Found problems: 85335
2017 Greece Junior Math Olympiad, 2
Let $x,y,z$ is positive. Solve:
$\begin{cases}{x\left( {6 - y} \right) = 9}\\
{y\left( {6 - z} \right) = 9}\\
{z\left( {6 - x} \right) = 9}\end{cases}$
2015 Sharygin Geometry Olympiad, 1
In trapezoid $ABCD$ angles $A$ and $B$ are right, $AB = AD, CD = BC + AD, BC < AD$. Prove that $\angle ADC = 2\angle ABE$, where $E$ is the midpoint of segment $AD$.
(V. Yasinsky)
2001 Tuymaada Olympiad, 7
Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers.
(The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)
2006 All-Russian Olympiad, 5
Let $a_1$, $a_2$, ..., $a_{10}$ be positive integers such that $a_1<a_2<...<a_{10}$. For every $k$, denote by $b_k$ the greatest divisor of $a_k$ such that $b_k<a_k$. Assume that $b_1>b_2>...>b_{10}$. Show that $a_{10}>500$.
2012 Hanoi Open Mathematics Competitions, 7
[b]Q7.[/b] Find all integers $n$ such that $60+2n-n^2$ is a perfect square.
Kvant 2021, M2665
The polynomials $f(x)$ and $g(x)$ are given. The points $A_1(f(1),g(1)),\ldots,A_n(f(n),g(n))$ are marked on the coordinate plane. It turns out that $A_1\ldots A_n$ is a regular $n{}$-gon. Prove that the degree of at least one of $f{}$ and $g{}$ is at least $n-1$.
[i]Proposed by V. Bragin[/i]
2019 Simon Marais Mathematical Competition, B2
For each odd prime number $p$, prove that the integer
$$1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rfloor$$is divisible by $p$
(Here, $e$ denotes the base of the natural logarithm and $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)
2023 BMT, 9
Let triangle $\vartriangle ABC$ be acute, and let point $M$ be the midpoint of $\overline{BC}$. Let $E$ be on line segment $\overline{AB}$ such that $\overline{AE} \perp \overline{EC}$. Then, suppose $T$ is a point on the other side of $\overleftrightarrow{BC}$ as $A$ is such that $\angle BTM = \angle ABC$ and $\angle TCA = \angle BMT$. If $AT = 14$, $AM = 9,$ and $\frac{AE}{AC} =\frac27$ , compute $BC$.
2018 Taiwan TST Round 2, 1
Let $A,B,C$ be the midpoints of the three sides $B'C', C'A', A'B'$ of the triangle $A'B'C'$ respectively. Let $P$ be a point inside $\Delta ABC$, and $AP,BP,CP$ intersect with $BC, CA, AB$ at $P_a,P_b,P_c$, respectively. Lines $P_aP_b, P_aP_c$ intersect with $B'C'$ at $R_b, R_c$ respectively, lines $P_bP_c, P_bP_a$ intersect with $C'A'$ at $S_c, S_a$ respectively. and lines $P_cP_a, P_cP_b$ intersect with $A'B'$ at $T_a, T_b$, respectively. Given that $S_c,S_a, T_a, T_b$ are all on a circle centered at $O$.
Show that $OR_b=OR_c$.
2010 IMAC Arhimede, 5
Different points $A_1, A_2,..., A_n$ in the plane ($n> 3$) are such that the triangle $A_iA_jA_k$ is obtuse for all the different $i,j,k \in\{1,2,...,n\}$. Prove that there is a point $A_{n + 1}$ in the plane, such that the triangle $A_iA_jA_{n + 1}$ is obtuse for all different $i,j \in\{1,2,...,n\}$
2005 Abels Math Contest (Norwegian MO), 2a
In an aquarium there are nine small fish. The aquarium is cube shaped with a side length of two meters and is completely filled with water. Show that it is always possible to find two small fish with a distance of less than $\sqrt3$ meters.
2006 Croatia Team Selection Test, 3
Let $ABC$ be a triangle for which $AB+BC = 3AC$. Let $D$ and $E$ be the points of tangency of the incircle with the sides $AB$ and $BC$ respectively, and let $K$ and $L$ be the other endpoints of the diameters originating from $D$ and $E.$ Prove that $C , A, L$, and $K$ lie on a circle.
2014 South East Mathematical Olympiad, 1
Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$
2011 Morocco National Olympiad, 2
Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system :
$\left\{\begin{matrix}
x+y+z+t=4\\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt}
\end{matrix}\right.$
2007 Turkey MO (2nd round), 3
If $a,b,c$ are three positive real numbers such that $a+b+c=3$, prove that
$ {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4 $
2010 Contests, 2
Show that
\[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \]
for all positive real numbers $a, \: b, \: c.$
2016 Hong Kong TST, 5
Let $ABCD$ be inscribed in a circle with center $O$. Let $E$ be the intersection of $AC$ and $BD$. $M$ and $N$ are the midpoints of the arcs $AB$ and $CD$ respectively (the arcs not containing any other vertices). Let $P$ be the intersection point of $EO$ and $MN$. Suppose $BC=5$, $AC=11$, $BD=12$, and $AD=10$. Find $\frac{MN}{NP}$
1997 Romania National Olympiad, 1
function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon:
1- $f(0,x)=x+1$
2- $f(x+1,0)=f(x,1)$
3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997)
find $f(3,1997)$
2014 Purple Comet Problems, 11
Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have.
2021 Romania Team Selection Test, 3
Let $\alpha$ be a real number in the interval $(0,1).$ Prove that there exists a sequence $(\varepsilon_n)_{n\geq 1}$ where each term is either $0$ or $1$ such that the sequence $(s_n)_{n\geq 1}$ \[s_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}\]verifies the inequality \[0\leq \alpha-2ns_n\leq\frac{2}{n+1}\] for any $n\geq 2.$
2023 Belarusian National Olympiad, 10.2
A positive integers has exactly $81$ divisors, which are located in a $9 \times 9$ table such that for any two numbers in the same row or column one of them is divisible by the other one.
Find the maximum possible number of distinct prime divisors of $n$
2007 Sharygin Geometry Olympiad, 5
A non-convex $n$-gon is cut into three parts by a straight line, and two parts are put together so that the resulting polygon is equal to the third part. Can $n$ be equal to:
a) five?
b) four?
2003 Purple Comet Problems, 10
How many gallons of a solution which is $15\%$ alcohol do we have to mix with a solution that is $35\%$ alcohol to make $250$ gallons of a solution that is $21\%$ alcohol?
2000 India Regional Mathematical Olympiad, 3
Suppose $\{ x_n \}_{n\geq 1}$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$, and for all $n$ \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 . \] Show that for all $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3. \]
2005 Today's Calculation Of Integral, 6
Calculate the following indefinite integrals.
[1] $\int \sin x\cos ^ 3 x dx$
[2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$
[3] $\int x^2 \sqrt{x^3+1}dx$
[4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$
[5] $\int (1-x^2)e^x dx$