In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

The function f is defined on the set of integers and satisfies $\bullet$ $f(n) = n - 2$, if $n \ge 2005$ $\bullet$ $f(n) = f(f(n+7))$, if $n < 2005$. Find $f(3)$.

Find all real numbers $k $ such that $x^2 - 2 x [x] + x - k = 0$ has at least two non-negative roots.

Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$

On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.

[b]8.[/b] Does there exist a Euclidean ring which is properly contained in the field $V$ of real numbers, and whose quotient field is $V$? [b](A.21)[/b]

A positive integer of at least two digits written in the base $ 10 $ is called 'ascending' if the digits increase in value from left to right. For example, $ 123 $ is 'ascending', but $ 132 $ and $ 122 $ is not. How many 'ascending' numbers are there?

Two circles, $\omega_1$ and $\omega_2$, centred at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meets $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1, O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $$f^{f^{f(x)}(y)}(z)=x+y+z+1$$ for all $x,y,z \in \mathbb{N}$. [i]Proposed by William Wang.[/i]

Consider the sequence $2, 3, 5, 6, 7, 8, 10, ...$ of all positive integers that are not perfect squares. Determine the $2011^{th}$ term of the sequence.

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

The real numbers $ a,b$ fulfil the conditions (i) $ 0<a<a\plus{}\frac12\le b$; (ii) $ a^{40}\plus{}b^{40}\equal{}1$. Prove that $ b$ has the first 12 digits after the decimal point equal to 9. [i]Mircea Fianu[/i]

Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]

At a party every woman dances with at least one man, and no man dances with every woman. Show that there are men M and M' and women W and W' such that M dances with W, M' dances with W', but M does not dance with W', and M' does not dance with W.

If $(1.0025)^{10}$ is evaluated correct to $5$ decimal places, then the digit in the fifth decimal place is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }5\qquad \textbf{(E) }8$

Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.

The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? [asy] size(6cm); for (int i=0; i<9; ++i) { draw((i,0)--(i,3),dotted); } for (int i=0; i<4; ++i){ draw((0,i)--(8,i),dotted); } for (int i=0; i<8; ++i) { for (int j=0; j<3; ++j) { if (j==1) { label("1",(i+0.5,1.5)); }}} [/asy] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$

Find all the functions $f : C \to R^+$ such that they fulfill simultaneously the following conditions: $$(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C$$ $$(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C$$ $$(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C$$

A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

What is the average (mean) of all $ 5$-digit numbers that can be formed by using each of the digits $ 1$, $ 3$, $ 5$, $ 7$, and $ 8$ exactly once? $ \textbf{(A)}\ 48000\qquad \textbf{(B)}\ 49999.5\qquad \textbf{(C)}\ 53332.8\qquad \textbf{(D)}\ 55555\qquad \textbf{(E)}\ 56432.8$

Does there exist a natural number which is not a divisor of any natural number whose decimal expression consists of zeros and ones, with no more than $1988$ ones?

Solve in natural the equation $9^x-3^x=y^4+2y^3+y^2+2y$ _____________________________ Azerbaijan Land of the Fire :lol:

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

For $i=1,2,$ let $T_i$ be a triangle with side length $a_i,b_i,c_i,$ and area $A_i.$ Suppose that $a_1\le a_2, b_1\le b_2, c_1\le c_2,$ and that $T_2$ is an acute triangle. Does it follow that $A_1\le A_2$?

Prove or disprove the following statements: (a) There exists a monotone function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$. (b) There exists a continuously differentiable function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$.