We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

Let $ABCDEF$ be a convex hexagon. Segments $AE$ and $BF$ intersect at $X$ and segments $BD$ and $CE$ intersect in $Y.$ It's known that $$ \angle XBC = \angle XDE = \angle YAB = \angle YEF = 80^\circ \text{ and } \angle XCB = \angle XED = \angle YBA = \angle YFE = \angle 70^\circ.$$ Let $P$ and $Q$ be such points on line $XY$ that segments $PX$ and $AF$ intersect, segments $QY$ and $CD$ intersect and $\angle APF = \angle CQD = 30 ^\circ.$ Estimate the sum: \[ \frac{BX}{BF} + \frac{BY}{BD} + \frac{EX}{EA} + \frac{EY}{EC} + \frac{PX}{PY} + \frac{QY}{QX} \] [i]Proposed by Gogi Khimshiashvili, Georgia [/i]

In triangle $ABC$, $\angle A=60^{\circ}$. The point $D$ changes on the segment $BC$. Let $O_1,O_2$ be the circumcenters of the triangles $\Delta ABD,\Delta ACD$, respectively. Let $M$ be the meet point of $BO_1,CO_2$ and let $N$ be the circumcenter of $\Delta DO_1O_2$. Prove that, by changing $D$ on $BC$, the line $MN$ passes through a constant point.

In a triangle, each median forms an angle with the side it is drawn to, which is less than $\alpha$. Prove that one of the angles of the triangle is greater than $180^\circ - \frac{4}{3}\alpha$. [i]Proposed by S. Berlov[/i]

$11$ carriers will carry $270$ kg of melons at one step where each melons weighs at most $7$ kg. Each carrier can carry at most $30$ kg in one step. Show that it is possible to carry all the melons at one step whatever a melon weighs.

Let \(n\) be a positive integer and consider an \(n\times n\) square grid. For \(1\le k\le n\), a [i]python[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single row, and no other cells. Similarly, an [i]anaconda[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single column, and no other cells. The grid contains at least one python or anaconda, and it satisfies the following properties: [list] [*]No cell is occupied by multiple snakes. [*]If a cell in the grid is immediately to the left or immediately to the right of a python, then that cell must be occupied by an anaconda. [*]If a cell in the grid is immediately to above or immediately below an anaconda, then that cell must be occupied by a python. [/list] Prove that the sum of the squares of the lengths of the snakes is at least \(n^2\). [i]Proposed by Linus Tang[/i]

Does there exist such a configuration of 22 circles and 22 point, that any circle contains at leats 7 points and any point belongs at least to 7 circles?

Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied: [b](i)[/b] each integer belongs to at least one of them; [b](ii)[/b] each progression contains a number which does not belong to other progressions. Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization. Prove that \[s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).\] [i]Proposed by Dierk Schleicher, Germany[/i]

Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

Let $ ABC $ be an acute triangle having $ AB<AC, I $ be its incenter, $ D,E,F $ be intersection of the incircle with $ BC, CA, $ respectively, $ AB, X $ be the middle of the arc $ BAC, $ which is an arc of the circumcicle of it, $ P $ be the projection of $ D $ on $ EF $ and $ Q $ be the projection of $ A $ on $ ID. $ [b]a)[/b] Show that $ IX $ and $ PQ $ are parallel. [b]b)[/b] If the circle of diameter $ AI $ intersects the circumcircle of $ ABC $ at $ Y\neq A, $ prove that $ XQ $ intersects $ PI $ at $ Y. $

Consider all non-empty subsets of the set $\{1,2\cdots,n\}$. For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as $S_n$. For example, \[S_3=\frac11+\frac12+\frac13+\frac1{1\cdot 2}+\frac1{1\cdot 3}+\frac1{2\cdot 3} +\frac1{1\cdot 2\cdot 3}\] [b](i)[/b] Show that $S_n=\frac1n+\left(1+\frac1n\right)S_{n-1}$. [b](ii)[/b] Hence or otherwise, deduce that $S_n=n$.

In triangle $ABC$, angle $B$ is obtuse and equal to $a$. The bisectors of angles $A$ and $C$ intersect opposite sides at points $P$ and $M$, respectively. On the side $AC$, points $K$ and $L$ are taken so that $\angle ABK = \angle CBL = 2a - 180^o$. What is the angle between straight lines $KP$ and $LM$?

Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros. [hide="Note"]Note. Here (and generally in MathLinks) natural numbers supposed to be positive.[/hide]

Let $\{n_{k}\}_{k \ge 1}$ be a sequence of natural numbers such that for $i<j$, the decimal representation of $n_{i}$ does not occur as the leftmost digits of the decimal representation of $n_{j}$. Prove that \[\sum^{\infty}_{k=1}\frac{1}{n_{k}}\le \frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{9}.\]

$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$

A natural number $a$ is said [i]to be contained[/i] in the natural number $b$ if it is possible to obtain a by erasing some digits from $b$ (in their decimal representations). For example, $123$ is contained in $901523$, but not contained in $3412$. Does there exist an infinite set of natural numbers such that no number in the set is contained in any other number from the set?

The triangle $ABC$ is right in $A$ and $R$ is the midpoint of the hypotenuse $BC$ . On the major leg $AB$ the point $P$ is marked such that $CP = BP$ and on the segment $BP$ the point $Q$ is marked such that the triangle $PQR$ is equilateral. If the area of triangle $ABC$ is $27$, calculate the area of triangle $PQR$ .

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

Let $ABCD$ be a rectangle with $AB = 8$ and $BC = 6$. Point $E$ is outside of the rectangle such that $CE = DE$. Point $D$ is reflected over line $AE$ so that its image, $D'$ , lies on the interior of the rectangle. Point $D'$ is then reflected over diagonal $AC$, and its image lies on side $AB$. What is the length of $DE$?

A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.

There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count $300$ runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?

Can we draw $\triangle ABC$ and points $X,Y$, such that $AX=BY=AB$, $ BX = CY = BC$, $CX = AY = CA$?

Of the students attending a school party, $ 60\%$ of the students are girls, and $ 40\%$ of the students like to dance. After these students are joined by $ 20$ more boy students, all of whom like to dance, the party is now $ 58\%$ girls. How many students now at the party like to dance?

Find all functions $f\colon \mathbb Z \to \mathbb Z$ for which \[ f(x)+f(y)+xy \quad \text{divides} \quad xf(x)-y^3 \] for all pairs of integers $(x, y)$. Here, we use the convention that $a$ divides $b$ if and only if there exists some integer $c$ such that $ac=b$. [i]Dennis Chen and Andrew Wen[/i]

How many three-digit numbers are there, which do not have a zero in their decimal representation and whose sum of digits is $7$?