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.

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Found problems: 876

1984 Miklós Schweitzer, 8
Tags: college contests , geometry , parallelogram
[b]8.[/b] Among all point lattices on the plane intersecting every closed convex region of unit width, which on's fundamental parallelogram has the largest area? ([b]G.36[/b]) [L. Fejes-Tóth]

1994 Putnam, 5
Tags: Putnam , limit , college contests
Let $(r_n)_{n\ge 0}$ be a sequence of positive real numbers such that $\lim_{n\to \infty} r_n = 0$. Let $S$ be the set of numbers representable as a sum \[ r_{i_1} + r_{i_2} +\cdots + r_{i_{1994}} ,\] with $i_1 < i_2 < \cdots< i_{1994}.$ Show that every nonempty interval $(a, b)$ contains a nonempty subinterval $(c, d)$ that does not intersect $S$.

1994 Putnam, 6
Tags: Putnam , function , college contests
Let $f_1,f_2,\cdots ,f_{10}$ be bijections on $\mathbb{Z}$ such that for each integer $n$, there is some composition $f_{\ell_1}\circ f_{\ell_2}\circ \cdots \circ f_{\ell_m}$ (allowing repetitions) which maps $0$ to $n$. Consider the set of $1024$ functions \[ \mathcal{F}=\{f_1^{\epsilon_1}\circ f_2^{\epsilon_2}\circ \cdots \circ f_{10}^{\epsilon_{10}}\} \] where $\epsilon _i=0$ or $1$ for $1\le i\le 10.\; (f_i^{0}$ is the identity function and $f_i^1=f_i)$. Show that if $A$ is a finite set of integers then at most $512$ of the functions in $\mathcal{F}$ map $A$ into itself.

1957 Miklós Schweitzer, 4
Tags: college contests
[b]4.[/b] Let $F_{\epsilon} (0<\epsilon<1)$ denote the class of non-negative piecewise continuous functions defined on $[0,\infty)$ which satisfy the following condition: $f(x)f(y)\leq \epsilon^{\mid x-y\mid} (x,y \geq 0)$. Find the value of $s_{\epsilon}= \sup_{f\in F_{\epsilon}} \int_{0}^{\infty} f(x) dx$ [b](R. 5)[/b]

1956 Miklós Schweitzer, 10
Tags: college contests
[b]10.[/b] In an urn there are balls of $N$ different colours, $n$ balls of each colour. Balls are drawn and not replaced until one of the colours turns up twice; denote by $V_{N,n} $ the number of the balls drawn and by $M_{N,n}$ the expectation of the random variable $v_{N,n}$. Find the limit distribution of the random variable $\frac{V_{N,n}}{M_{N,n}}$ if $N \to \infty$ and $n$ is a fixed number. [b](P. 8)[/b]

1954 Miklós Schweitzer, 1
Tags: Sequences , college contests , real analysis
[b]1.[/b] Given a positive integer $r>1$, prove that there exists an infinite number of infinite geometrical series, with positive terms, having the sum 1 and satisfying the following condition: for any positive real numbers $S_{1},S_{2},\dots,S_{r}$ such that $S_{1}+S_{2}+\dots+S_{r}=1$, any of these infinite geometrical series can be divided into $r$ infinite series(not necessarily geometrical) having the sums $S_{1},S_{2},\dots,S_{r}$, respectively. [b](S. 6)[/b]

1997 Putnam, 3
Tags: Putnam , number theory , greatest common divisor , college contests
For each positive integer $n$ write the sum $\sum_{i=}^{n}\frac{1}{i}=\frac{p_n}{q_n}$ with $\text{gcd}(p_n,q_n)=1$. Find all such $n$ such that $5\nmid q_n$.

2018 Brazil Undergrad MO, 2
Tags: functional equation , function , Olympiad , Brazilian Math Olympiad , algebra , college contests
Let $ f, g: \mathbb {R} \to \mathbb {R} $ function such that $ f (x + g (y)) = - x + y + 1 $ for each pair of real numbers $ x $ e $ y $. What is the value of $ g (x + f (y) $?

2005 IMC, 3
Tags: integration , inequalities , LaTeX , IMC , college contests
3) $f$ cont diff, $R\rightarrow ]0,+\infty[$, prove $|\int_{0}^{1}f^{3}-{f(0)}^{2}\int_{0}^{1}f| \leq \max_{[0,1]} |f'|(\int_{0}^{1}f)^{2}$

2022 District Olympiad, P4
Tags: matrix , rank , college contests , romania
Let $A\in\mathcal{M}_n(\mathbb{C})$ where $n\geq 2.$ Prove that if $m=|\{\text{rank}(A^k)-\text{rank}(A^{k+1})":k\in\mathbb{N}^*\}|$ then $n+1\geq m(m+1)/2.$

2018 IMC, 10
Tags: college contests , imc2018 , real analysis
For $R>1$ let $\mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\}$. Compute $$\lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}.$$ [i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]

1999 IMC, 2
Tags: inequalities , rearrangement inequality , IMC , college contests
Does there exist a bijective map $f:\mathbb{N} \rightarrow \mathbb{N}$ so that $\sum^{\infty}_{n=1}\frac{f(n)}{n^2}$ is finite?

