Found problems: 85335
2004 AIME Problems, 2
A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
PEN E Problems, 27
Prove that for each positive integer $n$, there exist $n$ consecutive positive integers none of which is an integral power of a prime number.
2020 Bangladesh Mathematical Olympiad National, Problem 2
How many integers $n$ are there subject to the constraint that $1 \leq n \leq 2020$ and $n^n$ is a perfect square?
2006 Oral Moscow Geometry Olympiad, 4
An arbitrary triangle $ABC$ is given. Construct a straight line passing through vertex $B$ and dividing it into two triangles, the radii of the inscribed circles of which are equal.
(M. Volchkevich)
2016 Turkey Team Selection Test, 8
All angles of the convex $n$-gon $A_1A_2\dots A_n$ are obtuse, where $n\ge5$. For all $1\le i\le n$, $O_i$ is the circumcenter of triangle $A_{i-1}A_iA_{i+1}$ (where $A_0=A_n$ and $A_{n+1}=A_1$). Prove that the closed path $O_1O_2\dots O_n$ doesn't form a convex $n$-gon.
2023 Thailand Online MO, 3
Let $a$ and $n$ be positive integers such that the greatest common divisor of $a$ and $n!$ is $1$. Prove that $n!$ divides $a^{n!}-1$.
2017 Iran Team Selection Test, 4
There are $6$ points on the plane such that no three of them are collinear. It's known that between every $4$ points of them, there exists a point that it's power with respect to the circle passing through the other three points is a constant value $k$.(Power of a point in the interior of a circle has a negative value.)
Prove that $k=0$ and all $6$ points lie on a circle.
[i]Proposed by Morteza Saghafian[/I]
1985 Federal Competition For Advanced Students, P2, 1
Determine all quadruples $ (a,b,c,d)$ of nonnegative integers satisfying:
$ a^2\plus{}b^2\plus{}c^2\plus{}d^2\equal{}a^2 b^2 c^2$.
2016 AMC 12/AHSME, 12
All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$
2022 Bulgarian Spring Math Competition, Problem 11.3
In every cell of a table with $n$ rows and $m$ columns is written one of the letters $a$, $b$, $c$. Every two rows of the table have the same letter in at most $k\geq 0$ positions and every two columns coincide at most $k$ positions. Find $m$, $n$, $k$ if
\[\frac{2mn+6k}{3(m+n)}\geq k+1\]
2017-IMOC, G6
A point $P$ lies inside $\vartriangle ABC$ such that the values of areas of $\vartriangle PAB, \vartriangle PBC, \vartriangle PCA$ can form a triangle. Let $BC = a,CA = b,AB = c, PA = x,PB = y, PC = z$, prove that
$$\frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{x + y + z} \le a + b + c$$
2000 Romania Team Selection Test, 3
Prove that for any positive integers $n$ and $k$ there exist positive integers $a>b>c>d>e>k$ such that
\[n=\binom{a}{3}\pm\binom{b}{3}\pm\binom{c}{3}\pm\binom{d}{3}\pm\binom{e}{3}\]
[i]Radu Ignat[/i]
2017 Canadian Open Math Challenge, C2
Source: 2017 Canadian Open Math Challenge, Problem C2
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A function $f(x)$ is periodic with period $T > 0$ if $f(x + T) = f(x)$ for all $x$. The smallest such number $T$ is called the least period. For example, the functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$.
$\qquad$(a) Let a function $g(x)$ be periodic with the least period $T = \pi$. Determine the least period of $g(x/3)$.
$\qquad$(b) Determine the least period of $H(x) = sin(8x) + cos(4x)$
$\qquad$(c) Determine the least periods of each of $G(x) = sin(cos(x))$ and $F(x) = cos(sin(x))$.
2009 Portugal MO, 1
A circumference was divided in $n$ equal parts. On each of these parts one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$ and $17$ were in consecutive positions. In how many parts was the circumference divided?
