Prove that for every real number $M$ there exists an infinite arithmetical progression of positive integers such that [list] [*] the common difference is not divisible by $10$, [*] the sum of digits of each term exceeds $M$. [/list]

Given an integer $n\ge 2$, given $n+1$ distinct points $X_0,X_1,\ldots,X_n$ in the plane, and a positive real number $A$, show that the number of triangles $X_0X_iX_j$ of area $A$ does not exceed $4n\sqrt n$.

Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$, $Q$, $R$, and $S$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB=15$, $BQ=20$, $PR=30$, and $QS=40$. Let $m/n$, in lowest terms, denote the perimeter of $ABCD$. Find $m+n$.

Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$. \\ \\ Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.

Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\ $\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\ $\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\ $\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true? \\ \\ [i] (Bogdan Blaga)[/i]

On a $5\times 5$ grid $\mathcal A$ of integers, each with absolute value $<10^9$, define a [i]flip[/i] to be the operation of negating each element in a row / column with negative sum. For example, $(-1,-4,3,-4,1) \to (1,4,-3,4,-1)$. Determine whether there exists an $\mathcal A$ so that it's possible to perform $90$ flips on it. [i]Alex Chen[/i]

For every positive integer $n$, let $S_n$ be the sum of the first $n$ prime numbers: $S_1 = 2, S_2 = 2 + 3 = 5, S_3 = 2 + 3 + 5 = 10$, etc. Can both $S_n$ and $S_{n+1}$ be perfect squares?

How many integral solutions are there of the equation $x^5 -31x+2015=0$ ? [list=1] [*] 2 [*] 4 [*] 1 [*] None of these [/list]

Prove that the plane dividing in equal proportions the surface area and volume of the circumscribed polyhedron passes through the center of the sphere inscribed in this polyhedron.

Compute $$\int_0^\pi\frac{1-\sin x}{1+\sin x}dx.$$

Prove that \[ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1) \] for all positive real numbers $a$ and $b.$

Find $n>0$ such that $\sqrt[3]{\sqrt[3]{5\sqrt2+n}+\sqrt[3]{5\sqrt2-n}}=\sqrt2$.

Let $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$ be two partitions of a set $M$ such that $|A_j\cup B_k|\ge n$ for any $j,k\in\{1,2,\ldots,n\}$. Prove that $|M|\ge n^2/2$.

Let $f(x) = 15x - 2016$. If $f(f(f(f(f(x))))) = f(x)$, find the sum of all possible values of $x$.

Increasing the radius of a cylinder by $ 6$ units increased the volume by $ y$ cubic units. Increasing the altitude of the cylinder by $ 6$ units also increases the volume by $ y$ cubic units. If the original altitude is $ 2$, then the original radius is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 6\pi \qquad\textbf{(E)}\ 8$

Let $n$ be a positive integer. What is the largest $k$ for which there exist $n\times n$ matrices $M_1,\dots,M_k$ and $N_1,\dots,N_k$ with real entries such that for all $i$ and $j,$ the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\ne j?$

The shapes in the image consist of six unit cubes. Which of the following 3D objects can be filled up with the aforementioned shapes: a) a cube with side length $3$, from which one edge has been removed (i.e. three layers of the shape [img]https://i.imgur.com/vUqgHS2.png[/img] )? b) a rectangular prism of size $5 \times 4 \times 3$, from which two edges of length $3$ have been removed from one of the $5 \times 3$ sides (i.e. three layers of the shape [img]https://imgur.com/W4pfEfz.png[/img] )? We can use each of shapes at most once, no two shapes can overlap, nor protrude from the 3D object and every unit cube of the 3D object must be covered by a unit cube of one of the constituent shapes. [center][img]https://imgur.com/evAmuep.png[/img][/center] [i]Proposed by Ilija Jovčeski[/i]

$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$

Given a triangle $ ABC $, $ AD $ is its angle bisector. Let $ E, F $ be the centers of the circles inscribed in the triangles $ ADC $ and $ ADB $, respectively. Denote by $ \omega $, the circle circumscribed around the triangle $ DEF $, and by $ Q $, the intersection point of $ BE $ and $ CF $, and $ H, J, K, M $ , respectively the second intersection point of the lines $ CE, CF, BE, BF $ with circle $ \omega $. Let $\omega_1, \omega_2 $ the circles be circumscribed around the triangles $ HQJ $ and $ KQM $ Prove that the intersection point of the circles $\omega_1, \omega_2 $ different from $ Q $ lies on the line $ AD $. (Kivva Bogdan)

If $\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3$, then (1) Find the range of $b$; (2) for every given $b$, find $a$.

Let $S_0=\{z\in\mathbb C:|z|=1,z\ne-1\}$ and $f(z)=\frac{\operatorname{Im}z}{1+\operatorname{Re}z}$. Prove that $f$ is a bijection between $S_0$ and $\mathbb R$. Find $f^{-1}$.

[b]p1.[/b] In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form a strictly increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to $94$. How many possible combinations of test scores could they have had? (Test scores at Greendale range between $0$ and $100$, inclusive.) [b]p2.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that $$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B1B1B1...}) + 1$$ Find the sum of all possible values of $10A + B$. [b]p3.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $\overline{AC}$ and $\overline{BC}$, respectively, such that triangles $\vartriangle ADB$ and $\vartriangle CDE$ are similar and $DE =EB$. If $\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}$ with $a$, $b$ positive integers and $a$ squarefree, then find $a + b$. [b]p4.[/b] Five bowling pins $P_1, P_2, ..., P_5$ are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is $\frac{a}{b}$ where $a$ and $b$ are relatively prime, find $a + b$. (Pins $P_i$ and $P_j$ are adjacent if and only if $|i - j| = 1$.) [b]p5.[/b] How many terms in the expansion of $$(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})$$ have coeffcients equal to $1011$? [b]p6.[/b] Suppose $f(x)$ is a monic quadratic polynomial with distinct nonzero roots $p$ and $q$, and suppose $g(x)$ is a monic quadratic polynomial with roots $p + \frac{1}{q}$ and $q + \frac{1}{p}$ . If we are given that $g(-1) = 1$ and $f(0)\ne -1$, then there exists some real number $r$ that must be a root of $f(x)$. Find $r$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The diagram below shows a large square with each of its sides divided into four equal segments. The shaded square whose sides are diagonals drawn to these division points has area $13$. Find the area of the large square. [img]https://cdn.artofproblemsolving.com/attachments/8/3/bee223ef39dea493d967e7ebd5575816954031.png[/img]

Given is a circular bus route with $n\geqslant2$ bus stops. The route can be frequented in both directions. The way between two stops is called [i]section[/i] and one of the bus stops is called [i]Zürich[/i]. A bus shall start at Zürich, pass through all the bus stops [b]at least once[/b] and drive along exactly $n+2$ sections before it returns to Zürich in the end. Assuming that the bus can chance directions at each bus stop, how many possible routes are there? EDIT: Sorry, there was a mistake...corrected now, thanks mavropnevma! :oops:

A point $P$ on the side $AB$ of a triangle $ABC$ and points $S$ and $T$ on the sides $AC$ and $BC$ are such that $AP=AS$ and $BP=BT$. The circumcircle of $PST$ meets the sides $AB$ and $BC$ again at $Q$ and $R$, respectively. The lines $PS$ and $QR$ meet at $L$. Prove that the line $CL$ bisects the segment $PQ$. [i]Proposed by A. Antropov[/i]