Found problems: 85335
2017 Kazakhstan National Olympiad, 2
For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds
$$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$
2020 Purple Comet Problems, 4
The gure below shows a large circle with area $120$ containing a circle with half of the radius of the large circle and six circles with a quarter of the radius of the large circle. Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/7/9/064a05feb9bd67896c079a5141bf7556d7165b.png[/img]
2006 Bulgaria Team Selection Test, 2
Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \]
[i] Nikolai Nikolov[/i]
2000 Czech And Slovak Olympiad IIIA, 5
Monika made a paper model of a tetrahedron whose base is a right-angled triangle. When she cut the model along the legs of the base and the median of a lateral face corresponding to one of the legs, she obtained a square of side a. Compute the volume of the tetrahedron.
2003 Tournament Of Towns, 4
Several squares on a $15 \times 15$ chessboard are marked so that a bishop placed on any square of the board attacks at least two of marked squares. Find the minimal number of marked squares.
2015 Middle European Mathematical Olympiad, 3
There are $n$ students standing in line positions $1$ to $n$. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position $i$ is now in position $j$, we say the student moved for $|i-j|$ steps. Determine the maximal sum of steps of all students that they can achieve.
2014 Danube Mathematical Competition, 3
Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.
2019 PUMaC Individual Finals A, B, B1
Find all pairs of nonnegative integers $(n, m)$ such that $2^n = 7^m + 9$.
2014 Contests, 3
Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$.
[i]Warut Suksompong, Thailand[/i]
1994 Portugal MO, 4
To date, in each Mathematics Olympiad Final, no participant has been able to solve all the problems, but every problem has been solved by at least one participant. Prove that in each Final, there was a participant $A$ who solved a problem $P_A$ and another participant $B$ who solved a problem $P_B$ such that $A$ did not solve $P_B$ and $B$ did not solve $P_A$.
1976 Spain Mathematical Olympiad, 6
Given a square matrix $M$ of order $n$ over the field of numbers real, find, as a function of $M$, two matrices, one symmetric and one antisymmetric, such that their sum is precisely $ M$.
PEN E Problems, 37
It's known that there is always a prime between $n$ and $2n-7$ for all $n \ge 10$. Prove that, with the exception of $1$, $4$, and $6$, every natural number can be written as the sum of distinct primes.
2013 Sharygin Geometry Olympiad, 7
Let $BD$ be a bisector of triangle $ABC$. Points $I_a$, $I_c$ are the incenters of triangles $ABD$, $CBD$ respectively. The line $I_aI_c$ meets $AC$ in point $Q$. Prove that $\angle DBQ = 90^\circ$.
2015 JHMT, 6
Consider the parallelogram $ABCD$ such that $CD = 8$ and $BC = 14$. The diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$ and $AC = 16$. Consider a point $F$ on the segment $\overline{ED}$ with $F D =\frac{\sqrt{66}}{3}$. Compute $CF$.
2014 Contests, 1
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$.
2012-2013 SDML (High School), 3
What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?
STEMS 2021 Phy Cat A, Q3
There are two semi-infinite plane mirrors inclined physically at a non-zero angle with inner surfaces being reflective.
[list]
[*] Prove that all lines of incident/reflected rays are tangential to a particular circle for any given incident ray being incident on a reflective side. Assume that the incident ray lies on one of the normal planes to the mirrors.[/*]
[*] Try to guess the radius of circle by the parameters you can observe. [/*]
[/list]
Champions Tournament Seniors - geometry, 2015.3
Given a triangle $ABC$. Let $\Omega$ be the circumscribed circle of this triangle, and $\omega$ be the inscribed circle of this triangle. Let $\delta$ be a circle that touches the sides $AB$ and $AC$, and also touches the circle $\Omega$ internally at point $D$. The line $AD$ intersects the circle $\Omega$ at two points $P$ and $Q$ ($P$ lies between $A$ and $Q$). Let $O$ and $I$ be the centers of the circles $\Omega$ and $\omega$. Prove that $OD \parallel IQ$.
2020 Peru IMO TST, 4
Find all functions $\,f: {\mathbb{N}}\rightarrow {\mathbb{N}}\,$ such that\[f(a)^{bf(b^2)}\le a^{f(b)^3}\hspace{0.2in}\text{for all}\,a,b\in \mathbb{N}. \]
2006 German National Olympiad, 2
Five points are on the surface of of a sphere of radius $1$. Let $a_{\text{min}}$ denote the smallest distance (measured along a straight line in space) between any two of these points. What is the maximum value for $a_{\text{min}}$, taken over all arrangements of the five points?
2005 Morocco TST, 2
Let $a,b,c$ be positive real numbers. Prove the inequality
\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\]
When does equality occur?
2024 TASIMO, 2
Find all positive integers $(r,s)$ such that there is a non-constant sequence $a_n$ os positive integers such that for all $n=1,2,\dots$
\[ a_{n+2}= \left(1+\frac{{a_2}^r}{{a_1}^s} \right ) \left(1+\frac{{a_3}^r}{{a_2}^s} \right ) \dots \left(1+\frac{{a_{n+1}}^r}{{a_n}^s} \right ).\]
Proposed by Navid Safaei, Iran
2017 Junior Regional Olympiad - FBH, 5
Points $K$ and $L$ are on side $AB$ of triangle $ABC$ such that $KL=BC$ and $AK=LB$. Let $M$ be a midpoint of $AC$. Prove that $\angle KML = 90^{\circ}$
1971 All Soviet Union Mathematical Olympiad, 157
a) Consider the function $$f(x,y) = x^2 + xy + y^2$$ Prove that for the every point $(x,y)$ there exist such integers $(m,n)$, that $$f((x-m),(y-n)) \le 1/2$$
b) Let us denote with $g(x,y)$ the least possible value of the $f((x-m),(y-n))$ for all the integers $m,n$. The statement a) was equal to the fact $g(x,y) \le 1/2$.
Prove that in fact, $$g(x,y) \le 1/3$$
Find all the points $(x,y)$, where $g(x,y)=1/3$.
c) Consider function $$f_a(x,y) = x^2 + axy + y^2 \,\,\, (0 \le a \le 2)$$
Find any $c$ such that $g_a(x,y) \le c$.
Try to obtain the closest estimation.
2019 LIMIT Category C, Problem 3
Which of the following series are convergent?
$\textbf{(A)}~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}}$
$\textbf{(B)}~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}}$
$\textbf{(C)}~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n$
$\textbf{(D)}~\text{None of the above}$