For every triangle $ABC$, denote by $D(ABC)$ the triangle whose vertices are the tangency points of the incircle of $\vartriangle ABC$ with the sides. Assume that $\vartriangle ABC$ is not equilateral. (a) Prove that $D(ABC)$ is also not equilateral. (b) Find in the sequence $T_1 = \vartriangle ABC, T_{k+1} = D(T_k)$ for $k \in N$ a triangle whose largest angle $\alpha$ satisfies $0 < \alpha -60^o < 0.0001^o$

How many 6-digit numbers have at least an even digit?

The figure shows a regular heptagon with sides of length 1. [asy] import geometry; unitsize(5); real R = 1/(2 sin(pi/7)); pair A = (0, R); pair B = rotate(360/7) * A; pair C = rotate(360/7) * B; pair D = rotate(360/7) * C; pair E = rotate(360/7) * D; pair F = rotate(360/7) * E; pair G = rotate(360/7) * F; pair X = B + G - A; pair Y = (D + E) / 2; draw(A -- B -- C -- D -- E -- F -- G -- cycle); draw("$1$", B -- X); draw("$1$", X -- G); draw("$d$", X -- Y); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(X); dot(Y); perpendicular(Y, NW, Y - A); [/asy] Determine the indicated length $d$. Express your answer in simplified radical form.

Two quadrangles have equal areas, perimeters and corresponding angles. Are such quadrilaterals necessarily congurent ?

Side lengths $AB,BC,CD,AD$ of convex quadrilateral $ABCD$ are equal $16,13,14,17$ respectively. Circles $w_1,w_2,w_3,w_4$ are drawn with centers $A,B,C,D$ and radii $2,6,3,9$ respectively. Common external tangents to circles $w_1,w_2$; $w_2,w_3$; $w_3,w_4$; $w_4,w_1$ intersect at $A_1,B_1,C_1,D_1$ respectively. Prove that lines $AA_1,BB_1,CC_1,DD_1$ are concurrent. [i]Aliaksei Vaidzelevich[/i]

Show that the unit sphere bundle of the $r$-fold direct sum of the tautological (universal) complex line bundle over the space $\mathbb{C}P^{\infty}$ is homotopically equivalent to $\mathbb{C}P^{r-1}$.

For a positive integer $n$, let $A_n$ be the set of quadruplets $(a,b,c,d)$ of integers, satisfying the following properties simultaneously: [list] [*] $0\le a\le c\le n,$ [*] $0\le b\le d\le n,$ [*] $c+d>n,$ and [*] $bc=ad+1.$ [/list] Moreover, define $$\alpha_n=\sum_{(a,b,c,d)\in A_n}\frac{1}{ab+cd}.$$ Find all real numbers $t$ such that $\alpha_n>t$ for every positive integer $n$.

Let $E$ be the projection of the vertex $C$ of a rectangle $ABCD$ to the diagonal $BD$. Prove that the common external tangents to the circles $AEB$ and $AED$ meet on the circle $AEC$.

The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

The median $AD$ of triangle $ABC$ intersects its inscribed circle (with center $O$) at the points $X$ and $Y$. Find the angle $XOY$ if $AC = AB + AD$. (A Fedotov)

Given the natural number N, consider triples of different positive integers $(a, b, c)$ such that $a + b + c = N$. Take the largest possible system of these triples such that no two triples of the system have any common elements. Denote the number of triples of this system by $K(N)$. Prove that: (a) $K(N) >\frac{N}{6}-1$ (b) $K(N) <\frac{2N}{9}$ (L.D. Kurliandchik, Leningrad)

There are $n$ parallel lines on a plane, and there is a set $S$ of distinct points. Each point in $S$ lies on one of the $n$ lines and is colored either red or blue. Determine the minimum value of $n$ such that if $S$ satisfies the following condition, it is guaranteed that there are infinitely many red points and infinitely many blue points. [list] [*] Each line contains at least one red point and at least one blue point from $S$. [*] Consider a triangle formed by three elements of $S$ located on three distinct lines. If two of the vertices of the triangle are red, there must exist a blue point, not one of the vertices, either inside or on the boundary of the triangle. Similarly, if two of the vertices are blue, there must exist a red point, not one of the vertices, either inside or on the boundary of the triangle. [/list]

Two 100-digit binary sequences are given. In one operation, one may insert (possibly at the beggining or end) or remove one or more identical digits from a sequence. What is the smallest $k{}$ for which we can transform the first sequence into the second one in no more than $k{}$ operations? [i]Proposed by V. Novikov[/i]

Let $ a_1,a_2,...,a_n$ with arithmetic mean equals zero; what is the value of: $ \sum_{j=1}^n{\frac{1}{a_j(a_j+a_{j+1})(a_j+a_{j+1}+a_{j+2})...(a_j+a_{j+1}+...+a_{j+n-2})}}$ , where $ a_{n+k}=a_k$ ?

Let's call a corner the figure that is obtained by removing one cell from a $2 \times 2$ square. Cut the $6 \times 6$ square into corners so that no two of them form a $2 \times 3$ or $3 \times 2$ rectangle together.

In how many ways can you color the six sides of a cube in black or white? (Do note that the cube is unchanged when rotated?) A. 7 B. 10 C. 20 D. 30 E. 36

Show that $1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}$ is divisible by $100$.

Area of $\triangle ABC$ is $\frac{1}{4}$, circumradius of $\triangle ABC$ is $1$. Let $s=\sqrt{a}+\sqrt{b}+\sqrt{c},t=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, then $\text{(A)}s>t\qquad\text{(B)}s=t\qquad\text{(C)}s<t\qquad\text{(D)}s>t$

For which even natural numbers $d{}$ does there exists a constant $\lambda>0$ such that any reduced polynomial $f(x)$ of degree $d{}$ with integer coefficients that does not have real roots satisfies the inequality $f(x) > \lambda$ for all real numbers?

For a given integer $n\ge 2$, let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$.

What is the value of $2-(-2)^{-2}$? $ \textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \qquad\textbf{(D) } \dfrac{9}{4} \qquad\textbf{(E) } 6 $

[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ [b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

Let $F$ be the set of all functions $f : N \to N$ which satisfy $f(f(x))-2 f(x)+x = 0$ for all $x \in N$. Determine the set $A =\{ f(1989) | f \in F\}$.

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$. Determine, justifying your answer, the set of all possible values for $f$.