1957 Moscow Mathematical Olympiad, 346
Find all isosceles trapezoids that are divided into $2$ isosceles triangles by a diagonal.
MBMT Team Rounds, 2020.1
Chris has a bag with 4 black socks and 6 red socks (so there are $10$ socks in total). Timothy reaches into the bag and grabs two socks [i]without replacement[/i]. Find the probability that he will not grab two red socks.
[i]Proposed by Chris Tong[/i]
2000 Manhattan Mathematical Olympiad, 4
An equilateral triangle $ABC$ is given, together with a point $P$ inside it.
[asy]
draw((0,0)--(4,0)--(2,3.464)--(0,0));
draw((1.3, 1.2)--(0,0));
draw((1.3, 1.2)--(2,3.464));
draw((1.3, 1.2)--(4,0));
label("$A$",(0,0),SW);
label("$B$",(4,0),SE);
label("$C$",(2,3.464),N);
label("$P$",(1.3,1.2),S);
[/asy]
Given that $PA = 3$ cm, $PB = 5$ cm, and $PC = 4$ cm, find the side of the equilateral triangle.
LMT Speed Rounds, 2016.3
The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.
[i]Proposed by Clive Chan
2018 Israel National Olympiad, 2
An [i]arithmetic sequence[/i] is an infinite sequence of the form $a_n=a_0+n\cdot d$ with $d\neq 0$.
A [i]geometric sequence[/i] is an infinite sequence of the form $b_n=b_0 \cdot q^n$ where $q\neq 1,0,-1$.
[list=a]
[*] Does every arithmetic sequence of [b]integers[/b] have an infinite subsequence which is geometric?
[*] Does every arithmetic sequence of [b]real numbers[/b] have an infinite subsequence which is geometric?
[/list]
2013 Balkan MO Shortlist, A1
Positive real numbers $a, b,c$ satisfy $ab + bc+ ca = 3$. Prove the inequality $$\frac{1}{4+(a+b)^2}+\frac{1}{4+(b+c)^2}+\frac{1}{4+(c+a)^2}\le \frac{3}{8}$$
Geometry Mathley 2011-12, 11.1
Let $ABCDEF$ be a hexagon with sides $AB,CD,EF$ being equal to $m$ units, sides $BC,DE, FA$ being equal to $n$ units. The diagonals $AD,BE,CF$ have lengths $x, y$, and $z$ units. Prove the inequality $$\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx} \ge \frac{3}{(m+ n)^2}$$
Nguyễn Văn Quý
2018 Caucasus Mathematical Olympiad, 4
By [i]centroid[/i] of a quadrilateral $PQRS$ we call a common point of two lines through the midpoints of its opposite sides. Suppose that $ABCDEF$ is a hexagon inscribed into the circle $\Omega$ centered at $O$. Let $AB=DE$, and $BC=EF$. Let $X$, $Y$, and $Z$ be centroids of $ABDE$, $BCEF$; and $CDFA$, respectively. Prove that $O$ is the orthocenter of triangle $XYZ$.
2010 Ukraine Team Selection Test, 3
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2015 AMC 12/AHSME, 2
Two of the three sides of a triangle are $20$ and $15$. Which of the following numbers is not a possible perimeter of the triangle?
$\textbf{(A) }52\qquad\textbf{(B) }57\qquad\textbf{(C) }62\qquad\textbf{(D) }67\qquad\textbf{(E) }72$
2002 AMC 12/AHSME, 25
Let $a$ and $b$ be real numbers such that $\sin a+\sin b=\dfrac{\sqrt2}2$ and $\cos a+\cos b=\dfrac{\sqrt6}2$. Find $\sin(a+b)$.
$\textbf{(A) }\dfrac12\qquad\textbf{(B) }\dfrac{\sqrt2}2\qquad\textbf{(C) }\dfrac{\sqrt3}2\qquad\textbf{(D) }\dfrac{\sqrt6}2\qquad\textbf{(E) }1$