On sides $ AB$ and $ AC$ of triangle $ ABC$ there are given points $ D,E$ such that $ DE$ is tangent of circle inscribed in triangle $ ABC$ and $ DE \parallel BC$. Prove $ AB\plus{}BC\plus{}CA\geq 8DE$

An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.

$S$ is natural, and $S=d_{1}>d_2>...>d_{1000000}=1$ are all divisors of $S$. What minimal number of divisors can have $d_{250}$?

[b]p1.[/b] One out of $12$ coins is counterfeited. It is known that its weight differs from the weight of a valid coin but it is unknown whether it is lighter or heavier. How to detect the counterfeited coin with the help of four trials using only a two-pan balance without weights? [b]p2.[/b] Below a $3$-digit number $c d e$ is multiplied by a $2$-digit number $a b$ . Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits. $\begin{tabular}{ccccc} & & c & d & e \\ x & & & a & b \\ \hline & & f & e & g \\ + & c & d & e & \\ \hline & b & b & c & g \\ \end{tabular}$ [b]p3.[/b] Find all integer $n$ such that $\frac{n + 1}{2n - 1}$is an integer. [b]p4[/b]. There are several straight lines on the plane which split the plane in several pieces. Is it possible to paint the plane in brown and green such that each piece is painted one color and no pieces having a common side are painted the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Four positive integers $a, b, c, d$ satisfy the condition: $a < b < c < d$. For what smallest possible value of $d$ could the following condition be true: the arithmetic mean of numbers $a, b, c$ is twice smaller than the arithmetic mean of numbers $a, b, c, d$?

The [b]Collaptz function[/b] is defined as $$C(n) = \begin{cases} 3n - 1 & n\textrm{~odd}, \\ \frac{n}{2} & n\textrm{~even}.\end{cases}$$ We obtain the [b]Collaptz sequence[/b] of a number by repeatedly applying the Collaptz function to that number. For example, the Collaptz sequence of $13$ begins with $13, 38, 19, 56, 28, \cdots$ and so on. Find the sum of the three smallest positive integers $n$ whose Collaptz sequences do not contain $1,$ or in other words, do not [b]collaptzse[/b]. [i]Proposed by Andrew Wu and Jason Wang[/i]

Carol places a king on a $5 \times 5$ chessboard. The king starts on the lower-left corner, and each move it steps one square to the right, up, up-right, up-left, or down-right. How many ways are there for the king to get to the top-right corner without visiting the same square twice?

Find all natural values of $k$ for which the system $$\begin{cases}x_1+x_2+\ldots+x_k=9\\\frac1{x_1}+\frac1{x_2}+\ldots+\frac1{x_k}=1\end{cases}$$ has solutions in positive numbers. Find these solutions. [i]I. Dimovski[/i]

Let $k, n$ be positive integers, and let $\alpha_1, \alpha_2, \ldots, \alpha_n$ all be $k$-th roots of unity, satisfying: \[ \alpha_1^j + \alpha_2^j + \cdots + \alpha_n^j = 0 \quad \text{for any } j (0 < j < k). \] Prove that among $\alpha_1, \alpha_2, \ldots, \alpha_n$, each $k$-th root of unity appears the same number of times.

Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.

Robin goes birdwatching one day. he sees three types of birds: penguins, pigeons, and robins. $\frac23$ of the birds he sees are robins. $\frac18$ of the birds he sees are penguins. He sees exactly $5$ pigeons. How many robins does Robin see?

The sequence $a_0,a_1,a_2,...$ is defined by $a_0 = 0, a_1 = a_2 = 1$ and $a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)$ for all $n \in N$. Show that all $a_n$ are perfect squares .

Prove that among $39$ consecutive natural numbers, there is always one whose sum of digits (in base $10$) is divisible by $11$.

Let ABCD be a convex quadrilateral and O a point inside it. Let the parallels to the lines BC, AB, DA, CD through the point O meet the sides AB, BC, CD, DA of the quadrilateral ABCD at the points E, F, G, H, respectively. Then, prove that $ \sqrt {\left|AHOE\right|} \plus{} \sqrt {\left|CFOG\right|}\leq\sqrt {\left|ABCD\right|}$, where $ \left|P_1P_2...P_n\right|$ is an abbreviation for the non-directed area of an arbitrary polygon $ P_1P_2...P_n$.

