(x)$ of the random variable $\xi$ bean even function; let further $f(x)$ be monotonically non-increasing for $x>0$. Suppose that
$D^{2}= \int_{-\infty }^{\infty }x^{2}f(x) dx$
exists. Prove that for every non negative $\lambda $
$P(\left |\xi \right |\geq \lambda D)\leq \frac{1}{1+\lambda ^{2}}$. [b](P. 7)[/b]
2010 Putnam, A4
Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.
2021 Alibaba Global Math Competition, 6
Let $M(t)$ be measurable and locally bounded function, that is,
\[M(t) \le C_{a,b}, \quad \forall 0 \le a \le t \le b<\infty\]
with some constant $C_{a,b}$, from $[0,\infty)$ to $[0,\infty)$ such that
\[M(t) \le 1+\int_0^t M(t-s)(1+t)^{-1}s^{-1/2} ds, \quad \forall t \ge 0.\]
Show that
\[M(t) \le 10+2\sqrt{5}, \quad \forall t \ge 0.\]
2006 IMC, 1
Let $f: \mathbb{R}\to \mathbb{R}$ be a real function. Prove or disprove each of the following statements.
(a) If f is continuous and range(f)=$\mathbb{R}$ then f is monotonic
(b) If f is monotonic and range(f)=$\mathbb{R}$ then f is continuous
(c) If f is monotonic and f is continuous then range(f)=$\mathbb{R}$
1990 Putnam, A1
Let \[T_0=2, T_1=3, T_2=6,\] and for $n\ge 3$, \[T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.\] The first few terms are \[2, 3, 6, 14, 40, 152, 784, 5158, 40576, 363392.\] Find a formula for $T_n$ of the form \[T_n=A_n+B_n,\] where $\{A_n\}$ and $\{B_n\}$ are well known sequences.
2003 IMC, 3
Let $A$ be a closed subset of $\mathbb{R}^{n}$ and let $B$ be the set of all those points $b \in \mathbb{R}^{n}$ for which there exists exactly one point $a_{0}\in A $ such that $|a_{0}-b|= \inf_{a\in A}|a-b|$.
Prove that $B$ is dense in $\mathbb{R}^{n}$; that is, the closure of $B$ is $\mathbb{R}^{n}$
2010 Putnam, A5
Let $G$ be a group, with operation $*$. Suppose that
(i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);
(ii) For each $\mathbf{a},\mathbf{b}\in G,$ either $\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3.$
Prove that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ for all $\mathbf{a},\mathbf{b}\in G.$
1995 Putnam, 5
Let $x_1,x_2,\cdots, x_n$ be real valued differentiable functions of a variable $t$ which satisfy
\begin{align*}
& \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\
& \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\
& \;\qquad \vdots \\
& \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\
\end{align*}
For some constants $a_{ij}>0$. Suppose that $\lim_{t \to \infty}x_i(t)=0$ for all $1\le i \le n$. Are the functions $x_i$ necessarily linearly dependent?
2014 Contests, 3
Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial
$$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$
has $n$ distinct real roots.
(Proposed by Stephan Neupert, TUM, München)
2017 Romania National Olympiad, 3
Let $G$ be a finite group with the following property:
If $f$ is an automorphism of $G$, then there exists $m\in\mathbb{N^\star}$, so that $f(x)=x^{m} $ for all $x\in G$.
Prove that G is commutative.
[i]Marian Andronache[/i]
MIPT student olimpiad spring 2024, 1
Find integral:
$\int_{x^2+y^2\leq 1}e^xcos(y)dxdy$
1955 Miklós Schweitzer, 10
[b]10.[/b] Show that if a convex polyhedron has vertices of regular distribution and congruent faces, then it is regular. (A system of points is said to be of regular distribution if every point of the system can be transformed into any other point by congruent transformations mapping the system onto itself.) [b](G. 11)[/b]
1997 Putnam, 5
Let $N_k$ denote number of ordered $n$-tuples of positive integers $(a_1,a_2, \cdots ,a_k)$ such that
\[ \frac{1}{a_1}+\frac{1}{a_2}+\ldots +\frac{1}{a_k}=1 \]
Determine $N_{10}$ is odd or even.