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: 329

PEN A Problems, 7
Tags: quadratics , LaTeX , modular arithmetic , Divisibility Theory
Let $n$ be a positive integer such that $2+2\sqrt{28n^2 +1}$ is an integer. Show that $2+2\sqrt{28n^2 +1}$ is the square of an integer.

PEN R Problems, 4
Tags: modular arithmetic , LaTeX , The Geometry of Numbers
The sidelengths of a polygon with $1994$ sides are $a_{i}=\sqrt{i^2 +4}$ $ \; (i=1,2,\cdots,1994)$. Prove that its vertices are not all on lattice points.

2006 AMC 8, 21
Tags: LaTeX , geometry
An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. The aquarium is filled with water to a depth of $ 37$ cm. A rock with volume $ 1000 \text{cm}^3$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise? $ \textbf{(A)}\ 0.25 \qquad \textbf{(B)}\ 0.5 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 1.25 \qquad \textbf{(E)}\ 2.5$

1969 Canada National Olympiad, 3
Tags: LaTeX
Let $c$ be the length of the hypotenuse of a right angle triangle whose two other sides have lengths $a$ and $b$. Prove that $a+b\le c\sqrt{2}$. When does the equality hold?

2002 China Team Selection Test, 2
Tags: LaTeX , number theory unsolved , number theory
Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?

2004 China Team Selection Test, 3
Tags: inequalities , LaTeX , inequalities unsolved
Let $k \geq 2, 1 < n_1 < n_2 < \ldots < n_k$ are positive integers, $a,b \in \mathbb{Z}^+$ satisfy \[ \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right) \] Prove that: \[ \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}. \]

1998 AMC 12/AHSME, 24
Tags: LaTeX , AMC
Call a $ 7$-digit telephone number $ d_1d_2d_3 \minus{} d_4d_5d_6d_7$ [i]memorable[/i] if the prefix sequence $ d_1d_2d_3$ is exactly the same as either of the sequences $ d_4d_5d_6$ or $ d_5d_6d_7$ (possibly both). Assuming that each $ d_i$ can be any of the ten decimal digits $ 0,1,2,\ldots9$, the number of different memorable telephone numbers is $ \textbf{(A)}\ 19,\!810 \qquad \textbf{(B)}\ 19,\!910 \qquad \textbf{(C)}\ 19,\!990 \qquad \textbf{(D)}\ 20,\!000 \qquad \textbf{(E)}\ 20,\!100$

2010 Contests, 2
Tags: algebra , polynomial , modular arithmetic , LaTeX , number theory , number theory unsolved
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

1982 IMO Longlists, 7
Tags: LaTeX , modular arithmetic , algorithm , floor function , number theory , greatest common divisor , Euclidean algorithm
Find all solutions $(x, y) \in \mathbb Z^2$ of the equation \[x^3 - y^3 = 2xy + 8.\]

2011 Balkan MO Shortlist, C2
Tags: inequalities , geometry , LaTeX , ratio , trigonometry , geometry proposed
Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

2002 China Team Selection Test, 2
Tags: LaTeX , number theory unsolved , number theory
Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?

2000 Harvard-MIT Mathematics Tournament, 36
Tags: geometry , LaTeX , inradius , trigonometry , trig identities , Law of Cosines
If, in a triangle of sides $a, b, c$, the incircle has radius $\frac{b+c-a}{2}$, what is the magnitude of $\angle A$?

2013 Czech-Polish-Slovak Match, 1
Tags: geometry , geometric transformation , homothety , LaTeX , cyclic quadrilateral , geometry unsolved
Suppose $ABCD$ is a cyclic quadrilateral with $BC = CD$. Let $\omega$ be the circle with center $C$ tangential to the side $BD$. Let $I$ be the centre of the incircle of triangle $ABD$. Prove that the straight line passing through $I$, which is parallel to $AB$, touches the circle $\omega$.

1994 Baltic Way, 7
Tags: LaTeX , number theory proposed , number theory
Let $p>2$ be a prime number and \[1+\frac{1}{2^3}+\frac{1}{3^3}+\ldots +\frac{1}{(p-1)^3}=\frac{m}{n}\] where $m$ and $n$ are relatively prime. Show that $m$ is a multiple of $p$.

2013 AIME Problems, 10
Tags: geometry , LaTeX , number theory , relatively prime , AMC , AIME
Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be written in the form $\tfrac{a-b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

2004 China Team Selection Test, 1
Tags: LaTeX , geometry unsolved , geometry
Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$

1989 AMC 12/AHSME, 18
Tags: LaTeX , function
The set of all numbers x for which \[x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}}\] is a rational number is the set of all: $\textbf{(A)}\ \text{ integers } x \qquad \textbf{(B)}\ \text{ rational } x \qquad \textbf{(C)}\ \text{ real } x\qquad \textbf{(D)}\ x \text{ for which } \sqrt{x^2+1} \text{ is rational} \qquad \textbf{(E)}\ x \text{ for which } x+\sqrt{x^2+1} \text{ is rational }$

2003 AMC 8, 5
Tags: percent , LaTeX
If $20\%$ of a number is $12$, what is $30\%$ of the same number? $\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 30$

2006 Peru IMO TST, 2
Tags: LaTeX , function , algebra proposed , algebra
[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 02[/b] Find all pairs $(a,b)$ real positive numbers $a$ and $b$ such that: $[a[bn]]= n-1,$ for all $n$ positive integer. Note: [x] denotes the integer part of $x$. ---------- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88510]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

2000 Tuymaada Olympiad, 1
Tags: analytic geometry , LaTeX , topology , set theory , combinatorics solved , combinatorics
Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?

2006 Peru IMO TST, 4
Tags: geometry , circumcircle , LaTeX , parallelogram , perpendicular bisector , similar triangles , geometry unsolved
[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 04[/b] In an actue-angled triangle $ABC$ draws up: its circumcircle $w$ with center $O$, the circumcircle $w_1$ of the triangle $AOC$ and the diameter $OQ$ of $w_1$. The points are chosen $M$ and $N$ on the straight lines $AQ$ and $AC$, respectively, in such a way that the quadrilateral $AMBN$ is a parallelogram. Prove that the intersection point of the straight lines $MN$ and $BQ$ belongs the circumference $w_1.$ --- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88513]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

2000 AIME Problems, 5
Tags: counting , distinguishability , function , LaTeX
Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n.$

2013 Moldova Team Selection Test, 3
Tags: geometry , geometric transformation , reflection , circumcircle , ratio , LaTeX , angle bisector
Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.

2004 China Team Selection Test, 2
Tags: LaTeX , number theory unsolved , number theory
Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$

2011 AMC 10, 23
Tags: modular arithmetic , Euler , function , number theory , least common multiple , search , LaTeX
What is the hundreds digit of $2011^{2011}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9 $