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: 85335

2010 Paenza, 3
Tags: limit , trigonometry , real analysis
Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

2014 ELMO Shortlist, 6
Tags: inequalities
Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]

2005 Spain Mathematical Olympiad, 1
Tags: algebra , number theory
Let $a$ and $b$ be integers. Demonstrate that the equation $$(x-a)(x-b)(x-3) +1 = 0$$ has an integer solution.

1999 Miklós Schweitzer, 3
Tags: graph theory
Prove that for any finite graph G there is a constant c(G)>0 such that for every n-point graph that does not have an induced subgraph isomorphic to G, there are two disjoint sets of vertices, each with at least $n^{c(G)}$ elements, between which either all edges are connected, or none of the edges are.

2011 Stars Of Mathematics, 3
Tags: induction , inequalities
For a given integer $n\geq 3$, determine the range of values for the expression \[ E_n(x_1,x_2,\ldots,x_n) := \dfrac {x_1} {x_2} + \dfrac {x_2} {x_3} + \cdots + \dfrac {x_{n-1}} {x_n} + \dfrac {x_n} {x_1}\] over real numbers $x_1,x_2,\ldots,x_n \geq 1$ satisfying $|x_k - x_{k+1}| \leq 1$ for all $1\leq k \leq n-1$. Do also determine when the extremal values are achieved. (Dan Schwarz)

2017 Princeton University Math Competition, A7
Tags: geometry , cyclic quadrilateral , circumcircle , number theory , relatively prime
Let $ACDB$ be a cyclic quadrilateral with circumcenter $\omega$. Let $AC=5$, $CD=6$, and $DB=7$. Suppose that there exists a unique point $P$ on $\omega$ such that $\overline{PC}$ intersects $\overline{AB}$ at a point $P_1$ and $\overline{PD}$ intersects $\overline{AB}$ at a point $P_2$, such that $AP_1=3$ and $P_2B=4$. Let $Q$ be the unique point on $\omega$ such that $\overline{QC}$ intersects $\overline{AB}$ at a point $Q_1$, $\overline{QD}$ intersects $\overline{AB}$ at a point $Q_2$, $Q_1$ is closer to $B$ than $P_1$ is to $B$, and $P_2Q_2=2$. The length of $P_1Q_1$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2012 China Team Selection Test, 1
Tags: circumcircle , trigonometry , geometry
In an acute-angled $ABC$, $\angle A>60^{\circ}$, $H$ is its orthocenter. $M,N$ are two points on $AB,AC$ respectively, such that $\angle HMB=\angle HNC=60^{\circ}$. Let $O$ be the circumcenter of triangle $HMN$. $D$ is a point on the same side with $A$ of $BC$ such that $\triangle DBC$ is an equilateral triangle. Prove that $H,O,D$ are collinear.

1979 Romania Team Selection Tests, 2.
Tags: geometry , 3d geometry , pyramid
Let $VA_1A_2A_3A_4$ be a pyramid with the vertex at $V$. Let $M,\, N,\, P$ be the midpoints of the segments $VA_1$, $VA_3$, and $A_2A_4$. Show that the plane $(MNP)$ cuts the pyramid into two parts with the same volume. [i]Radu Gologan[/i]

1981 Dutch Mathematical Olympiad, 2
Tags: equilateral , geometry
Given is the equilateral triangle $ABC$ with center $M$. On $CA$ and $CB$ the respective points $D$ and $E$ lie such that $CD = CE$. $F$ is such that $DMFB$ is a parallelogram. Prove that $\vartriangle MEF$ is equilateral.

2014 Saudi Arabia Pre-TST, 1.4
Tags: combinatorics , coloring , chessboard
Majid wants to color the cells of an $n\times n$ chessboard into white and black so that each $2\times 2$ subsquare contains two white cells and two black cells. In how many ways can Majid color this $n\times n$ chessboard?

