Found problems: 15925
2017 Harvard-MIT Mathematics Tournament, 5
Given that $a,b,c$ are integers with $abc = 60$, and that complex number $\omega \neq 1$ satisfies $\omega^3=1$, find the minimum possible value of $| a + b\omega + c\omega^2|$.
STEMS 2021-22 Math Cat A-B, A1
Let $f$ be an irreducible monic polynomial with integer coefficients such that $f(0)$ is
not equal to $1$. Let $z$ be a complex number that is a root of $f$. Show that if $w$ is another complex
root of $f$, then $\frac{z}{w}$ cannot be a positive integer greater than $1$.
2006 Turkey Team Selection Test, 3
If $x,y,z$ are positive real numbers and $xy+yz+zx=1$ prove that
\[ \frac{27}{4} (x+y)(y+z)(z+x) \geq ( \sqrt{x+y} +\sqrt{ y+z} + \sqrt{z+x} )^2 \geq 6 \sqrt 3. \]
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
2016 Federal Competition For Advanced Students, P2, 4
Let $a,b,c\ge-1$ be real numbers with $a^3+b^3+c^3=1$.
Prove that $a+b+c+a^2+b^2+c^2\le4$, and determine the cases of equality.
(Proposed by Karl Czakler)
2009 Moldova Team Selection Test, 1
Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.
1989 IMO Longlists, 13
Let $ n \leq 44, n \in \mathbb{N}.$ Prove that for any function $ f$ defined over $ \mathbb{N}^2$ whose images are in the set $ \{1, 2, \ldots , n\},$ there are four ordered pairs $ (i, j), (i, k), (l, j),$ and $ (l, k)$ such that \[ f(i, j) \equal{} f(i, k) \equal{} f(l, j) \equal{} f(l, k),\] in which $ i, j, k, l$ are chosen in such a way that there are natural numbers $ m, p$ that satisfy \[ 1989m \leq i < l < 1989 \plus{} 1989m\] and \[ 1989p \leq j < k < 1989 \plus{} 1989p.\]
2013 Dutch BxMO/EGMO TST, 4
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]
1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5
Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$.
A. 1
B. 3
C. 7
D. 12
E. None of these
1999 Mongolian Mathematical Olympiad, Problem 4
A forest grows up $p$ percent during a summer, but gets reduced by $x$ units between two summers. At the beginning of this summer, the size of the forest has been $a$ units. How large should $x$ be if we want the forest to increase $q$ times in $n$ years?
2021 CMIMC, 2.6 1.3
Let $a$ and $b$ be complex numbers such that $(a+1)(b+1)=2$ and $(a^2+1)(b^2+1)=32.$ Compute the sum of all possible values of $(a^4+1)(b^4+1).$
[i]Proposed by Kyle Lee[/i]
2021 Taiwan TST Round 2, 5
Let $\|x\|_*=(|x|+|x-1|-1)/2$. Find all $f:\mathbb{N}\to\mathbb{N}$ such that
\[f^{(\|f(x)-x\|_*)}(x)=x, \quad\forall x\in\mathbb{N}.\]
Here $f^{(0)}(x)=x$ and $f^{(n)}(x)=f(f^{(n-1)}(x))$ for all $n\in\mathbb{N}$.
[i]Proposed by usjl[/i]
2004 Thailand Mathematical Olympiad, 6
Let $a, b, c > 0$ satisfy $a + b + c \ge \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$. Prove that $a^3 + b^3 + c^3 \ge a + b + c$
2021 Israel Olympic Revenge, 1
Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to\mathbb N$ such that $$\frac{f(x)-f(y)+x+y}{x-y+1}$$ is an integer, for all positive integers $x,y$ with $x>y$.
1990 Vietnam Team Selection Test, 3
Prove that there is no real function $ f(x)$ satisfying $ f\left(f(x)\right) \equal{} x^2 \minus{} 2$ for all real number $ x$.
1983 Czech and Slovak Olympiad III A, 1
Let $n$ be a positive integer and $k\in[0,n]$ be a fixed real constant. Find the maximum value of $$\left|\sum_{i=1}^n\sin(2x_i)\right|$$ where $x_1,\ldots,x_n$ are real numbers satisfying $$\sum_{i=1}^n\sin^2(x_i)=k.$$
2007 Alexandru Myller, 1
[b]a)[/b] Show that $ n^2+2n+2007 $ is squarefree for any natural number $ n. $
[b]b)[/b] Prove that for any natural number $ k\ge 2 $ there is a nonnegative integer $ m $ such that $ m^2+2m+2k $ is a perfect square.
2013 Bangladesh Mathematical Olympiad, 2
Higher Secondary P2
Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.
1982 Spain Mathematical Olympiad, 7
Let $S$ be the subset of rational numbers that can be written in the form $a/b$, where $a$ is any integer and $b$ is an odd integer. Does the sum of two of its elements belong to the $S$ ? And the product? Are there elements in $S$ whose inverse belongs to $S$ ?
1973 IMO, 3
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.
2016 Dutch IMO TST, 1
Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$
1978 IMO Longlists, 26
For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$
1999 All-Russian Olympiad Regional Round, 11.8
For some polynomial there is an infinite set its values, each of which takes at least at two integer points. Prove that there is at most one the integer value that a polynomial takes at exactly one integer point.
2016 Czech And Slovak Olympiad III A, 4
For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$.
Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value
1951 Moscow Mathematical Olympiad, 193
Prove that the first 3 digits after the decimal point in the decimal expression of the number $\frac{0.123456789101112 . . . 495051}{0.515049 . . . 121110987654321}$ are $239$.