Found problems: 85335
2015 Cuba MO, 4
Let $A$ and $B$ be two subsets of $\{1, 2, 3, 4, ..., 100\}$, such that $|A| = |B|$ and $A\cap B =\emptyset$. If $n \in A$ implies that $2n + 2 \in B$, determine the largest possible value of $ |A \cup B|$.
1990 National High School Mathematics League, 5
Two non-zero-complex numbers $x,y$, satisfy that $x^2+xy+y^2=0$. Then the value of $(\frac{x}{x+y})^{1990}+(\frac{y}{x+y})^{1990}$ is
$\text{(A)}2^{-1989}\qquad\text{(B)}-1\qquad\text{(C)}1\qquad\text{(D)}$none above
2013 Stanford Mathematics Tournament, 23
Let $a$ and $b$ be the solutions to $x^2-7x+17=0$. Compute $a^4+b^4$.
2018 Greece Junior Math Olympiad, 3
Let $a$ and $b$ be positive integers with $b$ odd, such that the number $$\frac{(a+b)^2+4a}{ab}$$ is an integer. Prove that $a$ is a perfect square.
2021 LMT Spring, B10
Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$, and $f (2) =\frac19$, find $ f\left(\frac{1}{6}\right)$.
[i]Proposed by Zachary Perry[/i]
2014 Online Math Open Problems, 29
Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. It is known that $AB = 7$, $BC = 8$, $CA = 9$. Let $M$ denote the midpoint of major arc $\widehat{BAC}$ of $\Gamma$, and let $D$ denote the intersection of $\Gamma$ with the circumcircle of $\triangle IMO$ (other than $M$). Let $E$ denote the reflection of $D$ over line $IO$. Find the integer closest to $1000 \cdot \frac{BE}{CE}$.
[i]Proposed by Evan Chen[/i]
2016 South African National Olympiad, 2
Determine all pairs of real numbers $a$ and $b$, $b > 0$, such that the solutions to the two equations
$$x^2 + ax + a = b \qquad \text{and} \qquad x^2 + ax + a = -b$$
are four consecutive integers.
2021 South East Mathematical Olympiad, 6
Let $ABCD$ be a cyclic quadrilateral. The internal angle bisector of $\angle BAD$ and line $BC$ intersect at $E.$ $M$ is the midpoint of segment $AE.$ The exterior angle bisector of $\angle BCD$ and line $AD$ intersect at $F.$ The lines $MF$ and $AB$ intersect at $G.$ Prove that if $AB=2AD,$ then $MF=2MG.$
2017 Iran MO (2nd Round), 5
There are five smart kids sitting around a round table. Their teacher says: "I gave a few apples to some of you, and none of you have the same amount of apple. Also each of you will know the amount of apple that the person to your left and the person to your right has."
The teacher tells the total amount of apples, then asks the kids to guess the difference of the amount of apple that the two kids in front of them have.
$a)$ If the total amount of apples is less than $16$, prove that at least one of the kids will guess the difference correctly.
$b)$ Prove that the teacher can give the total of $16$ apples such that no one can guess the difference correctly.
1953 AMC 12/AHSME, 4
The roots of $ x(x^2\plus{}8x\plus{}16)(4\minus{}x)\equal{}0$ are:
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 0,4 \qquad\textbf{(C)}\ 0,4,\minus{}4 \qquad\textbf{(D)}\ 0,4,\minus{}4,\minus{}4 \qquad\textbf{(E)}\ \text{none of these}$
2006 Peru IMO TST, 2
[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 $x$.
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[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88510]Spanish version[/url]
$\text{\LaTeX}{}$ed by carlosbr
2004 Manhattan Mathematical Olympiad, 3
A prison has $2004$ cells, numbered $1$ through $2004$. A jailer, carrying out the terms of a partial amnesty, unlocked every cell. Next he locked every second cell. Then he turned the key in every third cell, locking the opened cells, and unlocking the locked ones. He continued this way, on $n^{\text{th}}$ trip, turning the key in every $n^{\text{th}}$ cell, and he finished his mission after $2004$ trips. How many prisoners were released?
2018 Irish Math Olympiad, 6
Find all real-valued functions $f$ satisfying $f(2x + f(y)) + f(f(y)) = 4x + 8y$ for all real numbers $x$ and $y$.
III Soros Olympiad 1996 - 97 (Russia), 11.8
Solve the system of equations:
$$ 2(3-2\cos y)^2+2(4-2\sin y)^2=2(3-x)^2+32=(x-2\cos y)^2+4\sin^2y$$
2018 Sharygin Geometry Olympiad, 8
Let $I$ be the incenter of fixed triangle $ABC$, and $D$ be an arbitrary point on $BC$. The perpendicular bisector of $AD$ meets $BI,CI$ at $F$ and $E$ respectively. Find the locus of orthocenters of $\triangle IEF$ as $D$ varies.
2006 China National Olympiad, 5
Let $\{a_n\}$ be a sequence such that: $a_1 = \frac{1}{2}$, $a_{k+1}=-a_k+\frac{1}{2-a_k}$ for all $k = 1, 2,\ldots$. Prove that
\[ \left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right). \]
2016 Online Math Open Problems, 3
In a rectangle $ABCD$, let $M$ and $N$ be the midpoints of sides $BC$ and $CD$, respectively, such that $AM$ is perpendicular to $MN$. Given that the length of $AN$ is $60$, the area of rectangle $ABCD$ is $m \sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$.
[i]Proposed by Yannick Yao[/i]
2021 Mediterranean Mathematics Olympiad, 1
Determine the smallest positive integer $M$ with the following property:
For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$.
[i]Proposed by Gerhard Woeginger, Austria[/i]
2009 Today's Calculation Of Integral, 474
Calculate the following indefinite integrals.
(1) $ \int \frac {3x \plus{} 4}{x^2 \plus{} 3x \plus{} 2}dx$
(2) $ \int \sin 2x\cos 2x\cos 4x\ dx$
(3) $ \int xe^{x}dx$
(4) $ \int 5^{x}dx$
2024 All-Russian Olympiad Regional Round, 10.9
Find all triplets $(a, b, c)$ of positive integers, such that $a+bc, b+ac, c+ab$ are primes and all divide $(a^2+1)(b^2+1)(c^2+1)$.
2009 USAMTS Problems, 1
Fill in the circles in the picture at right with the digits $1-8$, one digit in each circle with no digit repeated, so that no two circles that are connected by a line segment contain consecutive digits. In how many ways can this be done?
[asy]
defaultpen(linewidth(1)); real r = 0.2; pair[] dots = {(0,1),(1,0),(0,0),(-1,0),(1,-1),(0,-1),(-1,-1),(0,-2)}; draw((0,1)--(-1,0)--(-1,-1)--(0,-2)--(1,-1)--(1,0)--(0,1)--(0,-2)); draw((-1,0)--(1,0)--(0,-1)--cycle); draw((-1,-1)--(1,-1)); for(int i = 0; i < dots.length; ++i) filldraw(circle(dots[i], r), white);[/asy]
2012 Ukraine Team Selection Test, 1
Let $a, b, c$ be positive reals. Prove that $\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab}\ge 3 \sqrt{ab+bc+ca}$
2023 AMC 10, 22
How many distinct values of $x$ satisfy $\lfloor x \rfloor ^2 – 3x + 2 = 0$ where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$?
$\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0$
2003 France Team Selection Test, 3
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.
2005 Balkan MO, 3
Let $a,b,c$ be positive real numbers. Prove the inequality
\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\]
When does equality occur?