2015 VJIMC, 3
Tags: Summation , Convergence , infinite sum , college contests , real analysis
[b]Problem 3[/b] Determine the set of real values of $x$ for which the following series converges, and find its sum: $$\sum_{n=1}^{\infty} \left(\sum_{\substack{k_1, k_2,\ldots , k_n \geq 0\\ 1\cdot k_1 + 2\cdot k_2+\ldots +n\cdot k_n = n}} \frac{(k_1+\ldots+k_n)!}{k_1!\cdot \ldots \cdot k_n!} x^{k_1+\ldots +k_n} \right) \ . $$

1997 Putnam, 4
Tags: Putnam , function , college contests
Let $G$ be group with identity $e$ and $\phi :G\to G$ be a function such that : \[ \phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3) \] Whenever $g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3$ Show there exists $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism. (that is $\psi(x\cdot y)=\psi (x)\cdot \psi(y)$ for all $x,y\in G$ )

1965 Putnam, A1
Tags: Putnam , college contests
Let $ ABC$ be a triangle with angle $ A <$ angle $ C < 90^\circ <$ angle $ B$. Consider the bisectors of the external angles at $ A$ and $ B$, each measured from the vertex to the opposoite side (extended). Suppose both of these line-segments are equal to $ AB$. Compute the angle $ A$.

2016 Korea USCM, 6
Tags: matrix , linear algebra , trace , determinant , commutator , common eigenvector , college contests
$A$ and $B$ are $2\times 2$ real valued matrices satisfying $$\det A = \det B = 1,\quad \text{tr}(A)>2,\quad \text{tr}(B)>2,\quad \text{tr}(ABA^{-1}B^{-1}) = 2$$ Prove that $A$ and $B$ have a common eigenvector.

2014 Contests, 4
Tags: analytic geometry , college contests
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.

1966 Putnam, A3
Tags: college contests
Let $0<x_1<1$ and $x_{n+1}=x_n(1-x_n), n=1,2,3, \dots$. Show that $$\lim_{n \to \infty} nx_n=1.$$

1998 Putnam, 4
Tags: Putnam , floor function , college contests
Find necessary and sufficient conditions on positive integers $m$ and $n$ so that \[\sum_{i=0}^{mn-1}(-1)^{\lfloor i/m\rfloor+\lfloor i/n\rfloor}=0.\]

1998 Putnam, 5
Tags: Putnam , search , LaTeX , college contests
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the thousandth digit after the decimal point of $\sqrt N$.

1966 Putnam, B4
Tags: college contests
Let $0<a_1<a_2< \dots < a_{mn+1}$ be $mn+1$ integers. Prove that you can select either $m+1$ of them no one of which divides any other, or $n+1$ of them each dividing the following one.

1999 Putnam, 2
Tags: Putnam , algebra , polynomial , college contests
Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),f_2(x),\ldots,f_k(x)$ such that \[p(x)=\sum_{j=1}^k(f_j(x))^2.\]

MIPT student olimpiad autumn 2022, 3
Tags: powers , MIPT , college contests , ratio
How many ways are there (in terms of power) to represent the number 1 as a finite number or an infinite sum of some subset of the set: {$\phi^{-n} | n \in Z^+$} $\phi=\frac{1+\sqrt5}{2}$

2001 Miklós Schweitzer, 1
Tags: Miklos Schweitzer , function , set theory , real analysis , college contests
Let $f\colon 2^S\rightarrow \mathbb R$ be a function defined on the subsets of a finite set $S$. Prove that if $f(A)=F(S\backslash A)$ and $\max \{ f(A), f(B)\}\geq f(A\cup B)$ for all subsets $A, B$ of $S$, then $f$ assumes at most $|S|$ distinct values.

2011 IMC, 4
Tags: algebra , polynomial , induction , IMC , college contests
Let $f$ be a polynomial with real coefficients of degree $n$. Suppose that $\displaystyle \frac{f(x)-f(y)}{x-y}$ is an integer for all $0 \leq x<y \leq n$. Prove that $a-b | f(a)-f(b)$ for all distinct integers $a,b$.