Ice cream costs $2000$ rubles. Petya has $$400^5 - 399^2\cdot (400^3 + 2\cdot 400^2 + 3\cdot 400 + 4)$$rubles. Does Petya have enough money for ice cream?

Let's call a ticket with a number from $000000$ to $999999$ excellent if the difference between some two adjacent digits is $5$. Find the number of excellent tickets.

Does such a convex (all angles less than $180^o$) pentagon $ABCDE$, such that all angles $ABD$, $BCE$, $CDA$, $DEB$ and $EAC$ are obtuse?

There are $n$ matches on the table ($n > 1$). Two players take turns shooting them from the table. On the first move, the player removes any number of matches from the table from $1$ to $n - 1$, and then each time you can take no more matches from the table, than the partner took with the previous move. The one who took the last match wins.. Find all $n$ for which the first player can provide win for yourself.

Is it possible to arrange the chips in the cells of an $8 \times 8$ board so that in any two columns the number of chips is the same, and in any two lines are different?

Spot spotlight located at vertex $B$ of an equilateral triangle $ABC$, illuminates angle $\alpha$. Find all such values of $\alpha$, not exceeding $60^o$, which at any position of the spotlight, when the illuminated corner is entirely located inside the angle $ABC$, from the illuminated and two unlit segments of side $AC$ can be formed into a triangle.

Dunno wrote several different natural numbers on the board and divided (in his head) the sum of these numbers by their product. After this, Dunno erased the smallest number and divided (again in his mind) the amount of the remaining numbers by their product. The second result was $3$ times greater than the first. What number did Dunno erase?

There are 4 coins, 3 of which are real, which weigh the same, and one is fake, which differs in weight from the rest. Cup scales without weights are such that if equal weights are placed on their cups, then any of the cups can outweigh, but if the loads are different in mass, then the cup with a heavier load is sure to pull. How to definitely identify a counterfeit coin in three weighings and easily establish what is it or is it heavier than the others?

Find all pairs of square trinomials $x^2 + ax + b$, $ x^2 + cx + d$ such that $a$ and $b$ are the roots of the second trinomial, $c$ and $d$ are the roots of the first.

In triangle $ABC$, in which $AB = BC$, on side $AB$ is selected point $D$, and the ciscumcircles of triangles $ADC$ and $BDC$ , $S1$ and $S2$ respectively. The tangent drawn to $S_1$ at point $D$ intersects $S_2$ for second time at point $M$. Prove that $BM \parallel AC$.

Let $a, b$ and $c$ be pairwise relatively prime natural numbers. Find all possible values of $\frac{(a + b)(b + c)(c + a)}{abc}$ if known what it is integer.

There is a token in one of the nodes of a hexagon with side $n$, divided into regular triangles (see figure). Two players take turns moving it to one of the neighboring nodes, and it is forbidden to go to a node that the token has already visited. The one who loses who can't make a move. Who wins with the right game?

Find all natural numbers that have exactly six divisors whose sum is $3500$.

Prove that if $0 < a, b < 1,$ then $$\frac{ab(1 - a)(1 - b)}{(1- ab)^2 }< \frac14.$$

There are 8 coins, 7 of which are real, which weigh the same, and one is fake, which differs in weight from the rest. Cup scales without weights mean that if you put equal weights on their cups, then any of the cups can outweigh, but if the loads are different in mass, then the cup with a heavier load is definitely overpowered. How to definitely identify a counterfeit coin in four weighings and establish is it lighter or heavier than the others?

Prove that if $a, b, c$ are positive numbers and $ab + bc + ca > a+ b + c$, then $a + b + c > 3$.

Is it true that from an arbitrary triangle you can cut three equal figures, the area of each of which is more than a quarter of the area triangle?

Given an angle with vertex $B$. Construct point $M$ as follows. Let us take an arbitrary isosceles trapezoid whose sides lie on the sides of a given angle. Through two opposite ones draw tangents to the vertices of the circle circumscribed around it. Let $M$ denote the point of intersection of these tangents. What figure do all such points $M$ form?

In each cell of a square table of size $n \times n$ cells ($n \ge 3$) the number $1$ or $-1$ is written. If you take any two lines, multiply numbers standing above each other in them and add the n resulting products, then the sum will be equal to $0$. Prove that the number $n$ is divisible by $4$.

Given triangle $A_0B_0C_0$. On the segment $A_0B_0$ points $A_1$, $A_2$, $...$, $A_n$, and on the segment $B_0C_0$ - points $C_1$, $C_2$, $...$, $Cn$ so that all segments $A_iC_{i+1}$ ($i = 0$, $1$, $...$,$n-1$) are parallel to each other and all segments $ C_iA_{i+1}$ ($i = 0$, $1$, $...$,$n-1$) are too. Segments $C_0A_1$, $A_1C_2$, $A_2C_1$ and $C_1A_0$ bound a certain parallelogram, segments $C_1A_2$, $A_2C_3$, $A_3C_2$ and $C_2A_1$ too, etc. Prove that the sum of the areas of all $n -1$ resulting parallelograms less than half the area of triangle $A_0B_0C_0$.

There are $1996$ points marked on a straight line at regular intervals. Petya colors half of them red and the rest blue. Then Vasya divides them into pairs ''red'' - ''blue'' so that the sum distances between points in pairs was maximum. Prove that this maximum does not depend on what coloring Petya made.

Let us call the median of a system of $2n$ points of a plane a straight line passing through exactly two of them, on both sides of which there are points of this system equally. What is the smallest number of medians that a system of $2n$ points, no three of which lie on the same line?

The length of the longest side of a triangle is $1$. Prove that three circles of radius $\frac{1}{\sqrt3}$ with centers at the vertices cover the entire triangle.

A polynomial $P(x)$ of degree $n$ has $n$ different real roots. What is the largest number of its coefficients that can be zero?

Given the function $f(x) =|4 - 4|x||- 2$. How many solutions does the equation $f(f(x)) = x$ have?

Find all natural $n$ such that for some different natural $a, b, c$ and $d$ among numbers $$\frac{(a-c)(b-d)}{(b-c)(a-d)} , \frac{(b-c)(a-d)}{(a-c)(b-d)} , \frac{(a-b)(d-c)}{(a-d)(b-c)} , \frac{(a-c)(b-d)}{(a-b)(c-d)} ,$$there are at least two numbers equal to $n$.

In triangle $ABC$, a point $O$ is taken such that $\angle COA = \angle B + 60^o$, $\angle COB = \angle A + 60^o$, $\angle AOB = \angle C + 60^o$.Prove that if a triangle can be formed from the segments $AO$, $BO$, $CO$, then a triangle can also be formed from the altitudes of triangle $ABC$ and these triangles are similar.

Is there an infinite periodic sequence consisting of the letters $a$ and$ b$, such that if all letters are replaced simultaneously $a$ to $aba$ and letters $b$ to $bba$ does it transform into itself (possibly with a shift)? (A sequence is called periodic if there is such natural number $n$, which for every $i = 1, 2, . . . i$-th member of this sequence is equal to the ($i + n$)- th.)