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

2006 IMC, 2
Tags: modular arithmetic , IMC , college contests
Find the number of positive integers x satisfying the following two conditions: 1. $x<10^{2006}$ 2. $x^{2}-x$ is divisible by $10^{2006}$

1956 Miklós Schweitzer, 3
Tags: college contests
[b]3.[/b] A triangulation of a convex closed polygon is the division into triangles of this poilygon by diagonals not intersecting in the interior of the polygon. Find the number of all triangulations fo a conves n-gon and also the number of those triangulations in which every triangle has at least one side in common with the given n-gon. [b](C. 4)[/b]

2017 Korea USCM, 1
Tags: linear algebra , matrix , Determinants , Summation , college contests
$n(\geq 2)$ is a given integer and $T$ is set of all $n\times n$ matrices whose entries are elements of the set $S=\{1,\cdots,2017\}$. Evaluate the following value. \[\sum_{A\in T} \text{det}(A)\]

ICMC 4, 4
Tags: analysis , ICMC , college contests , geometry
Let \(\mathbb R^2\) denote the Euclidean plane. A continuous function \(f : \mathbb R^2 \to \mathbb R^2\) maps circles to circles. (A point is not a circle.) Prove that it maps lines to lines. [i]Proposed by Tony Wang[/i]

2004 Putnam, B3
Tags: Putnam , function , geometry , perimeter , college contests
Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region $R=\{(x,y): 0\le x\le a, 0\le y\le f(x)\}$ has perimeter $k$ units and area $k$ square units for some real number $k$.

2004 Putnam, A1
Tags: Putnam , induction , inequalities , percent , college contests , Putnam easy
Basketball star Shanille O'Keal's team statistician keeps track of the number, $S(N),$ of successful free throws she has made in her first $N$ attempts of the season. Early in the season, $S(N)$ was less than 80% of $N,$ but by the end of the season, $S(N)$ was more than 80% of $N.$ Was there necessarily a moment in between when $S(N)$ was exactly 80% of $N$?

2014 Putnam, 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, 14
Tags: modulo , algebra , Brazilian Math Olympiad , college contests
What is the arithmetic mean of all values of the expression $ | a_1-a_2 | + | a_3-a_4 | $ Where $ a_1, a_2, a_3, a_4 $ is a permutation of the elements of the set {$ 1,2,3,4 $}?

2013 IMC, 5
Tags: IMC , college contests
Consider a circular necklace with $\displaystyle{2013}$ beads. Each bead can be paintes either green or white. A painting of the necklace is called [i]good[/i] if among any $\displaystyle{21}$ successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd. [b]Note.[/b] Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings. [i]Proposed by Vsevolod Bykov and Oleksandr Rybak, Kiev.[/i]

1954 Miklós Schweitzer, 8
Tags: Ring Theory , college contests
[b]8.[/b] Prove the following generalization of the well-known Chinese remainder theorem: Let $R$ be a ring with unit element and let $A_{1},A_{2},\dots . A_{n} (n\geqslant 2)$ be pairwise relative prime ideals of $R$. Then, for arbitrary elements $c_{1},c_{2}, \dots , c_{n}$ of $R$, there exists an element $x\in R$ such that $x-c_{k} \in A_{k} (k= 1,2, \dots , n)$. [b](A. 17)[/b]

2014 Putnam, 1
Tags: Putnam , modular arithmetic , induction , college contests , Putnam 2014
A [i]base[/i] 10 [i]over-expansion[/i] of a positive integer $N$ is an expression of the form $N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0$ with $d_k\ne 0$ and $d_i\in\{0,1,2,\dots,10\}$ for all $i.$ For instance, the integer $N=10$ has two base 10 over-expansions: $10=10\cdot 10^0$ and the usual base 10 expansion $10=1\cdot 10^1+0\cdot 10^0.$ Which positive integers have a unique base 10 over-expansion?

2008 Miklós Schweitzer, 5
Tags: college contests , Miklos Schweitzer , asymptotics , set theory , real analysis
Let $A$ be an infinite subset of the set of natural numbers, and denote by $\tau_A(n)$ the number of divisors of $n$ in $A$. Construct a set $A$ for which $$\sum_{n\le x}\tau_A(n)=x+O(\log\log x)$$ and show that there is no set for which the error term is $o(\log\log x)$ in the above formula. (translated by Miklós Maróti)

1976 Putnam, 4
Tags: college contests
Let $r$ be a root of $P(x)=x^3+ax^2+bx-1=0$ and $r+1$ be a root of $y^3+cy^2+dy+1=0,$ where $a,b,c$ and $d$ are integers. Also let $P(x)$ be irreducible over the rational numbers. Express another root $s$ of $P(x)=0$ as a function of $r$ which does not explicitly involve $a,b,c$ or $d.$