1981 Miklós Schweitzer, 1
Tags: advanced fields
We are given an infinite sequence of $ 1$'s and $ 2$'s with the following properties: (1) The first element of the sequence is $ 1$. (2) There are no two consecutive $ 2$'s or three consecutive $ 1$'s. (3) If we replace consecutive $ 1$'s by a single $ 2$, leave the single $ 1$'s alone, and delete the original $ 2$'s, then we recover the original sequence. How many $ 2$'s are there among the first $ n$ elements of the sequence? [i]P. P. Palfy[/i]

2014 Tuymaada Olympiad, 8
Tags: combinatorics
There are $m$ villages on the left bank of the Lena, $n$ villages on the right bank and one village on an island. It is known that $(m+1,n+1)>1$. Every two villages separated by water are connected by ferry with positive integral number. The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong. [i](K. Kokhas)[/i]

2016 NZMOC Camp Selection Problems, 4
Tags: number theory , divisible
A quadruple $(p, a, b, c)$ of positive integers is a[i] karaka quadruple[/i] if $\bullet$ $p$ is an odd prime number $\bullet$ $a, b$ and $c$ are distinct, and $\bullet$ $ab + 1$, $bc + 1$ and $ca + 1$ are divisible by $p$. (a) Prove that for every karaka quadruple $(p, a, b, c)$ we have $p + 2 \le\frac{a + b + c}{3}$. (b) Determine all numbers $p$ for which a karaka quadruple $(p, a, b, c)$ exists with $p + 2 =\frac{a + b + c}{3}$

1949 Putnam, A5
Tags: root
How many roots of the equation $z^6 +6z +10=0$ lie in each quadrant of the complex plane?

1966 All Russian Mathematical Olympiad, 077
Tags: algebra
Given the numbers $a_1, a_2, ... , a_n$ such that $$0\le a_1\le a_2\le 2a_1 , a_2\le a_3\le 2a_2 , ... , a_{n-1}\le a_n\le 2a_{n-1}$$ Prove that in the sum $s=\pm a1\pm a2\pm ...\pm a_n$ You can choose appropriate signs to make $0\le s\le a_1$.

2022 Purple Comet Problems, 17
Tags:
There are real numbers $x, y,$ and $z$ such that the value of $$x+y+z-\left(\frac{x^2}{5}+\frac{y^2}{6}+\frac{z^2}{7}\right)$$ reaches its maximum of $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n + x + y + z.$

2010 AMC 8, 7
Tags:
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than one dollar? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 99 $

2022 IMC, 4
Tags: combinatorics , graph coloring , logarithm , set
Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of $\{1, 2, \ldots , n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever $1\leq a<b<c<d \leq n$, the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$.

2014 Saudi Arabia BMO TST, 3
Tags: algebra
Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.

MBMT Guts Rounds, 2015.2
Tags:
Evaluate $\frac{1}{2+\frac{3}{1+\frac{2}{3+x}}}$ if $x = 1$.

2013 AMC 8, 14
Tags: probability
Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match? $\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23$

2011 Today's Calculation Of Integral, 747
Tags: calculus , integration , trigonometry , inequalities , calculus computations
Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$

2020 BMT Fall, 3
Tags: geometry
Right triangular prism $ABCDEF$ with triangular faces $\vartriangle ABC$ and $\vartriangle DEF$ and edges $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ has $\angle ABC = 90^o$ and $\angle EAB = \angle CAB = 60^o$ . Given that $AE = 2$, the volume of $ABCDEF$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/4/7/25fbe2ce2df50270b48cc503a8af4e0c013025.png[/img]

2004 Postal Coaching, 5
Tags: combinatorics
How many paths from $(0,0)$ to $(n,n)$ of length $2n$ are there with exactly $k$ steps. A step is an occurence of the pair $EN$ in the path

2011 India Regional Mathematical Olympiad, 2
Tags:
Let $(a_1,a_2,a_3,...,a_{2011})$ be a permutation of the numbers $1,2,3,...,2011$. Show that there exist two numbers $j,k$ such that $1\leq{j}<k\leq2011$ and $|a_j-j|=|a_k-k|$