Let $ABCD$ be a convex quadrilateral such that $\angle A, \angle B, \angle C$ are acute. $AB$ and $CD$ meet at $E$ and $BC,DA$ meet at $F$. Let $K,L,M,N$ be the midpoints of $AB,BC,CD,DA$ repectively. $KM$ meets $BC,DA$ at $X$ and $Y$, and $LN$ meets $AB,CD$ at $Z$ and $W$. Prove that the line passing $E$ and the midpoint of $ZW$ is parallel to the line passing $F$ and the midpoint of $XY$.

A group of $n$ people play a board game with the following rules: 1) In each round of the game exactly $3$ people play 2) The game ends after exactly $n$ rounds 3) Every pair of players has played together at least at one round Find the largest possible value of $n$

Find all triplets of natural numbers $(a,b,c)$ that satisfy the equation $abc=a+b+c+1$.

The triangle $ABC$ is inscribed in a circle. At points $A$ and $B$ are tangents to this circle, which intersect at point $T$. A line drawn through the point $T$ parallel to the side $AC$ intersects the side $BC$ at the point $D$. Prove that $AD = CD$.

Solve the system of equation for $(x,y) \in \mathbb{R}$ $$\left\{\begin{matrix} \sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\ 3x^2+4xy=24 \end{matrix}\right.$$ \\ Explain your answer

Given a strictly increasing infinite sequence $\{a_n\}$ of positive real numbers such that for any $n\in N$: $$a_{n+2}=(a_{n+1}-a_{n})^{\sqrt{n}}+n^{-\sqrt{n}}$$ Prove that for any $C>0$ there exist a positive integer $m(C)$ (depended on $C$) such that $a_{m(C)}>C$.

Find all positive integers $n$ such that a positive integer power of $2n^2+4n-1$ equals to a positive integer power of $3n+4$.

Suppose that the vertices of a polygon all lie on a rectangular lattice of points where adjacent points on the lattice are a distance 1 apart. Then the area of the polygon can be found using Pick’s Formula: $I + \frac{B}{2}$ −1, where I is the number of lattice points inside the polygon, and B is the number of lattice points on the boundary of the polygon. Pat applied Pick’s Formula to find the area of a polygon but mistakenly interchanged the values of I and B. As a result, Pat’s calculation of the area was too small by 35. Using the correct values for I and B, the ratio n = $\frac{I}{B}$ is an integer. Find the greatest possible value of n.

Consider a rectangle $A_1A_2A_3A_4$ and a circle $\mathcal{C}_i$ centered at $A_i$ with radius $r_i$ for $i=1,2,3,4$. Suppose that $r_1+r_3=r_2+r_4<d$, where $d$ is the diagonal of the rectangle. The two pairs of common outer tangents of $\mathcal{C}_1$ and $\mathcal{C}_3$, and of $\mathcal{C}_2$ and $\mathcal{C}_4$ form a quadrangle. Prove that this quadrangle has an inscribed circle.

$a,b,c$ are three non-zero-complex numbers, and $\frac{a}{b}=\frac{b}{c}=\frac{c}{a}$, then the value of $\frac{a+b-c}{a-b+c}$ is ($\omega=-\frac{1}{2}+\frac{\sqrt3}{2}\text{i}$) $\text{(A)}1\qquad\text{(B)}\pm\omega\qquad\text{(C)}1,\omega,\omega^2\qquad\text{(D)}1,-\omega,-\omega^2$

Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on the next flip can be expressed in the form $\tfrac{p}{q}$ for positive integers $p, q$ satisfying $\gcd(p, q) = 1$, what is $p + q$?

Let $P_n = (19 + 92)(19^2 +92^2) \cdots(19^n +92^n)$ for each positive integer $n$. Determine, with proof, the least positive integer $m$, if it exists, for which $P_m$ is divisible by $33^{33}.$

Prove that any permutation group of an order equal to a power of $ 2 $ contains a commutative subgroup whose order is the square of the exponent of the order of the group.

