Found problems: 15925
2014 Iran MO (3rd Round), 5
We say $p(x,y)\in \mathbb{R}\left[x,y\right]$ is [i]good[/i] if for any $y \neq 0$ we have $p(x,y) = p\left(xy,\frac{1}{y}\right)$ . Prove that there are good polynomials $r(x,y) ,s(x,y)\in \mathbb{R}\left[x,y\right]$ such that for any good polynomial $p$ there is a $f(x,y)\in \mathbb{R}\left[x,y\right]$ such that \[f(r(x,y),s(x,y))= p(x,y)\]
[i]Proposed by Mohammad Ahmadi[/i]
2016 Balkan MO Shortlist, A4
The positive real numbers $a, b, c$ satisfy the equality $a + b + c = 1$. For every natural number $n$ find the minimal possible value of the expression $$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$
2023 Euler Olympiad, Round 1, 8
Let $a$, $b$, $c$, and $d$ be positive integers such that the following two inequalities hold: $a < 10^{20} \cdot c$ and $b > 10^{23} \cdot d$.
Determine the minimum possible value of the total number of positive integer pairs $(n, m)$ for which $n \cdot m = 2^{2023}$ and
$$ \frac {ab}{n} + \frac{cd}{m} < \frac{(a + c)(b + d)}{n + m}$$
[i]Proposed by Stijn Cambie, Belgium[/i]
2006 Estonia National Olympiad, 2
Find the smallest possible distance of points $ P$ and $ Q$ on a $ xy$-plane, if $ P$ lies on the line $ y \equal{} x$ and $ Q$ lies on the curve $ y \equal{} 2^x$.
2021 Balkan MO, 2
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$.
[i]Proposed by Athanasios Kontogeorgis, Greece[/i]
2008 Princeton University Math Competition, A7
Suppose $x^9 = 1$ but $x^3 \ne 1$. Find a polynomial of minimal degree equal to $\frac{1}{1+x}$ .
2019 Belarus Team Selection Test, 6.2
The numbers $1,2,\ldots,49,50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a,b,c$ from the board, replaces them by their sum $a+b+c$ and writes $(a+b)(b+c)(c+a)$ to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all $24$ numbers written in the notebook. Let $A$ and $B$ be the maximum and the minimum possible sums that Ann san obtain.
Find the value of $\frac{A}{B}$.
[i](I. Voronovich)[/i]
2014 AIME Problems, 12
Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.
2013 All-Russian Olympiad, 2
Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?
2002 Mongolian Mathematical Olympiad, Problem 2
Prove that for each $n\in\mathbb N$ the polynomial $(x^2+x)^{2^n}+1$ is irreducible over the polynomials with integer coefficients.
2010 China Girls Math Olympiad, 7
For given integer $n \geq 3$, set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$. Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$, one of these number does not lie in between the other two numbers in every permutations $p_i$ ($1 \leq i \leq m$). (For example, in the permutation $(1, 3, 2, 4)$, $3$ lies in between $1$ and $4$, and $4$ does not lie in between $1$ and $2$.) Determine the maximum value of $m$.
2010 Tournament Of Towns, 5
$N$ horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time? Consider the cases:
$(a) N = 3;$
$(b) N = 10.$
1949-56 Chisinau City MO, 35
The numbers $a^2, b^2, c^2$ form an arithmetic progression. Show that the numbers $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ also form arithmetic progression.
2024 Spain Mathematical Olympiad, 4
Let $a,b,c,d$ be real numbers satisfying \[abcd=1\quad \text{and}\quad a+\frac1a+b+\frac1b+c+\frac1c+d+\frac1d=0.\] Prove that at least one of the numbers $ab$, $ac$, $ad$ equals $-1$.
2016 Uzbekistan National Olympiad, 5
Solve following system equations:
\[\left\{ \begin{array}{c}
3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array}
\right.\ \ \]
2011 AIME Problems, 15
For some integer $m$, the polynomial $x^3-2011x+m$ has the three integer roots $a$, $b$, and $c$. Find $|a|+|b|+|c|$.
2006 Taiwan National Olympiad, 3
$f(x)=x^3-6x^2+17x$. If $f(a)=16, f(b)=20$, find $a+b$.
2011 District Olympiad, 4
Find the sum of the elements of the set
$$M = \left\{ \frac{n}{2}+\frac{m}{5} \,\, | m, n = 0, 1, 2,..., 100\right\}$$
2006 Thailand Mathematical Olympiad, 6
Let $a, b, c$ be positive reals. Show that $$1 +\frac{3}{ab + bc + ca}\ge \frac{6}{a + b + c}$$
1955 Poland - Second Round, 1
Calculate the sum $ x^4 + y^4 + z^4 $ knowing that $ x + y + z = 0 $ and $ x^2 + y^2 + z^2 = a $, where $ a $ is a given positive number.
2009 International Zhautykov Olympiad, 1
Find all pairs of integers $ (x,y)$, such that
\[ x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0
\]
2021 JHMT HS, 10
A sequence of real numbers $a_1, a_2, a_3, \dots$ satisfies $0 \leq a_1 \leq 1$ and $a_{n+1} = \tfrac{1 + \sqrt{a_n}}{2}$ for all positive integers $n$. If $a_1 + a_{2021} = 1$, then the product $a_1a_2a_3\cdots a_{2020}$ can be written in the form $m^k$, where $k$ is an integer and $m$ is a positive integer that is not divisible by any perfect square greater than $1$. Compute $m + k$.
1991 IberoAmerican, 6
Let $M$, $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.
1971 Bulgaria National Olympiad, Problem 2
Prove that the equation
$$\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$$
has no real solutions.
2022 Balkan MO Shortlist, A5
Find all functions $f: (0, \infty) \to (0, \infty)$ such that
\begin{align*}
f(y(f(x))^3 + x) = x^3f(y) + f(x)
\end{align*}
for all $x, y>0$.
[i]Proposed by Jason Prodromidis, Greece[/i]