Call number $A$ as interesting if $A$ is divided by every number that can be received from $A$ by crossing some last digits. Find maximum interesting number with different digits.

In the plane there is a quadrilateral $ ABCD $ and a point $ M $. Construct a parallelogram with center $ M $ and its vertices lying on the lines $ AB $, $ BC $, $ CD $, $ DA $.

Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime.

Suppose $P,Q\in \mathbb R[x]$ that $deg\ P=deg\ Q$ and $PQ'-QP'$ has no real root. Prove that for each $\lambda \in \mathbb R$ number of real roots of $P$ and $\lambda P+(1-\lambda)Q$ are equal.

In unit cube $ABCDEFGH$ (with faces $ABCD$, $EFGH$ and connecting vertices labeled so that $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, $\overline{DH}$ are edges of the cube), $L$ is the midpoint of $GH$. The area of $\vartriangle CAL$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

A circle with diameter $23$ is cut by a chord $AC$. Two different circles can be inscribed between the large circle and $AC$. Find the sum of the two radii.

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

A table $2022 \times 2022$ is divided onto the tiles of two types: $L$-tetromino and $Z$-tetromino. Determine the least amount of $Z$-tetromino one needs to use.

The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, \ldots$ consists of 1’s separated by blocks of 2’s with n 2’s in the nth block. The sum of the first $1234$ terms of this sequence is $\text{(A)}\ 1996 \qquad \text{(B)}\ 2419 \qquad \text{(C)}\ 2429 \qquad \text{(D)}\ 2439 \qquad \text{(E)}\ 2449$

10. Michael is playing basketball. He makes $10\%$ of his shots, and gets the ball back after $90\%$ of his missed shots. If he does not get the ball back he stops playing. What is the probability that Michael eventually makes a shot? 11. How many subsets $S$ of the set $\{1, 2, \ldots , 10\}$ satisfy the property that, for all $i \in [1, 9]$, either $i$ or $i + 1$ (or both) is in S? 12. A positive integer $\overline{ABC}$, where $A, B, C$ are digits, satisfies $$\overline{ABC} = B^C - A$$ Find $\overline{ABC}$.

1. If $5x+3y-z=4$, $x=y$, and $z=4$, find $x+y+z$. 2. How many ways are there to pick three distinct vertices of a regular hexagon such that the triangle with those three points as its vertices shares exactly one side with the hexagon? 3. Sirena picks five distinct positive primes, $p_1 < p_2 < p_3 < p_4 < p_5$, and finds that they sum to $192$. If the product $p_1p_2p_3p_4p_5$ is as large as possible, what is $p_1 - p_2 + p_3 - p_4 + p_5$?

Consider the regular $101$-gon. A line $l$ does not contain any vertex of this polygon. Prove that line $l$ intersects even number of the diagonals of this polygon.

Given an acute triangle $ABC$, point $D$ is chosen on the side $AB$ and a point $E$ is chosen on the extension of $BC$ beyond $C$. It became known that the line through $E$ parallel to $AB$ is tangent to the circumcircle of $\triangle ADC$. Prove that one of the tangents from $E$ to the circumcircle of $\triangle BCD$ cuts the angle $\angle ABE$ in such a way that a triangle similar to $\triangle ABC$ is formed.

$x_1=\frac{1}{2}$ and $x_{k+1}=\frac{x_k}{x_1^2+...+x_k^2}$ Prove that $\sqrt{x_k^4+4\frac{x_{k-1}}{x_{k+1}}}$ is rational

Square table $5 \times 5$ is filled with numbers in a following way. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8zLzQ0Y2M1NjdiNjQ3NjhlYTAwMWQ0MTg2ZjIwZWE4NzkwYzcwYWFkLnBuZw==&rn=dGFiZWxpY2EucG5n[/img] We can change the table in a way we take two arbitrary numbers from the table and we decrease both of them with value of smaller of those two. Can we get to the table with all zeros?

For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals? $\textbf{(A) }1\qquad \textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad \textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad \textbf{(D) }\dfrac{2014}{2013}\qquad \textbf{(E) }2014^{\frac1{2014}}\qquad$

An isosceles triangle $ABC$ ($AB = BC$) is given. Circles $\omega_1$ and $\omega_2$ with centres $O_1$ and $O_2$ lie in the angle $ABC$ and touch the sides $AB$ and $CB$ at $A$ and $C$ respectively, and touch each other externally at point $X$. The side $AC$ meets the circles again at points $Y$ and $Z$. $O$ is the circumcenter of the triangle $XYZ$. Lines $O_2 O$ and $O_1 O$ intersect lines $AB$ and $BC$ at points $C_1$ and $A_1$ respectively. Prove that $B$ is the circumcentre of the triangle $A_1 OC_1$.