Let $N$ be the set of natural numbers. Suppose $f: N \to N$ is a function satisfying the following conditions: (a) $f(mn) =f(m)f(n)$ (b) $f(m) < f(n)$ if $m < n$ (c) $f(2) = 2$ What is the sum of $\Sigma_{k=1}^{20}f(k)$?

The lines $AD , BE$ and $CF$ intersect in $S$ within a triangle $ABC$ . It is given that $AS: DS = 3: 2$ and $BS: ES = 4: 3$ . Determine the ratio $CS: FS$ . [asy] unitsize (1 cm); pair A, B, C, D, E, F, S; A = (0,0); B = (5,0); C = (1,4); S = (14*A + 15*B + 6*C)/35; D = extension(A,S,B,C); E = extension(B,S,C,A); F = extension(C,S,A,B); draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, N); dot("$D$", D, NE); dot("$E$", E, W); dot("$F$", F, dir(270)); dot("$S$", S, NE); [/asy]

Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of $ 3 : 2 : 1$, respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be his correct share of candy, what fraction of the candy goes unclaimed? $ \textbf{(A)}\ \frac {1}{18} \qquad \textbf{(B)}\ \frac {1}{6} \qquad \textbf{(C)}\ \frac {2}{9} \qquad \textbf{(D)}\ \frac {5}{18} \qquad \textbf{(E)}\ \frac {5}{12}$

Given natural $a,b,n$. It is known, that for every natural $k$ ($k\ne b$) the number $a-k^n$ is divisible by $b-k$. Prove that $$a=b^n$$

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

Let $P$ be a point in the plane of triangle $ABC$. Denote the reflections of $A,B,C$ over $P$ by $A',B'$ and $C'$, respectively. Let $A'',B'',C''$ be the reflection of $A',B',C'$ over $BC,CA$ and $AB$, respectively. Let the line $A''B''$ intersects $AC$ at $A_b$ and let $A''C''$ intersects $AB$ at $A_c$. Denote by $\omega_A$ the circle through the points $A,A_b,A_c$. The circles $\omega_B,\omega_C$ are defined similarly. Prove that $\omega_A ,\omega_B ,\omega_C$ are coaxial, i.e., they share a common radical axis. [i]Proposed by Navneel Singhal, Delhi and K. V. Sudharshan, Chennai, India[/i]

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

Given a finite string $S$ of symbols $X$ and $O$, we denote $\Delta(s)$ as the number of$X'$s in $S$ minus the number of $O'$s (For example, $\Delta(XOOXOOX)=-1$). We call a string $S$ [b]balanced[/b] if every sub-string $T$ of (consecutive symbols) $S$ has the property $-1\leq \Delta(T)\leq 2.$ (Thus $XOOXOOX$ is not balanced, since it contains the sub-string $OOXOO$ whose $\Delta$ value is $-3.$ Find, with proof, the number of balanced strings of length $n$.

For positive integers $a$ and $b$, define \[a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i\] Let $p$ be a prime and $n>3$ a positive integer. Show that there exist at least 2 different positive integers $t$ such that $1<t<p^n$ and $t!_p\equiv 1\pmod {p^n}$.

Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

For some integer $n > 0$, a square paper of side length $2^{n}$ is repeatedly folded in half, right-to-left then bottom-to-top, until a square of side length 1 is formed. A hole is then drilled into the square at a point $\tfrac{3}{16}$ from the top and left edges, and then the paper is completely unfolded. The holes in the unfolded paper form a rectangular array of unevenly spaced points; when connected with horizontal and vertical line segments, these points form a grid of squares and rectangles. Let $P$ be a point chosen randomly from \textit{inside} this grid. Suppose the largest $L$ such that, for all $n$, the probability that the four segments $P$ is bounded by form a square is at least $L$ can be written in the form $\tfrac mn$ where $m$ and $n$ are positive relatively prime integers. Find $m+n$.