Found problems: 876
2007 Miklós Schweitzer, 5
Let $D=\{ (x,y) \mid x>0, y\neq 0\}$ and let $u\in C^1(\overline {D})$ be a bounded function that is harmonic on $D$ and for which $u=0$ on the $y$-axis. Prove that $u$ is identically zero.
(translated by Miklós Maróti)
2003 Putnam, 2
Let $a_1, a_2, \cdots , a_n$ and $b_1, b_2,\cdots, b_n$ be nonnegative real numbers. Show that \[(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}\]
1961 Miklós Schweitzer, 1
[b]1.[/b] Let $a$ ( $\neq e$, the unit element) be an element of finite order of a group $G$ and let $t$ ($\geq 2$) be a positive integer. Show: if the complex $A= \{ e,a,a^2, \dots , a^{t-1} \} $ is not a group, then for every positive integer $k$( $2 \leq k \leq t$) the complex $B= \{ e. a^k, a^{2k}, \dots , a^{(t-1)k} \} $ differs from $A$. [b](A. 16)[/b]
2021 Alibaba Global Math Competition, 15
Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that
\[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\]
Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.
2013 IMC, 3
There are $\displaystyle{2n}$ students in a school $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$. Each week $\displaystyle{n}$ students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?
[i]Proposed by Oleksandr Rybak, Kiev, Ukraine.[/i]
2016 Korea USCM, 8
For a $n\times n$ complex valued matrix $A$, show that the following two conditions are equivalent.
(i) There exists a $n\times n$ complex valued matrix $B$ such that $AB-BA=A$.
(ii) There exists a positive integer $k$ such that $A^k = O$. ($O$ is the zero matrix.)
2024 SEEMOUS, P1
Let $(x_n)_{n\geq 1}$ be the sequence defined by $x_1\in (0,1)$ and $x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}$ for all $n\geq 1$. Find the values of $\alpha\in\mathbb{R}$ for which the series $\sum_{n=1}^{\infty}x_n^{\alpha}$ is convergent.
2017 Brazil Undergrad MO, 3
Let $X = \{(x,y) \in \mathbb{R}^2 | y \geq 0, x^2+y^2 = 1\} \cup \{(x,0),-1\leq x\leq 1\} $ be the edge of the closed semicircle with radius 1.
a) Let $n>1$ be an integer and $P_1,P_2,\dots,P_n \in X$. Show that there exists a permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$ such that
\[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8\].
Where $\sigma(n+1) = \sigma(1)$.
b) Find all sets $\{P_1,P_2,\dots,P_n \} \subset X$ such that for any permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$,
\[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2 \geq 8\].
Where $\sigma(n+1) = \sigma(1)$.
2014 Paenza, 4
Let $\mathcal{C}$ be the family of circumferences in $\mathbb{R}^2$ that satisfy the following properties:
(i) if $C_n$ is the circumference with center $(n,1/2)$ and radius $1/2$, then $C_n\in \mathcal{C}$, for all $n\in \mathbb{Z}$.
(ii) if $C$ and $C'$, both in $\mathcal{C}$, are externally tangent, then the circunference externally tangent to $C$ and $C'$ and tanget to $x$-axis also belongs to $\mathcal{C}$.
(iii) $\mathcal{C}$ is the least family which these properties.
Determine the set of the real numbers which are obtained as the first coordinate of the points of intersection between the elements of $\mathcal{C}$ and the $x$-axis.
2007 IMC, 2
Let $ x$, $ y$ and $ z$ be integers such that $ S = x^{4}+y^{4}+z^{4}$ is divisible by 29. Show that $ S$ is divisible by $ 29^{4}$.
2023 SG Originals, Q3
Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started.
[i]Proposed by Dylan Toh[/i]
1960 Miklós Schweitzer, 10
[b]10.[/b] A car is used by $n$ drivers. Every morning the drivers choose by drawing that one of them who will drive the car that day. Let us define the random variable $\mu (n)$ as the least positive integer such that each driver drives at least one day during the first $\mu (n)$ days. Find the limit distribution of the random variable
$\frac {\mu (n) -n \log n}{n}$
as $n \to \infty$. [b](P. 9)[/b]
2018 IMC, 5
Let $p$ and $q$ be prime numbers with $p<q$. Suppose that in a convex polygon $P_1,P_2,…,P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that
$$P_1P_2+P_2P_3+\cdots +P_kP_{k+1}\geqslant \frac{k^3+k}{2}$$holds for every integer $k$ with $1\leqslant k\leqslant p$.