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. Compute the real value of $a$ such that $$\sqrt{a(101b + 1)} - 1 = \sqrt{b(c - 1)}+ 10\sqrt{(a - c)b}.$$ [b]p20.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. For some triangle $\vartriangle ABC$, let $\omega$ and $\omega_A$ be the incircle and $A$-excircle with centers $I$ and $I_A$, respectively. Suppose $AC$ is tangent to $\omega$ and $\omega_A$ at $E$ and $E'$, respectively, and $AB$ is tangent to $\omega$ and $\omega_A$ at $F$ and $F'$ respectively. Furthermore, let $P$ and $Q$ be the intersections of $BI$ with $EF$ and $CI$ with $EF$, respectively, and let $P'$ and $Q'$ be the intersections of $BI_A$ with $E'F'$ and $CI_A$ with $E'F'$, respectively. Given that the circumradius of $\vartriangle ABC$ is a, compute the maximum integer value of $BC$ such that the area $[P QP'Q']$ is less than or equal to $1$. [b]p21.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. Let $c$ be a positive integer such that $gcd(b, c) = 1$. From each ordered pair $(x, y)$ such that $x$ and $y$ are both integers, we draw two lines through that point in the $x-y$ plane, one with slope $\frac{b}{c}$ and one with slope $-\frac{c}{b}$ . Given that the number of intersections of these lines in $[0, 1)^2$ is a square number, what is the smallest possible value of $ c$? Note that $[0, 1)^2$ refers to all points $(x, y)$ such that $0 \le x < 1$ and $ 0 \le y < 1$.

Set $p\ge 5$ be a prime number and $n$ be a natural number. Let $f$ be a function $ f: \mathbb{Z_{ \neq }}_0 \rightarrow \mathbb{ N }_0 $ satisfy the following conditions: i) For all sequences of integers satisfy $ a_i \not\in \{0, 1\} $, and $ p $ $\not |$ $ a_i-1 $, $ \forall $ $ 1 \le i \le p-2 $,\\ $$ \displaystyle \sum^{p-2}_{i=1}f(a_i)=f(a_1a_2 \cdots a_{p-2}) $$ ii) For all coprime integers $ a $ and $ b $, $ a \equiv b \pmod p \Rightarrow f(a)=f(b) $ iii) There exist $k \in \mathbb{Z}_{\neq 0} $ that satisfy $ f(k)=n $ Prove that the number of such functions is $ d(n) $, where $ d(n) $ denotes the number of divisors of $ n $.

7) A train, originally of mass $M$, is traveling on a frictionless straight horizontal track with constant speed $v$. Snow starts to fall vertically and sticks to the train at a rate of $\rho$, where $\rho$ has units of kilograms per second. The train's engine keeps the train moving at constant speed $v$ as snow accumulates on the train. What is the rate at which the kinetic energy of the train and snow increases? A) $0$ B) $Mgv$ C) $\frac{1}{2}Mv^2$ D) $\frac{1}{2}pv^2$ E) $\rho v^2$

In a mathematical competition $n=10\,000$ contestants participate. During the final party, in sequence, the first one takes $1/n$ of the cake, the second one takes $2/n$ of the remaining cake, the third one takes $3/n$ of the cake that remains after the first and the second contestant, and so on until the last one, who takes all of the remaining cake. Determine which competitor takes the largest piece of cake.

Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. What is the perimeter of $\triangle{APQ}$? (A) 84 (B) 86 (C) 88 (D)90 (E)92

Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on $\left[AB\right]$. $Q$ is the point on $BC$ such that $PQ$ is perpendicular to $AB$. $R$ is the point on $AC$ such that $QR$ is perpendicular to $BC$. And $S$ is the point on $AB$ such that $RS$ is perpendicular to $AC$. $Q'$ is the point on $BC$ such that $PQ'$ is perpendicular to $BC$. $R'$ is the point on $AC$ such that $Q'R'$ is perpendicular to $AC$. And $S'$ is the point on $AB$ such that $R'S'$ is perpendicular to $AB$. Determine $\frac{|PB|}{|AB|}$ if $S=S'$.

Prove that for every natural number $n > 1$ there exists a suquence $a_1$,$a_2$, $...$, $a_n$ of the numbers $1,2,...,n$ such that for each $k \in \{1,2,...,n-1\}$ the number $a_{k+1}$ divides $a_1+a_2+...+a_k$.

Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.