Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system : $\left\{\begin{matrix} x+y+z+t=4\\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt} \end{matrix}\right.$

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 $

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 $a, \: b, \: c.$

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 $\frac{MN}{NP}$

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 $f(3,1997)$

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.

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 $n\geq 2.$

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 $n$

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?

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?

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 $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3. \]

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] $\int (1-x^2)e^x dx$

Let the number of ways for a rook to return to its original square on a $4\times 4$ chessboard in 8 moves if it starts on a corner be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path. [i]Proposed by Bradley Guo[/i]

Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$. Show that $\gamma$ is the second-largest angle in the triangle $ABC$.

Isosceles $\triangle{ABC}$ has interior point $O$ such that $AO = \sqrt{52}$, $BO = 3$, and $CO = 5$. Given that $\angle{ABC}=120^{\circ}$, find the length $AB$. [i]Proposed by Powell Zhang[/i]

Let $a{}$ and $b{}$ be positive integers, whose difference is a prime number. Prove that $(a^n+a+1)(b^n+b+1)$ is not a perfect square for infinitely many positive integers $n{}$. [i]Proposed by Vlad Matei[/i]

Two circles lie outside regular hexagon $ ABCDEF$. The first is tangent to $ \overline{AB}$, and the second is tangent to $ \overline{DE}$. Both are tangent to lines $ BC$ and $ FA$. What is the ratio of the area of the second circle to that of the first circle? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 81\qquad\textbf{(E)}\ 108$

Let $ABC$ be a triangle and $X,Y,Z$ be the tangency points of its inscribed circle with the sides $BC, CA, AB$, respectively. Suppose that $C_1, C_2, C_3$ are circle with chords $YZ, ZX, XY$, respectively, such that $C_1$ and $C_2$ intersect on the line $CZ$ and that $C_1$ and $C_3$ intersect on the line $BY$. Suppose that $C_1$ intersects the chords $XY$ and $ZX$ at $J$ and $M$, respectively; that $C_2$ intersects the chords $YZ$ and $XY$ at $L$ and $I$, respectively; and that $C_3$ intersects the chords $YZ$ and $ZX$ at $K$ and $N$, respectively. Show that $I, J, K, L, M, N$ lie on the same circle.

Find all real $(x,y)$ such that $x + {y^2} = {y^3}$ $y + {x^2} = {x^3}$

A Sudoku matrix is defined as a $ 9\times9$ array with entries from $ \{1, 2, \ldots , 9\}$ and with the constraint that each row, each column, and each of the nine $ 3 \times 3$ boxes that tile the array contains each digit from $ 1$ to $ 9$ exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of the squares in this matrix, as shown. What is the probability that the square marked by ? contains the digit $ 3$? $ \setlength{\unitlength}{6mm} \begin{picture}(9,9)(0,0) \multiput(0,0)(1,0){10}{\line(0,1){9}} \multiput(0,0)(0,1){10}{\line(1,0){9}} \linethickness{1.2pt} \multiput(0,0)(3,0){4}{\line(0,1){9}} \multiput(0,0)(0,3){4}{\line(1,0){9}} \put(0,8){\makebox(1,1){1}} \put(1,7){\makebox(1,1){2}} \put(3,6){\makebox(1,1){?}} \end{picture}$

The square polynomial $x^2+ax+b+1$ has natural roots. Prove that $(a^2+b^2)$ is a composite number.

How many triples $(x,y,z)$ of positive integers are there such that $xyz=510510$ and $x^2y+y^2z+z^2x = xy^2 + yz^2 + zx^2$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{None of above} $

Two bicyclists traveled the distance from $A$ to $B$, which is $100$ km, with speed $30$ km/h and it is known that the first started $30$ minutes before the second. $20$ minutes after the start of the first bicyclist from $A$, there is a control car started whose speed is $90$ km/h and it is known that the car is reached the first bicyclist and is driving together with him for $10$ minutes, went back to the second and was driving for $10$ minutes with him and after that the car is started again to the first bicyclist with speed $90$ km/h and etc. to the end of the distance. How many times will the car drive together with the first bicyclist? [i]K. Dochev[/i]

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

Let $G$ be an abelian group with $n$ elements, and let \[\{g_1=e,g_2,\dots,g_k\}\subsetneq G\] be a (not necessarily minimal) set of distinct generators of $G.$ A special die, which randomly selects one of the elements $g_1,g_2,\dots,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g\in G.$ Prove that there exists a real number $b\in(0,1)$ such that \[\lim_{m\to\infty}\frac1{b^{2m}}\sum_{x\in G}\left(\mathrm{Prob}(g=x)-\frac1n\right)^2\] is positive and finite.