A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985. [asy] size(200); pair A=(0,1), B=(1,1), C=(1,0), D=origin; draw(A--B--C--D--A--(1,1/6)); draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1)); pair point=( 0.5 , 0.5 ); //label("$A$", A, dir(point--A)); //label("$B$", B, dir(point--B)); //label("$C$", C, dir(point--C)); //label("$D$", D, dir(point--D)); label("$1/n$", (11/12,1), N, fontsize(9));[/asy]

Solve the equation $|x+1|-|x-1|=2$.

If ${\cos^5 x}-{\sin^5 x}<7({\sin^3 x}-{\cos ^3 x}) $ (for $x\in [ 0,2\pi) $), then find the range of $x$.

Students $ A $ and $ B $ play according to the following rules: student $ A $ selects a vector $ \overrightarrow{a_1} $ of length 1 in the plane, then student $ B $ gives the number $ s_1 $, equal to $ 1 $ or $ - $1; then the student $ A $ chooses a vector $ \overrightarrow{a_1} $ of length $ 1 $, and in turn the student $ B $ gives a number $ s_2 $ equal to $ 1 $ or $ -1 $ etc. $ B $ wins if for a certain $ n $ vector $ \sum_{j=1}^n \varepsilon_j \overrightarrow{a_j} $ has a length greater than the number $ R $ determined before the start of the game. Prove that student $B$ can achieve a win in no more than $R^2 + 1$ steps regardless of partner $A$'s actions.

Calculate the product: \[A=\sin 1^\circ \times \sin 2^\circ \times \sin 3^\circ \times \cdots \times \sin 89^\circ\]

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

Show that there exists a degree $58$ monic polynomial $$P(x) = x^{58} + a_1x^{57} + \cdots + a_{58}$$ such that $P(x)$ has exactly $29$ positive real roots and $29$ negative real roots and that $\log_{2017} |a_i|$ is a positive integer for all $1 \leq i \leq 58$.

Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$: $a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$

Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that \[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\] for all $x,y \in \mathbb{R} $.

At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges? [i]A. Shapovalov[/i]

For all integers $0 \leq k \leq 16$, let \[S_k = \sum_{j=0}^{k}(-1)^j {\binom{16}{j}}.\] Compute $\max(S_0,S_1, \ldots S_{16})$. [i]2021 CCA Math Bonanza Tiebreaker Round #4[/i]

Find the number of positive integers (m,n) which follows the following: 1) m<n 2) The sum of even numbers between 2m and 2n is 100 greater than the sum of odd numbers between 2m and 2n.

Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $ $ [] $ and $ \{\} $ [i]denote the floor, respectively, the fractional part.[/i]

Nine lattice points (i.e. with integer coordinates) $P_1,P_2,...,P_9$ are given in space. Show that the midpoint of at least one of the segments $P_iP_j$ , where $1 \le i < j \le 9$, is a lattice point as well.

Let $P(x)$ be a polynomial with integer coefficients satisfying $$(x^2+1)P(x-1)=(x^2-10x+26)P(x)$$ for all real numbers $x.$ Find the sum of all possible values of $P(0)$ between $1$ and $5000,$ inclusive.

$P(x)$ is a polynomial in $x$ with non-negative integer coefficients. If $P(1)=5$ and $P(P(1))=177$, what is the sum of all possible values of $P(10)$?

Let $ P$ be a polynomial with integer coefficients such that there are two distinct integers at which $ P$ takes coprime values. Show that there exists an infinite set of integers, such that the values $ P$ takes at them are pairwise coprime.

In the right trapezoid $ABCD$ with $AB \parallel CD, \angle B = 90^o$ and $AB = 2DC$. At points $A$ and $D$ there is therefore a part of the plane $(ABC)$ perpendicular to the plane of the trapezoid, on which the points $N$ and $P$ are taken, ($AP$ and $PD$ are perpendicular to the plane) such that $DN = a$ and $AP = \frac{a}{2}$ . Knowing that $M$ is the midpoint of the side $BC$ and the triangle $MNP$ is equilateral, determine: a) the cosine of the angle between the planes $MNP$ and $ABC$. b) the distance from $D$ to the plane $MNP$

A positive interger number $k$ is called “$t-m$”-property if forall positive interger number $a$, there exists a positive integer number $n$ such that ${{1}^{k}}+{{2}^{k}}+{{3}^{k}}+...+{{n}^{k}} \equiv a (\bmod m).$ a) Find all positive integer numbers $k$ which has $t-20$-property. b) Find smallest positive integer number $k$ which has $t-{{20}^{15}}$-property.