[i]Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin[/i]
2022 District Olympiad, P3
Find all values of $n\in\mathbb{N}^*$ for which \[I_n:=\int_0^\pi\cos(x)\cdot\cos(2x)\cdot\ldots\cdot\cos(nx) \ dx=0.\]
1994 Putnam, 3
Show that if the points of an isosceles right triangle of side length $1$ are each colored with one of four colors, then there must be two points of the same color which are at least a distance $2-\sqrt 2$ apart.
1960 Miklós Schweitzer, 6
[b]6.[/b] Let $\{ n_k \}_{k=1}^{\infty}$ be a stricly increasing sequence of positive integers such that
$\lim_{k \to \infty} n_k^{\frac {1}{2^k}}= \infty$
Show that the sum of the series $\sum_{k=1}^{\infty} \frac {1}{n_k} $ is an irrational number. [b](N. 19)[/b]
ICMC 5, 3
A set of points has [i]point symmetry[/i] if a reflection in some point maps the set to itself. Let $\cal P$ be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that $\cal P$ has point symmetry.
[i]Proposed by Ethan Tan[/i]
1959 Miklós Schweitzer, 6
[b]6.[/b] Let $T$ be a one-to-one mapping of the unit square $E$ of the plane into itself. Suppose that $T$ and $T^{-1}$ are measure-preserving (i.e. if $M \subseteq E$ is a measurable set, then $TM$ and $T^{-1}M$ are also measurable and $\mu (M)= \mu (TM)= \mu (T^{-1}M)$, where $\mu$ denotes the Lebesgue measure) and, furthermore, that if $Tx \in N$ for almost all points $x$ of a measurable set $N \subseteq E$, then either $n$ or $ E \setminus N$ is of measure 0.
Prove that, for any measurable set $A \subseteq E$, with $\mu (A)>0$, the function $n(x)$ defined by
$$n(x)=\begin{cases}
0, \mbox{if} \quad T^k x \notin A \quad (k=1, 2, \dots),\\
\min (k: T^k x \in A; k=1,2, \dots ) &\mbox{otherwise}
\end{cases}
$$
is measurable and
$\int_{A}n(x) d\mu(x) =1$
[b](R. 18)[/b]
2020 IMC, 4
A polynomial $p$ with real coefficients satisfies $p(x+1)-p(x)=x^{100}$ for all $x \in \mathbb{R}.$ Prove that $p(1-t) \ge p(t)$ for $0 \le t \le 1/2.$
1979 Putnam, B1
Prove or disprove: there is at least one straight line normal to the graph of $y=\cosh x$ at a point $(a,\cosh a)$ and also normal to the graph of $y=$ $\sinh x$ at a point $(c,\sinh c).$
2011 IMC, 5
Let $F=A_0A_1...A_n$ be a convex polygon in the plane. Define for all $1 \leq k \leq n-1$ the operation $f_k$ which replaces $F$ with a new polygon $f_k(F)=A_0A_1..A_{k-1}A_k^\prime A_{k+1}...A_n$ where $A_k^\prime$ is the symmetric of $A_k$ with respect to the perpendicular bisector of $A_{k-1}A_{k+1}$. Prove that $(f_1\circ f_2 \circ f_3 \circ...\circ f_{n-1})^n(F)=F$.
ICMC 8, 6
A set of points in the plane is called rigid if each point is equidistant from the three (or more) points nearest to it.
(a) Does there exist a rigid set of $9$ points?
(b) Does there exist a rigid set of $11$ points?
2006 Putnam, B2
Prove that, for every set $X=\{x_{1},x_{2},\dots,x_{n}\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
\[\left|m+\sum_{s\in S}s\right|\le\frac1{n+1}\]
2009 IMC, 3
In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from $1$ to $n$ and let $a_i$ be the number of friends of the $i^{\text{th}}$ resident. Suppose that
\[ \sum_{i=1}^{n}a_i^2=n^2-n \]
Let $k$ be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of $k.$
2013 Putnam, 4
For any continuous real-valued function $f$ defined on the interval $[0,1],$ let \[\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|.\] Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1],$ then \[\text{Var}(fg)\le 2\text{Var}(f)M(g)^2+2\text{Var}(g)M(f)^2.\]