Find the last two digits of $11^{1998}$

Let $S$ be the incenter of the triangle $ABC$ and let the line $AS$ intersect the circumcircle of triangle $ABC$ at point $D$ ($D\ne A$). Prove that the segments $BD, CD$ and $SD$ are of equal length.

On a closed track, clockwise, there are five boxes $A, B, C, D$ and $E$, and the length of the track section between boxes $A$ and $B$ is $1$ km, between $B$ and $C$ - $5$ km, between $C$ and $D$ - $2$ km, between $D$ and $E$ - $10$ km, and between $E$ and $A$ - $3$ km. On the track, they drive in a clockwise direction, the race always begins and ends in the box. What box did you start from if the length of the race was exactly $1998$ km?

For real numbers $x, y$ and $z$ it is known that $$\begin{cases} x + y = 2 \\ xy = z^2 + 1\end {cases}$$Find the value of the expression $x^2 + y^2+ z^2$.

Thirteen children are sitting at a round table, each holding two cards. Each card has one of the numbers $1, 2, ..., 13$ written on it, and each number is written on exactly two cards. On a signal, each child gives the card with the lower number to his neighbor on the right (and at the same time receives his card with the lower number from the neighbor on the left). Prove that after a finite number of such exchanges, a situation arises when at least one of the children will have two cards with the same number.

Prove that for any reals $a> b> c$, the inequality $a^2(b - c) + b^2(c - a) + c^2(a - b)> 0$.

Let $C$ and $D$ be two distinct points on a semicircle of diameter $AB$. Let $E$ be the intersection of $AC$ and $BD$, $F$ be the intersection of $AD$ and $BC$ and $X, Y$, and $Z$ are the midpoints of $AB, CD$, and $EF$, respectively. Prove that the points $X, Y,$ and $Z$ are collinear.

The hotel has $13$ rooms with rooms from $1$ to $13$, located on one side of a straight corridor in ascending order of numbers. During the tourist season, which lasts from May $1$st to October $1$st, the hotel visitor has the opportunity to rent either one room for two days in a row, or two adjacent rooms together for one day. How much could a hotel owner earn in a season if it is known that on October $1$, rooms $1$ and $13$ were empty, and the payment for one room was one tugrik per day?

Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

The paper is marked with the finite number of blue and red dots and some these points are connected by lines. Let's name a point $P$ special if more than half of the points connected with $P$ has a color other than point $P$. Juku selects one special point and reverses its color. Then Juku selects a new special point and changes its color, etc. Prove that by a finite number of integers Juku ends up in a situation where the paper has not made a special point.

Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.

In a triangle $ABC, A_1,B_1,C_1$ are the midpoints of segments $BC,CA,AB, A_2,B_2,C_2$ are the midpoints of segments $B_1C_1,C_1A_1,A_1B_1$, and $A_3,B_3,C_3$ are the incenters of triangles $B_1AC_1,C_1BA_1,A_1CB_1$, respectively. Show that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrent.

A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.

A real number $a$ satisfies the equality $\frac{1}{a} = a - [a]$. Prove that $a$ is irrational.

A circle is divided into $n$ equal arcs by $n$ points. Assume that, no matter how we color the $n$ points in two colors, there always exists an axis of symmetry of the set of points such that any two of the $n$ points which are symmetric with respect to that axis have the same color. Find all possible values of $n$.

Solve the equation $x^2+1 = log_2(x+2)- 2x$.

Find all prime numbers of the form $10101...01$.

In a triangle $ABC$, the bisector of the largest angle $\angle A$ meets $BC$ at point $D$. Let $E$ and $F$ be the feet of perpendiculars from $D$ to $AC$ and $AB$, respectively. Let $R$ denote the ratio between the areas of triangles $DEB$ and $DFC$. (a) Prove that, for every real number $r > 0$, one can construct a triangle ABC for which $R$ is equal to $r$. (b) Prove that if $R$ is irrational, then at least one side length of $\vartriangle ABC$ is irrational. (c) Give an example of a triangle $ABC$ with exactly two sides of irrational length, but with rational $R$.

Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .

From an $n\times n$ square divided into $n^2$ unit squares, one corner unit square is cut off. Find all positive integers $n$ for which it is possible to tile the remaining part of the square with $L$-trominos.