Let $ABC$ be a triangle and $D$ a point on $BC$ such that $AB =\sqrt2$, $AC =\sqrt3$, $\angle BAD = 30^o$, and $\angle CAD = 45^o$. Find $AD$.

Let $a$ and $b$ be two real numbers. Find these numbers given that the graphs of $f:\mathbb{R} \to \mathbb{R} , f(x)=2x^4-a^2x^2+b-1$ and $g:\mathbb{R} \to \mathbb{R} ,g(x)=2ax^3-1$ have exactly two points of intersection.

Does the bellow depicted figure fit into a square $5\times5$.

Suppose there are $n\geq{5}$ different points arbitrarily arranged on a circle, the labels are $1, 2,\dots $, and $n$, and the permutation is $S$. For a permutation , a “descending chain” refers to several consecutive points on the circle , and its labels is a clockwise descending sequence (the length of sequence is at least $2$), and the descending chain cannot be extended to longer .The point with the largest label in the chain is called the "starting point of descent", and the other points in the chain are called the “non-starting point of descent” . For example: there are two descending chains $5, 2$and $4, 1$ in $5, 2, 4, 1, 3$ arranged in a clockwise direction, and $5$ and $4$ are their starting points of descent respectively, and $2, 1$ is the non-starting point of descent . Consider the following operations: in the first round, find all descending chains in the permutation $S$, delete all non-starting points of descent , and then repeat the first round of operations for the arrangement of the remaining points, until no more descending chains can be found. Let $G(S)$ be the number of all descending chains that permutation $S$ has appeared in the operations, $A(S)$ be the average value of $G(S)$of all possible n-point permutations $S$. (1) Find $A(5)$. (2)For $n\ge{6}$ , prove that $\frac{83}{120}n-\frac{1}{2} \le A(S) \le \frac{101}{120}n-\frac{1}{2}.$

Let $a$ and $b$ be real numbers greater than $1$ for which there exists a positive real number $c$, different from $1$, such that \[2(\log_ac+\log_bc)=9\log_{ab}c.\] Find the largest possible value of $\log_ab$. $\textbf{(A) }\sqrt2\qquad\textbf{(B) }\sqrt3\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt6\qquad\textbf{(E) }3$

The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.

Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.

An infinite network is partitioned with dominos. Prove there exist three other tilings with dominos, have neither common domino with the existing tiling nor with each other. Clarifications for network: It means an infinite board consisting of square cells.

The six offices of the city of Salavaara are to be connected to each other by a communication network which utilizes modern picotechnology. Each of the offices is to be connected to all the other ones by direct cable connections. Three operators compete to build the connections, and there is a separate competition for every connection. When the network is finished one notices that the worst has happened: the systems of the three operators are incompatible. So the city must reject connections built by two of the operators, and these are to be chosen so that the damage is minimized. What is the minimal number of offices which still can be connected to each other, possibly through intermediate offices, in the worst possible case.

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

Evaluate $1001^3 - 1000^3$

What positive number is equal to twice its square? $\text{(A) }\frac{1}{4}\qquad\text{(B) }\frac{1}{2}\qquad\text{(C) }1\qquad\text{(D) }2\qquad\text{(E) }\frac{5}{2}$

Let $f(x)$ be a real-valued function defined on the positive reals such that (1) if $x < y$, then $f(x) < f(y)$, (2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$. Show that $f(x) < 0$ for some value of $x$.

Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]

For any positive integer $a$, let $\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest possible value of $4\tau(n)-n$ over all positive integers $n$.

For a multiple of $ kb$ of $ b$ let $ a \% kb$ be the greatest number such that $ a \% kb \equal{} a \bmod b$ which is smaller than $ kb$ and not greater than $ a$ itself. Let $ n \in \mathbb{Z}^ \plus{} .$ Determine all integer pairs $ (a,b)$ with: \[ a\%b \plus{} a\%2b \plus{} a\%3b \plus{} \ldots \plus{} a\%nb \equal{} a \plus{} b \]

$200n$ diagonals are drawn in a convex $n$-gon. Prove that one of them intersects at least 10000 others. [b]proposed by D. Karpov, S. Berlov[/b]

Consider the recurrence relation $x_{n+2}=2x_{n+1}+x_n,$ with $x_0=0, x_1=1.$ What is the greatest common divisor of $x_{2023}$ and $x_{721}$?

Given positive integers $k,m, n$ with $km \leq n$ and non-negative real numbers $x_1, \ldots , x_k$, prove that \[n \left( \prod_{i=1}^k x_i^m -1 \right) \leq m \sum_{i=1}^k (x_i^n-1).\]

$ABC$ is an equilateral triangle and $l$ is a line such that the distances from $A, B,$ and $C$ to $l$ are $39, 35,$ and $13$, respectively. Find the largest possible value of $AB$. [i]Team #6[/i]

A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).

Let $a,b,c,d$ be real numbers such that $abcd>0$. Prove that:There exists a permutation $x,y,z,w$ of $a,b,c,d$ such that $$2(xy+zw)^2>(x^2+y^2)(z^2+w^2)$$.

Let $a,b,c>0$ be real numbers with the property that $a+b+c=1$. Prove that \[\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq\frac{7}{1+abc}.\]

$\frac{1}{2^{1990}}(1-3\text{C}_{1990}^2+3^2\text{C}_{1990}^4-3^3\text{C}_{1990}^6+\cdots+3^{994}\text{C}_{1990}^{1988}-3^{995}\text{C}_{1990}^{1990})=$________.

Let $(a_{n})_{n\ge 0}$ and $(b_{n})_{n\ge 0}$ be two sequences with arbitrary real values $a_0, a_1, b_0, b_1$. For $n\ge 1$, let $a_{n+1}, b_{n+1}$ be defined in this way: $$a_{n+1}=\dfrac{b_{n-1}+b_{n}}{2}, b_{n+1}=\dfrac{a_{n-1}+a_{n}}{2}$$ Prove that for any constant $c>0$ there exists a positive integer $N$ s.t. for all $n>N$, $|a_{n}-b_{n}|<c$.