Found problems: 85335
1976 Poland - Second Round, 5
Prove that if $ \cos \pi x =\frac{1}{3} $ then $ x $ is an irrational number.
2024 Australian Mathematical Olympiad, P5
The sequence of positive integers $a_1, a_2, \ldots, a_{2025}$ is defined as follows: $a_1=2^{2024}+1$ and $a_{n+1}$ is the greatest prime factor of $a_n^2-1$ for $1 \leq n \leq 2024$. Find the value of $a_{2024}+a_{2025}$.
2014 Iran Team Selection Test, 5
if $x,y,z>0$ are postive real numbers such that $x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}$
prove that \[((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})\]
2019 Romania National Olympiad, 1
a) Prove that for $x,y \ge 1$, holds $$x+y - \frac{1}{x}- \frac{1}{y} \ge 2\sqrt{xy} -\frac{2}{\sqrt{xy}}$$
b) Prove that for $a,b,c,d \ge 1$ with $abcd=16$ , holds $$a+b+c+d-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}\ge 6$$
2007 Hungary-Israel Binational, 2
Let $ a,b,c,d$ be real numbers, such that $ a^2\le 1, a^2 \plus{} b^2\le 5, a^2 \plus{} b^2 \plus{} c^2\le 14, a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2\le 30$. Prove that $ a \plus{} b \plus{} c \plus{} d\le 10$.
2003 All-Russian Olympiad Regional Round, 8.1
The numbers from $1$ to $10$ were divided into two groups so that the product of the numbers in the first group is completely divisible by the product of the numbers in the second. Which the smallest value can be for the quotient of the first product money for the second?
2011 Pre-Preparation Course Examination, 4
represent a way to calculate $\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^3}=1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+...$.
2015 Junior Balkan Team Selection Tests - Romania, 3
Prove that if $a,b,c>0$ and $a+b+c=1,$ then $$\frac{bc+a+1}{a^2+1}+\frac{ca+b+1}{b^2+1}+\frac{ab+c+1}{c^2+1}\leq \frac{39}{10}$$
2005 VJIMC, Problem 2
Let $f:A^3\to A$ where $A$ is a nonempty set and $f$ satisfies:
(a) for all $x,y\in A$, $f(x,y,y)=x=f(y,y,x)$ and
(b) for all $x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3\in A$,
$$f(f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3))=f(f(x_1,y_1,z_1),f(x_2,y_2,z_2),f(x_3,y_3,z_3)).$$
Prove that for an arbitrary fixed $a\in A$, the operation $x+y=f(x,a,y)$ is an Abelian group addition.
2006 Abels Math Contest (Norwegian MO), 4
Let $\gamma$ be the circumscribed circle about a right-angled triangle $ABC$ with right angle $C$. Let $\delta$ be the circle tangent to the sides $AC$ and $BC$ and tangent to the circle $\gamma$ internally.
(a) Find the radius $i$ of $\delta$ in terms of $a$ when $AC$ and $BC$ both have length $a$.
(b) Show that the radius $i$ is twice the radius of the inscribed circle of $ABC$.
1997 German National Olympiad, 2
For a positive integer $k$, let us denote by $u(k)$ the greatest odd divisor of $k$.
Prove that, for each $n \in N$, $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$.
2013 Iran MO (3rd Round), 4
Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation:
\[x^2 - (n^2 +1)y^2 = n^2.\]
(25 points)
1977 All Soviet Union Mathematical Olympiad, 236
Given several points, not all lying on one straight line. Some number is assigned to every point. It is known, that if a straight line contains two or more points, than the sum of the assigned to those points equals zero. Prove that all the numbers equal to zero.
2020 Regional Olympiad of Mexico Center Zone, 6
Let $n,k$ be integers such that $n\geq k\geq3$. Consider $n+1$ points in a plane (there is no three collinear points) and $k$ different colors, then, we color all the segments that connect every two points. We say that an angle is good if its vertex is one of the initial set, and its two sides aren't the same color. Show that there exist a coloration such that the \\ total number of good angles is greater than $n \binom{k}{2} \lfloor(\frac{n}{k})\rfloor^2$
2018 Portugal MO, 3
How many ways are there to paint an $m \times n$ board, so that each square is painted blue, white, brown or gold, and in each $2 \times 2$ square there is one square of each color?
2017 Saint Petersburg Mathematical Olympiad, 1
It’s allowed to replace any of three coefficients of quadratic trinomial by its discriminant. Is it true that from any quadratic trinomial that does not have real roots, we can perform such operation several times to get a quadratic trinomial that have real roots?
2012 Sharygin Geometry Olympiad, 6
Let $\omega$ be the circumcircle of triangle $ABC$. A point $B_1$ is chosen on the prolongation of side $AB$ beyond point B so that $AB_1 = AC$. The angle bisector of $\angle BAC$ meets $\omega$ again at point $W$. Prove that the orthocenter of triangle $AWB_1$ lies on $\omega$ .
(A.Tumanyan)
2010 South East Mathematical Olympiad, 1
Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.
2002 AMC 12/AHSME, 6
Suppose that $ a$ and $ b$ are are nonzero real numbers, and that the equation $ x^2\plus{}ax\plus{}b\equal{}0$ has solutions $ a$ and $ b$. Then the pair $ (a,b)$ is
$ \textbf{(A)}\ (\minus{}2,1) \qquad
\textbf{(B)}\ (\minus{}1,2) \qquad
\textbf{(C)}\ (1,\minus{}2) \qquad
\textbf{(D)}\ (2,\minus{}1) \qquad
\textbf{(E)}\ (4,4)$
1995 Tournament Of Towns, (447) 3
Given the equilateral triangle $ABC$, find the locus of all points $P$ such that the segments of the lines $AP$ and $BP$ lying inside the triangle are equal.
2010 Abels Math Contest (Norwegian MO) Final, 4a
Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.
2020 LMT Fall, B1
Four $L$s are equivalent to three $M$s. Nine $M$s are equivalent to fourteen $T$ s. Seven $T$ s are equivalent to two $W$ s. If Kevin has thirty-six $L$s, how many $W$ s would that be equivalent to?
2018 Harvard-MIT Mathematics Tournament, 1
What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?
2014 Junior Regional Olympiad - FBH, 1
If $a$ and $b$ are digits, how many are there $4$ digit numbers $\overline{3ab4}$ divisible with $9$ . Which numbers are they ($4$ digit numbers)?
2016 Tournament Of Towns, 4
There are $2016$ red and $2016$ blue cards each having a number written on it. For some $64$ distinct positive real numbers, it is known that the set of numbers on cards of a particular color happens to be the set of their pairwise sums and the other happens to be the set of their pairwise products. Can we necessarily determine which color corresponds to sum and which to product?
[i](B. Frenkin)[/i]
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])