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 \[ 2\cdot f(x)-g(x)=f(y)-y \textrm{ and } f(x)\cdot g(x) \geq x+1. \]

Some $ m$ squares on the chessboard are marked. If among four squares at the intersection of some two rows and two columns three squares are marked then it is allowed to mark the fourth square. Find the smallest $ m$ for which it is possible to mark all squares after several such operations.

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

Given an integer number $n \ge 2$, show that there exists a function $f : R \to R$ such that $f(x) + f(2x) + ...+ f(nx) = 0$, for all $x \in R$, and $f(x) = 0$ if and only if $x = 0$.