This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 876

ICMC 6, 1
Tags: college contests , combinatorial geometry , ICMC
Two straight lines divide a square of side length $1$ into four regions. Show that at least one of the regions has a perimeter greater than or equal to $2$. [i]Proposed by Dylan Toh[/i]

1961 Miklós Schweitzer, 5
Tags: college contests
[b]5.[/b] Determine the functions $G$ defined on the set of all non-zero real numbers the values of which are regular matrices of order $2$, and the functions $f$ mapping the two-dimensional real vector space $E_2$ into itself, such that for any vector $y \in E_2$ and for any regular matrix $X$ of order $2$, $f(X_y)= G(det X)Xf(y)$ ($det X $ denotes the determinant of $X$).[b](A. 5)[/b]

2016 SEEMOUS, Problem 4
Tags: college contests
SEEMOUS 2016 COMPETITION PROBLEMS

2004 Miklós Schweitzer, 10
Tags: college contests , Miklos Schweitzer , probability , normal distribution
Let $\mathcal{N}_p$ stand for a $p$ dimensional random variable of standard normal distribution. For $a\in\mathbb{R}^p$, let $H_p(a)$ stand for the expectation $E|\mathcal{N}_p+a|$. For $p>1$, prove that $$H_p(a)=(p-1)\int_0^{\infty} H_1\left( \frac{|a|}{\sqrt{r^2+1}}\right) \frac{r^{p-2}}{\sqrt{(r^2+1)^p}} \mathrm{d}r$$

2018 Brazil Undergrad MO, 4
Tags: Sets , college contests , Brazilian Undergrad MO
Consider the property that each a element of a group $G$ satisfies $a ^ 2 = e$, where e is the identity element of the group. Which of the following statements is not always valid for a group $G$ with this property? (a) $G$ is commutative (b) $G$ has infinite or even order (c) $G$ is Noetherian (d) $G$ is vector space over $\mathbb{Z}_2$

2017 VJIMC, 3
Tags: college contests , geometry
Let $P$ be a convex polyhedron. Jaroslav writes a non-negative real number to every vertex of $P$ in such a way that the sum of these numbers is $1$. Afterwards, to every edge he writes the product of the numbers at the two endpoints of that edge. Prove that the sum of the numbers at the edges is at most $\frac{3}{8}$.

2011 IMC, 1
Tags: induction , IMC , college contests
Let $(a_n)\subset (\frac{1}{2},1)$. Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$. Is this sequence convergent? If yes find the limit.

2016 Korea USCM, 3
Tags: linear algebra , matrix , trace , invertible , college contests
Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries. (1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.) (2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.

1995 Putnam, 3
Tags: Putnam , linear algebra , matrix , function , college contests
To each number with $n^2$ digits, we associate the $n\times n$ determinant of the matrix obtained by writing the digits of the number in order along the rows. For example : $8617\mapsto \det \left(\begin{matrix}{\;8}& 6\;\\ \;1 &{ 7\;}\end{matrix}\right)=50$. Find, as a function of $n$, the sum of all the determinants associated with $n^2$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n = 2$, there are $9000$ determinants.)

1959 Miklós Schweitzer, 2
Tags: geometry , college contests , topology , real analysis
[b]2.[/b] Omit the vertices of a closed rectangle; the configuration obtained in such a way will be called a reduced rectangle. Prove tha the set-union of any system of reduced rectangles with parallel sides is equal to the union of countably many elements of the system. [b](St. 3)[/b]

1986 Miklós Schweitzer, 1
Tags: Miklos Schweitzer , college contests , set theory
If $(A, <)$ is a partially ordered set, its dimension, $\dim (A, <)$, is the least cardinal $\kappa$ such that there exist $\kappa$ total orderings $\{ <_{\alpha} \colon \alpha < \kappa \}$ on $A$ with $<=\cap_{\alpha < \kappa} <_\alpha$. Show that if $\dim (A, <)>\aleph_0$, then there exist disjoint $A_0, A_1\subseteq A$ with $\dim (A_0, <)$, $\dim (A_1, <)>\aleph_0$. [D. Kelly, A. Hajnal, B. Weiss]

1956 Miklós Schweitzer, 2
Tags: complex numbers , college contests
[b]2.[/b] Find the minimum of $max ( |1+z|, |1+z^{2}|)$ if $z$ runs over all complex numbers. [b](F. 2)[/b]

2017 Korea USCM, 2
Tags: polynomial , linear algebra , algebra , college contests
Show that any real coefficient polynomial $f(x,y)$ is a linear combination of polynomials of the form $(x+ay)^k$. ($a$ is a real number and $k$ is a non-negative integer.)