Let $P$ be a point in the interior of an equilateral triangle with height $1$, and let $x,y,z$ denote the distances from $P$ to the three sides of the triangle. Prove that \[ x^2+y^2+z^2 ~\ge~ x^3+y^3+z^3 +6xyz \]

Group of $10$ people are buying books. We know the following: $i)$ Every person bought four different books $ii)$ Every two persons bought at least one book common for both of them Taking in consideration book which was bought by maximum number of people, determine minimal value of that number

grade IX P4, X P3 The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the circumscirbed circle of this triangle at the points $ A_0 $ and $ C_0 $, respectively. The straight line passing through the center of the inscribed circle of triangle $ ABC $ parallel to the side of $ AC $, intersects with the line $ A_0C_0 $ at $ P $. Prove that the line $ PB $ is tangent to the circumcircle of the triangle $ ABC $. grade XI P4 The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the sides at the points $ A_1 $ and $ C_1 $, and the circumcircle of this triangle at points $ A_0 $ and $ C_0 $ respectively. Straight lines $ A_1C_1 $ and $ A_0C_0 $ intersect at point $ P $. Prove that the segment connecting $ P $ with the center inscribed circles of triangle $ ABC $, parallel to $ AC $.

Let $ABC$ be a scalene triangle and $\omega$ is your incircle. The sides $BC,CA$ and $AB$ are tangents to $\omega$ in $X,Y,Z$ respectively. Let $M$ be the midpoint of $BC$ and $D$ is the intersection point of $BC$ with the angle bisector of $\angle BAC$. Prove that $\angle BAX=\angle MAC$ if and only if $YZ$ passes by the midpoint of $AD$.

The altitudes $AA_1$ and $BB_1$ of an acute-angled triangle ABC meet at point $O$. Let $A_1A_2$ and $B_1B_2$ be the altitudes of triangles $OBA_1$ and $OAB_1$ respectively. Prove that $A_2B_2$ is parallel to $AB$. (L.Steingarts)

Let $LOV ER$ be a convex pentagon such that $LOV E$ is a rectangle. Given that $OV = 20$ and $LO =V E = RE = RL = 23$, compute the radius of the circle passing through $R$, $O$, and $V$ .

Call a tuple $(b_m, b_{m+1},..., b_n)$ of integers perfect if both following conditions are fulfilled: 1. There exists an integer $a > 1$ such that $b_k = a^k + 1$ for all $k = m, m + 1,..., n$ 2. For all $k = m, m + 1,..., n,$ there exists a prime number $q$ and a non-negative integer $t$ such that $b_k = q^t$. Prove that if $n - m$ is large enough then there is no perfect tuples, and find all perfect tuples with the maximal number of components.

Inscribe a triangle in a given circle, if its smallest side is known, as well as the point of intersection of altitudes lying outside the circle.

A steak temperature $5^\circ$ is put into an oven. After $15$ minutes, it has temperature $45^\circ$. After another $15$ minutes it has temperature $77^\circ$. The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.

We have many $\text{three-element}$ subsets of a $1000\text{-element}$ set. We know that the union of every $5$ of them has at least $12$ elements. Find the most possible value for the number of these subsets.

Let $f(x)$ be a non-constant polynomial with integer coefficients and $n,k$ be natural numbers. Show that there exist $n$ consecutive natural numbers $a,a+1,\ldots,a+n-1$ such that the numbers $f(a),f(a+1),\ldots,f(a+n-1)$ all have at least $k$ prime factors. (We say that the number $p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ has $\alpha_1+\ldots+\alpha_s$ prime factors.)

In triangle ABC, $\angle$ A equals 120 degrees. A point D is inside the triangle such that $\angle$DBC = 2 $\times \angle $ABD and $\angle$DCB = 2 $\times \angle$ACD. Determine the measure, in degrees, of $\angle$ BDC. [asy] pair A = (5,4); pair B = (0,0); pair C = (10,0); pair D = (5,2.5) ; draw(A--B); draw(B--C); draw(C--A); draw (B--D--C); label ("A", A, dir(45)); label ("B", B, dir(45)); label ("C", C, dir(45)); label ("D", D, dir(45)); [/asy]