Found problems: 15925
2019 USA TSTST, 7
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$
for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.
[i]Ankan Bhattacharya[/i]
1983 AIME Problems, 13
For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$.
2017 AIME Problems, 8
Find the number of positive integers $n$ less than $2017$ such that
\[ 1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!} \]
is an integer.
2015 AIME Problems, 10
Let $f(x)$ be a third-degree polynomial with real coefficients satisfying
\[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\] Find $|f(0)|$.
2007 Hanoi Open Mathematics Competitions, 15
Let $p = \overline{abc}$ be the 3-digit prime number. Prove that the equation $ax^2 + bx + c = 0$ has no rational roots.
2021 CHMMC Winter (2021-22), 3
Suppose $a, b, c$ are complex numbers with $a + b + c = 0$, $a^2 + b^2 + c^2 = 0$, and $|a|,|b|,|c| \le 5$. Suppose further at least one of $a, b, c$ have real and imaginary parts that are both integers. Find the number of possibilities for such ordered triples $(a, b, c)$.
TNO 2008 Senior, 7
Find all pairs of prime numbers $p$ and $q$ such that:
\[
p(p + q) = q^p+ 1.
\]
1962 Vietnam National Olympiad, 5
Solve the equation $ \sin^6x \plus{} \cos^6x \equal{} \frac{1}{4}$.
Revenge ELMO 2023, 5
Complex numbers $a,b,w,x,y,z,p$ satisfy
\begin{align*}
\frac{(x-w)\lvert a-w \rvert}{(a-w)\lvert x-w \rvert}&=\text{(cyclic variants)};\\
\frac{(z-w)\lvert b-w \rvert}{(b-w)\lvert z-w \rvert}&=\text{(cyclic variants)};\\
p &= \frac{\sum_{\text{cyc}} \frac w{\lvert p-w \rvert}}{\sum_{\text{cyc}}\frac1{\lvert p-w \rvert}};
\end{align*}
where cyclic sums, equations, etc. are wrt $w,x,y,z$.
Prove that there exists a real $k$ such that
\[\sum_{\text{cyc}} \frac{(x-w)(a-w)}{\lvert x-w\rvert (p-w)}
=k\sum_{\text{cyc}} \frac{(z-w)(b-w)}{\lvert z-w\rvert(p-w)}.\]
[i]Neal Yan[/i]
2009 Germany Team Selection Test, 3
Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If
\[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\]
then
\[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]
2013 ELMO Problems, 5
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$?
[i]Proposed by Andre Arslan[/i]
2022 Estonia Team Selection Test, 1
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the following condition for any real numbers $x{}$ and $y$ $$f(x)+f(x+y) \leq f(xy)+f(y).$$
2000 Romania National Olympiad, 3
A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is [i]olympic[/i] if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon.
Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.
1974 IMO Longlists, 21
Let $M$ be a nonempty subset of $\mathbb Z^+$ such that for every element $x$ in $M,$ the numbers $4x$ and $\lfloor \sqrt x \rfloor$ also belong to $M.$ Prove that $M = \mathbb Z^+.$
2023 Poland - Second Round, 3
Given positive integers $k,n$ and a real number $\ell$, where $k,n \geq 1$. Given are also pairwise different positive real numbers $a_1,a_2,\ldots, a_k$. Let $S = \{a_1,a_2,\ldots,a_k, -a_1, -a_2,\ldots, -a_k\}$.
Let $A$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = 0,$$
where $x_1,x_2,\ldots, x_{2n} \in S$. Let $B$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = \ell,$$
where $x_1,x_2,\ldots,x_{2n} \in S$. Prove that $A \geq B$.
Solutions of an equation with only difference in the permutation are different.
2000 AIME Problems, 13
The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r,$ where $m, n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0.$ Find $m+n+r.$
2010 Junior Balkan Team Selection Tests - Moldova, 2
The positive real numbers $x$ and $y$ satisfy the relation $x + y = 3 \sqrt{xy}$. Find the value of the numerical expression $$E=\left| \frac{x-y}{x+y}+\frac{x^2-y^2}{x^2+y^2}+\frac{x^3-y^3}{x^3+y^3}\right|.$$
2019 ELMO Shortlist, A4
Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$
[i]Proposed by Carl Schildkraut[/i]
2017 Taiwan TST Round 2, 1
Determine all surjective functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(xyz+xf\left(y\right)+yf\left(z\right)+zf\left(x\right)\right)=f\left(x\right)f\left(y\right)f\left(z\right) $$ for all $ x,y,z $ in $ \mathbb{Z} $
2009 IberoAmerican, 5
Consider the sequence $ \{a_n\}_{n\geq1}$ defined as follows: $ a_1 \equal{} 1$, $ a_{2k} \equal{} 1 \plus{} a_k$ and $ a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}}$ for every $ k\geq 1$. Prove that every positive rational number appears on the sequence $ \{a_n\}$ exactly once.
1992 Czech And Slovak Olympiad IIIA, 4
Solve the equation $\cos 12x = 5\sin 3x+9\ tan ^2x+\ cot ^2x$
1896 Eotvos Mathematical Competition, 1
If $k$ is the number of distinct prime divisors of a natural number $n$, prove that log $n \geq k$ log $2$.
1998 Akdeniz University MO, 5
Solve the equation system for real numbers:
$$x_1+x_2=x_3^2$$
$$x_2+x_3=x_4^2$$
$$x_3+x_4=x_1^2$$
$$x_4+x_1=x_2^2$$
2016 Saudi Arabia GMO TST, 2
Find all functions $f : Z \to Z$ such that $f (2m + f (m) + f (m)f (n)) = nf (m) + m$ for any integers $m, n$
2018 Tuymaada Olympiad, 1
Do there exist three different quadratic trinomials $f(x), g(x), h(x)$ such that the roots of the equation $f(x)=g(x)$ are $1$ and $4$, the roots of the equation $g(x)=h(x)$ are $2$ and $5$, and the roots of the equation $h(x)=f(x)$ are $3$ and $6$?
[i]Proposed by A. Golovanov[/i]