2015 IMC, 10
Tags: college contests , Polynomials , Inequality , IMC2015
Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$ \max_{0\le x\le1} \big|p(x)\big| &gt; \frac1{e^n}. $$ Proposed by Géza Kós, Eötvös University, Budapest

1996 Miklós Schweitzer, 6
Tags: real analysis , college contests , Miklos Schweitzer
Let $\{a_n\}$ be a bounded real sequence. (a) Prove that if X is a positive-measure subset of $\mathbb R$, then for almost all $x\in X$, there exist a subsequence $\{y_n\}$ of X such that $$\sum_{n=1}^\infty (n(y_n-x)-a_n)=1$$ (b) construct an unbounded sequence $\{a_n\}$ for which the above equation is also true.

1994 IMC, 5
Tags: vector , induction , IMC , college contests
[b]problem 5.[/b] Let $x_1, x_2,\ldots, x_k$ be vectors of $m$-dimensional Euclidean space, such that $x_1+x_2+\ldots + x_k=0$. Show that there exists a permutation $\pi$ of the integers $\{ 1, 2, \ldots, k \}$ such that: $$\left\lVert \sum_{i=1}^n x_{\pi (i)}\right\rVert \leq \left( \sum_{i=1}^k \lVert x_i \rVert ^2\right)^{1/2}$$for each $n=1, 2, \ldots, k$. Note that $\lVert \cdot \rVert$ denotes the Euclidean norm. (18 points).

2009 IMC, 1
Tags: IMC , college contests
Suppose that $f,g:\mathbb{R}\to \mathbb{R}$ satisfying \[ f(r)\le g(r)\quad \forall r\in \mathbb{Q} \] Does this imply $f(x)\le g(x)\quad \forall x\in \mathbb{R}$ if [list] (a)$f$ and $g$ are non-decreasing ? (b)$f$ and $g$ are continuous?[/list]

2022 District Olympiad, P3
Tags: Sequence , limit , college contests , romania
Let $(x_n)_{n\geq 1}$ be the sequence defined recursively as such: \[x_1=1, \ x_{n+1}=\frac{x_1}{n+1}+\frac{x_2}{n+2}+\cdots+\frac{x_n}{2n} \ \forall n\geq 1.\]Consider the sequence $(y_n)_{n\geq 1}$ such that $y_n=(x_1^2+x_2^2+\cdots x_n^2)/n$ for all $n\geq 1.$ Prove that [list=a] [*]$x_{n+1}^2<y_n/2$ and $y_{n+1}<(2n+1)/(2n+2)\cdot y_n$ for all $n\geq 1;$ [*]$\lim_{n\to\infty}x_n=0.$ [/list]

2015 VJIMC, 4
Tags: polynomial , college contests , number theory
[b]Problem 4 [/b] Let $m$ be a positive integer and let $p$ be a prime divisor of $m$. Suppose that the complex polynomial $a_0 + a_1x + \ldots + a_nx^n$ with $n < \frac{p}{p-1}\varphi(m)$ and $a_n \neq 0$ is divisible by the cyclotomic polynomial $\phi_m(x)$. Prove that there are at least $p$ nonzero coefficients $a_i\ .$ The cyclotomic polynomial $\phi_m(x)$ is the monic polynomial whose roots are the $m$-th primitive complex roots of unity. Euler’s totient function $\varphi(m)$ denotes the number of positive integers less than or equal to $m$ which are coprime to $m$.

2014 Contests, 2
Tags: Putnam , linear algebra , matrix , induction , college contests , Putnam 2014 , Putnam matrices
Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$

2018 Brazil Undergrad MO, 9
Tags: function , Brazilian Math Olympiad , college contests , algebra
How many functions $f: \left\{1,2,3\right\} \to \left\{1,2,3 \right\}$ satisfy $f(f(x))=f(f(f(x)))$ for every $ x $?

2018 Brazil Undergrad MO, 6
Tags: geometry , college contests , Brazilian Math Olympiad
Given an equilateral triangle $ABC$ in the plane, how many points $P$ in the plane are such that the three triangles $AP B, BP C $ and $CP A$ are isosceles and not degenerate?

1976 Putnam, 2
Tags: college contests
Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\dots B^{n_k},$ where $n_1,\dots n_k$ are any integers, and $A^0=B^0=1$ as usual. Also suppose that $A^4=B^7=ABA^{-1}B=1, A^2\neq 1,$ and $B\neq 1.$ (a) How many elements of $G$ are of the form $C^2$ with $C$ in $G$? (b) Write each such square as a word in $A$ and $B.$

2006 IMC, 2
Tags: function , IMC , college contests
Find all functions $f: \mathbb{R}\to{R}$ such that for any $a<b$, $f([a,b])$ is an interval of length $b-a$

2012 IMC, 2
Tags: limit , IMC , college contests
Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$, $a_1=\frac{1}{2}$, and \[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\] Show that the series $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value. [i]Proposed by Christophe Debry, KU Leuven, Belgium.[/i]