$f(x)$ is a function defined on $\mathbb{R}$, satisfying: $f(10+x)=f(10-x),f(20-x)=-f(20+x)$. Then, $f(x)$ is $\text{(A)}$even function, but not periodic function $\text{(B)}$even function, and periodic function $\text{(C)}$odd function, but not periodic function $\text{(D)}$odd function, and periodic function

$N$ cossacks split into $3$ groups to discuss various issues with their friends. Cossack Taras moved from the first group to the second, cossack Andriy moved from the second to the third, and cossack Ostap - from the third group to the first. It turned out that the average height of the cossacks in the first group decreased by $8$ cm, while in the second and third groups it increased by $5$ cm and $8$ cm, respectively. What is $N$, if it is known that there were $9$ cossacks in the first group?

Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$. $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$. $(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.

Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose \[(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.\] Compute the product of all possible values of $a$.

An acute triangle $\triangle{ABC}$ is inscribed in a circle with centre $O$. The altitudes of the triangle are $AD,BE$ and $CF$. The line $EF$ cut the circumference on $P$ and $Q$. a) Show that $OA$ is perpendicular to $PQ$. b) If $M$ is the midpoint of $BC$, show that $AP^2=2AD\cdot{OM}$.

[b]10.[/b] An Abelian group $G$ is said to have the property $(A)$ if torsion subgroup of $G$ is a direct summand of $G$. Show that if $G$ is an Abelian group such that $nG$ has the property $(A)$ for some positive integer $n$, then $G$ itself has the property $(A)$. [b](A. 13)[/b]

Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\frac{1}{2}$ times the exterior angle at the same vertex. Then $ \textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad$ $\textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad$ $\textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad$ $\textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad$ $\textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular} $

Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.

Each of $27$ bricks (right rectangular prisms) has dimensions $a \times b \times c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28$[sup]th[/sup] brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is $1$ unit taller, $1$ unit wider, and $1$ unit deeper than the old one. What is $a + b + c$? [img]https://cdn.artofproblemsolving.com/attachments/2/d/b18d3d0a9e5005c889b34e79c6dab3aaefeffd.png[/img] $ \textbf{(A) }88 \qquad \textbf{(B) }89 \qquad \textbf{(C) }90 \qquad \textbf{(D) }91 \qquad \textbf{(E) }92 \qquad $

The numbers from $1$ to $100$ are arranged in a $10\times 10$ table so that any two adjacent numbers have sum no larger than $S$. Find the least value of $S$ for which this is possible. [i]D. Hramtsov[/i]

Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$, each once, on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made. [i]Proposed by Andrew Wen[/i]

Given an acute-angled triangle $ABC$ ($AB \neq AC$) with orthocenter $H$, circumcenter $O$, and midpoint $M$ of side $BC$. The line $AM$ intersects the circumcircle of triangle $BHC$ at point $K$, with $M$ between $A$ and $K$. The segments $HK$ and $BC$ intersect at point $N$. If $\angle BAM = \angle CAN$, prove that the lines $AN$ and $OH$ are perpendicular.

Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.

Mr. A owns a home worth $ \$ 10000$. He sells it to Mr. B at a $ 10\%$ profit based on the worth of the house. Mr. B sells the house back to Mr. A at a $ 10\%$ loss. Then: $ \textbf{(A)}\ \text{A comes out even} \qquad\textbf{(B)}\ \text{A makes }\$ 1100\text{ on the deal}\qquad \textbf{(C)}\ \text{A makes }\$ 1000\text{ on the deal}$ $ \textbf{(D)}\ \text{A loses }\$ 900\text{ on the deal} \qquad\textbf{(E)}\ \text{A loses }\$ 1000\text{ on the deal}$

What is the value of $\dfrac{11!-10!}{9!}$? $\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$

(a) Solve for $\theta\in\mathbb{R}$: $\cos(4\theta) = \cos(3\theta)$ (b) $\cos\left(\frac{2\pi}{7}\right)$, $\cos\left(\frac{4\pi}{7}\right)$ and $\cos\left(\frac{6\pi}{7}\right)$ are the roots of an equation of the form $ax^3+bx^2+cx+d = 0$ where $a, b, c, d$ are integers. Determine $a, b, c$ and $d$.

How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit?

It is stated that $2^{2010}$ is a $606$-digit number that begins with $1$. How many of the numbers $1, 2,2^2,2^3, ..., 2^{2009}$ start with $4$?

Let $a$ be a fixed real number. Consider the equation $$ (x+2)^{2}(x+7)^{2}+a=0, x \in R $$ where $R$ is the set of real numbers. For what values of $a$, will the equ have exactly one double-root?

Triangle $ABC$ has $AB = 8, AC = 12, BC = 10$. Let $D$ be the intersection of the angle bisector of angle $A$ with $BC$. Let $M$ be the midpoint of $BC$. The line parallel to $AC$ passing through $M$ intersects $AB$ at $N$. The line parallel to $AB$ passing through $D$ intersects $AC$ at $P$. $MN$ and $DP$ intersect at $E$. Find the area of $ANEP$.

Real numbers $x$ and $y$ satisfy the equations $x^2 - 12y = 17^2$ and $38x - y^2 = 2 \cdot 7^3$. Compute $x + y$.

Simplify $\left(\sqrt[6]{27} - \sqrt{6 \frac{3}{4} }\right)^2$ $ \textbf{(A)}\ \frac{3}{4} \qquad \textbf{(B)}\ \frac{\sqrt 3}{2} \qquad \textbf{(C)}\ \frac{3 \sqrt 3}{4} \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \frac{3 \sqrt 3}{2} $

How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $

Determine all $f: \mathbb{R}\rightarrow\mathbb{R}$ for which \[ 2\cdot f(x)-g(x)=f(y)-y \textrm{ and } f(x)\cdot g(x) \geq x+1. \]