Found problems: 1269
2010 Tournament Of Towns, 3
Consider a composition of functions $\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos$, applied to the number $1$. Each function may be applied arbitrarily many times and in any order. (ex: $\sin \cos \arcsin \cos \sin\cdots 1$). Can one obtain the number $2010$ in this way?
2001 Romania Team Selection Test, 2
Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that
\[f(x+y)\ge f(x)+yf(f(x)) \]
for every $x,y\in (0,\infty )$.
2014 Contests, 3
Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.
1987 Vietnam National Olympiad, 1
Let $ u_1$, $ u_2$, $ \ldots$, $ u_{1987}$ be an arithmetic progression with $ u_1 \equal{} \frac {\pi}{1987}$ and the common difference $ \frac {\pi}{3974}$. Evaluate
\[ S \equal{} \sum_{\epsilon_i\in\left\{ \minus{} 1, 1\right\}}\cos\left(\epsilon_1 u_1 \plus{} \epsilon_2 u_2 \plus{} \cdots \plus{} \epsilon_{1987} u_{1987}\right)
\]
2006 Estonia National Olympiad, 3
Let there be $ n \ge 2$ real numbers such that none of them is greater than the arithmetic mean of the other numbers. Prove that all the numbers are equal.
1979 IMO Longlists, 72
Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$, then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$.
2005 China National Olympiad, 4
The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]
2002 USA Team Selection Test, 4
Let $n$ be a positive integer and let $S$ be a set of $2^n+1$ elements. Let $f$ be a function from the set of two-element subsets of $S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any elements $x, y, z$ of $S$, one of $f(\{x,y\}), f(\{y,z\}), f(\{z, x\})$ is equal to the sum of the other two. Show that there exist $a, b, c$ in $S$ such that $f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$ are all equal to 0.
1989 IMO Longlists, 46
Let S be the point of intersection of the two lines $ l_1 : 7x \minus{} 5y \plus{} 8 \equal{} 0$ and $ l_2 : 3x \plus{} 4y \minus{} 13 \equal{} 0.$ Let $ P \equal{} (3, 7), Q \equal{} (11, 13),$ and let $ A$ and $ B$ be points on the line $ PQ$ such that $ P$ is between $ A$ and $ Q,$ and $ B$ is between $ P$ and $ Q,$ and such that \[ \frac{PA}{AQ} \equal{} \frac{PB}{BQ} \equal{} \frac{2}{3}.\] Without finding the coordinates of $ B$ find the equations of the lines $ SA$ and $ SB.$
1986 IMO Longlists, 40
Find the maximum value that the quantity $2m+7n$ can have such that there exist distinct positive integers $x_i \ (1 \leq i \leq m), y_j \ (1 \leq j \leq n)$ such that the $x_i$'s are even, the $y_j$'s are odd, and $\sum_{i=1}^{m} x_i +\sum_{j=1}^{n} y_j=1986.$
1989 IMO Longlists, 11
Given the equation \[ y^4 \plus{} 4y^2x \minus{} 11y^2 \plus{} 4xy \minus{} 8y \plus{} 8x^2 \minus{} 40x \plus{} 52 \equal{} 0,\] find all real solutions.
1989 Irish Math Olympiad, 4
Let $a$ be a positive real number and let
$b= \sqrt[3] {a+ \sqrt {a^{2}+1}} + \sqrt[3] {a- \sqrt {a^{2}+1}}$.
Prove that $b$ is a positive integer if, and only if, $a$ is a positive integer of the form $\frac{1}{2} n(n^{2}+3)$, for some positive integer $n$.
2006 China Second Round Olympiad, 3
Solve the system of equations in real numbers:
\[ \begin{cases} x-y+z-w=2 \\ x^2-y^2+z^2-w^2=6 \\ x^3-y^3+z^3-w^3=20 \\ x^4-y^4+z^4-w^4=66 \end{cases} \]
1985 Austrian-Polish Competition, 1
Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.
1993 Turkey MO (2nd round), 4
$a_{n}$ is a sequence of positive integers such that, for every $n\geq 1$, $0<a_{n+1}-a_{n}<\sqrt{a_{n}}$. Prove that for every $x,y\in{R}$ such that $0<x<y<1$ $x< \frac{a_{k}}{a_{m}}<y$ we can find such $k,m\in{Z^{+}}$.
1984 Vietnam National Olympiad, 2
The sequence $(u_n)$ is defined by $u_1 = 1, u_2 = 2$ and $u_{n+1} = 3u_n - u_{n-1}$ for $n \ge 2$. Set $v_n =\sum_{k=1}^n \text{arccot }u_k$. Compute $\lim_{n\to\infty} v_n$.
2012 Nordic, 3
Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \dots , x_n$ (not necessarily different), with $1\le x_k\le n$, $1\le k\le n$, and such that
\[x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,\]
but $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , n\}$.
1996 China Team Selection Test, 3
Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?
1974 IMO Longlists, 39
Let $n$ be a positive integer, $n \geq 2$, and consider the polynomial equation
\[x^n - x^{n-2} - x + 2 = 0.\]
For each $n,$ determine all complex numbers $x$ that satisfy the equation and have modulus $|x| = 1.$
2006 APMO, 2
Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden ratio.
1998 Polish MO Finals, 1
Find all solutions in positive integers to:
\begin{eqnarray*} a + b + c = xyz \\ x + y + z = abc \end{eqnarray*}
2018 Spain Mathematical Olympiad, 6
Find all functions such that $ f: \mathbb{R}^\plus{} \rightarrow \mathbb{R}^\plus{}$ and $ f(x\plus{}f(y))\equal{}yf(xy\plus{}1)$ for every $ x,y\in \mathbb{R}^\plus{}$.
2014 Indonesia MO, 4
Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)
2010 Tournament Of Towns, 4
Assume that $P(x)$ is a polynomial with integer non negative coefficients, different from constant. Baron Munchausen claims that he can restore $P(x)$ provided he knows the values of $P(2)$ and $P(P(2))$ only. Is the baron's claim valid?
2012 Philippine MO, 4
Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$,
(i) $(x + 1)\star 0 = (0\star x) + 1$
(ii) $0\star (y + 1) = (y\star 0) + 1$
(iii) $(x + 1)\star (y + 1) = (x\star y) + 1$.
If $123\star 456 = 789$, find $246\star 135$.