Ben picks a positive number $n$ less than $2017$ uniformly at random. Then Rex, starting with the number $ 1$, repeatedly multiplies his number by $n$ and then finds the remainder when dividing by $2017$. Rex does this until he gets back to the number $ 1$. What is the probability that, during this process, Rex reaches every positive number less than $2017$ before returning back to $ 1$?

Let $\bigtriangleup ABC$ be an acute triangle with a point $D$ on side $BC$. Let $J$ be a point on side $AC$ such that $\angle BAD = 2\angle ADJ$, and $\omega$ be the circumcircle of triangle $\bigtriangleup CDJ$. The line $AD$ intersects $\omega$ again at a point $P$, and $Q$ is the feet of the altitude from $J$ to $AB$.\\ Prove that if $JP = JQ$, then the line perpendicular to $DJ$ through $A$ is tangent to $\omega$. [i]Proposed by Ivan Chan - Malaysia[/i]

A baker with access to a number of different spices bakes ten cakes. He uses more than half of the different kinds of spices in each cake, but no two of the combinations of spices are exactly the same. Show that there exist three spices $a,b,c$ such that every cake contains at least one of these.

Tetris is a Soviet block game developed in $1984$, probably to torture misbehaving middle school children. Nowadays, Tetris is a game that people play for fun, and we even have a mini-event featuring it, but it shall be used on this test for its original purpose. The $7$ Tetris pieces, which will be used in various problems in this theme, are as follows: [img]https://cdn.artofproblemsolving.com/attachments/b/c/f4a5a2b90fcf87968b8f2a1a848ad32ef52010.png[/img] [b]p1.[/b] Each piece has area $4$. Find the sum of the perimeters of each of the $7$ Tetris pieces. [b]p2.[/b] In a game of Tetris, Qinghan places $4$ pieces every second during the first $2$ minutes, and $2$ pieces every second for the remainder of the game. By the end of the game, her average speed is $3.6$ pieces per second. Find the duration of the game in seconds. [b]p3.[/b] Jeff takes all $7$ different Tetris pieces and puts them next to each other to make a shape. Each piece has an area of $4$. Find the least possible perimeter of such a shape. [b]p4.[/b] Qepsi is playing Tetris, but little does she know: the latest update has added realistic physics! She places two blocks, which form the shape below. Tetrominoes $ABCD$ and $EFGHI J$ are both formed from $4$ squares of side length $1$. Given that $CE = CF$, the distance from point $I$ to the line $AD$ can be expressed as $\frac{A\sqrt{B}-C}{D}$ . Find $1000000A+10000B +100C +D$. [img]https://cdn.artofproblemsolving.com/attachments/9/a/5e96a855b9ebbfd3ea6ebee2b19d7c0a82c7c3.png[/img] [b]p5.[/b] Using the following tetrominoes: [img]https://cdn.artofproblemsolving.com/attachments/3/3/464773d41265819c4f452116c1508baa660780.png[/img] Find the number of ways to tile the shape below, with rotation allowed, but reflection disallowed: [img]https://cdn.artofproblemsolving.com/attachments/d/6/943a9161ff80ba23bb8ddb5acaf699df187e07.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a right triangle $ABC $ with a right angle $\angle C $, n the sides $AC$ and $AB$, the points $M$ and $N$ are selected, respectively, that $CM = MN$ and $\angle MNB = \angle CBM$. Let the point $K$ be the projection of the point $C $ on the segment $MB $. Prove that the line $NK$ passes through the midpoint of the segment $BC$. (Alex Klurman)

Can a $5\times 7$ checkerboard be covered by L's (figures formed from a $2\times2$ square by removing one of its four $1\times1$ corners), not crossing its borders, in several layers so that each square of the board is covered by the same number of L's? [i]M. Evdokimov[/i]