ICMC 3, 2
Tags: college contests , complex numbers , Sum , roots of unity
Find integers \(a\) and \(b\) such that \[a^b=3^0\binom{2020}{0}-3^1\binom{2020}{2}+3^2\binom{2020}{4}-\cdots+3^{1010}\binom{2020}{2020}.\] [i]proposed by the ICMC Problem Committee[/i]

1998 Putnam, 1
Tags: Putnam , geometry , 3D geometry , rectangle , college contests , Putnam easy
A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?

1955 Miklós Schweitzer, 4
Tags: number theory , prime numbers , college contests
[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]

2018 Brazil Undergrad MO, 17
Tags: geometry , Brazilian Math Olympiad , college contests
In the figure, a semicircle is folded along the $ AN $ string and intersects the $ MN $ diameter in $ B $. $ MB: BN = 2: 3 $ and $ MN = 10 $ are known to be. If $ AN = x $, what is the value of $ x ^ 2 $?

2025 VJIMC, 1
Tags: Sequences , Recurrence , characteristic polynomial , limits , college contests , VJIMC
Let $x_0=a, x_1= b, x_2 = c$ be given real numbers and let $x_{n+2} = \frac{x_n + x_{n-1}}{2}$ for all $n\geq 1$. Show that the sequence $(x_n)_{n\geq 0}$ converges and find its limit.

1986 Miklós Schweitzer, 7
Tags: Miklos Schweitzer , college contests , function , Summation
Prove that the series $\sum_p c_p f(px)$, where the summation is over all primes, unconditionally converges in $L^2[0,1]$ for every $1$-periodic function $f$ whose restriction to $[0,1]$ is in $L^2[0,1]$ if and only if $\sum_p |c_p|<\infty$. ([i]Unconditional convergence[/i] means convergence for all rearrangements.) [G. Halasz]

1965 Putnam, B1
Tags: Putnam , limit , integration , trigonometry , calculus , college contests
Evaluate $ \lim_{n\to\infty} \int_0^1 \int_0^1 \cdots \int_0^1 \cos ^ 2 \left\{\frac{\pi}{2n}(x_1\plus{}x_2\plus{}\cdots \plus{}x_n)\right\} dx_1dx_2\cdots dx_n.$

2009 IMC, 4
Tags: algebra , polynomial , IMC , college contests
Let $p(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a complex polynomial. Suppose that $1=c_0\ge c_1\ge \cdots \ge c_n\ge 0$ is a sequence of real numbers which form a convex sequence. (That is $2c_k\le c_{k-1}+c_{k+1}$ for every $k=1,2,\cdots ,n-1$ ) and consider the polynomial \[ q(z)=c_0a_0+c_1a_1z+c_2a_2z^2+\cdots +c_na_nz^n \] Prove that : \[ \max_{|z|\le 1}q(z)\le \max_{|z|\le 1}p(z) \]

ICMC 8, 3
Tags: college contests
Let $V$ be a subspace of the vector space $\mathbb{R}^{2 \times 2}$ of $2$-by-$2$ real matrices. We call $V$ nice if for any linearly independent $A, B \in V$, $AB \neq BA$. Find the maximum dimension of a nice subspace of $\mathbb{R}^{2 \times 2}$.

2005 IMC, 5
Tags: limit , integration , calculus , IMC , college contests
5) f twice cont diff, $|f''(x)+2xf'(x)+(x^{2}+1)f(x)|\leq 1$. prove $\lim_{x\rightarrow +\infty} f(x) = 0$

ICMC 4, 3
Tags: expectation , ICMC , college contests , Inequality
Let $f,g,h : \mathbb R \to \mathbb R$ be continuous functions and \(X\) be a random variable such that $E(g(X)h(X))=0$ and $E(g(X)^2) \neq 0 \neq E(h(X)^2)$. Prove that $$E(f(X)^2) \geq \frac{E(f(X)g(X))^2}{E(g(X)^2)} + \frac{E(f(X)h(X))^2}{E(h(X)^2)}.$$ You may assume that all expected values exist. [i]Proposed by Cristi Calin[/i]

ICMC 8, 5
Tags: college contests
A positive integer is a non-trivial perfect power if it can be expressed as $n^k$ where $n$ and $k$ are positive integers and $k>1$. Show that there exist arbitrarily large consecutive square numbers with no other non-trivial perfect powers between them.