Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. What is $\tfrac{AB}{BC}$? $\textbf{(A) \ } 1+\sqrt{2} \qquad \textbf{(B) \ } 2+\sqrt{2}\qquad \textbf{(C) \ } \sqrt{17}\qquad \textbf{(D) \ } 2+\sqrt{5} \qquad \textbf{(E) \ } 1+2\sqrt{3}$

Given a quadrilateral, the midpoints $A, B, C, D$ of its consecutive sides, and the midpoints of its diagonals, $P$ and $Q$. Prove that $\vartriangle BCP = \vartriangle ADQ$.

Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \] at least one is $\leq 2$ and at least one is $\geq 2$

In a chess festival that is held in a school with $2017$ students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most $22$ matches. What is the maximum possible number of matches in this event?

$\triangle ABC$ and $\triangle ADE$ $(\angle ABC=\angle ADE=\frac{\pi}{2})$ are two isosceles right triangle that are not congruent. Fix $\triangle ABC$, but rotate $\triangle ADE$ on the plane. Prove that there exists point $M\in BC$, satisfying that $\triangle BMD$ is an isosceles right triangle.

The incircle of $ABC$ touches the sides $BC,CA,AB$ at $A' ,B' ,C'$ respectively. The line $A' C'$ meets the angle bisector of $\angle A$ at $D$. Find $\angle ADC$.

Two circles, $\omega_1$ and $\omega_2,$ intersect at $X$ and $Y.$ The segment between their centers intersects $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively, such that $AB=2.$ Given that the radii of $\omega_1$ and $\omega_2$ are $3$ and $4,$ respectively, find $XY.$

Suppose that $ R(z)= \sum_{n=-\infty}^{\infty} a_nz^n$ converges in a neighborhood of the unit circle $ \{ z : \;|z|=1\ \}$ in the complex plane, and $ R(z)=P(z) / Q(z)$ is a rational function in this neighborhood, where $ P$ and $ Q$ are polynomials of degree at most $ k$. Prove that there is a constant $ c$ independent of $ k$ such that \[ \sum_{n=-\infty} ^{\infty} |a_n| \leq ck^2 \max_{|z|=1} |R(z)|.\] [i]H. S. Shapiro, G. Somorjai[/i]

A hexagon inscribed in a circle has three consecutive sides each of length $3$ and three consecutive sides each of length $5$. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length $3$ and the other with three sides each of length $5$, has length equal to $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $\text{(A)}\ 309 \qquad \text{(B)}\ 349 \qquad \text{(C)}\ 369 \qquad \text{(D)}\ 389\qquad \text{(E)}\ 409$

In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.

How many ordered integer pairs of $ (a,b)$ satisfying $ a^2b \plus{} ab^2 \equal{} 2009201020092010$ ? $\textbf{(A)} 4 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None}$

Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time.

Find the $2016$th smallest positive integer that is a solution to $x^x\equiv x\pmod{5}$.

A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.

Let for all $k \in {\mathbb N}$ $k$'s sum of the digits is $T(k)$. If a natural number $n$ such that $T(n)=T(1997n)$, prove that $$9\mid n$$

Let $a$ and $b$ be two numbers in the interval $(0,1)$ such that $a$ is rational and [center]$\{na\} \ge \{nb\},$ for every nonnegative integer $n.$[/center] Prove that $a=b.$ (Note: $\{x\}$ is the fractional part of $x.$)

[b]p1.[/b] At $8$ AM Black Widow and Hawkeye began to move towards each other from two cities. They were planning to meet at the midpoint between two cities, but because Black Widow was driving $100$ mi/h faster than Hawkeye, they met at the point that is located $120$ miles from the midpoint. When they met Black Widow said ”If I knew that you drive so slow I would have started one hour later, and then we would have met exactly at the midpoint”. Find the distance between cities. [b]p2.[/b] Solve the inequality: $\frac{x-1}{x-2} \le \frac{x-2}{x-1}$. [b]p3.[/b] Solve the equation: $(x - y - z)^2 + (2x - 3y + 2z + 4)^2 + (x + y + z - 8)^2 = 0$. [b]p4.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is $50$ miles. Can you help Captain America to evaluate the distances between the camps. [b]p5.[/b] $N$ regions are located in the plane, every pair of them have a nonempty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions? [b]p6.[/b] Numbers $1, 2, . . . , 100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1$, $a_2$, $...$ , $a_{50}$. In the second group the numbers are written in decreasing order and denoted $b_1$, $b_2$, $...$, $b_{50}$. Thus, $a_1 < a_2 < ... < a_{50}$ and $b_1 > b2_ > ... > b_{50}$. Evaluate $|a_1 - b_1| + |a_2 - b_2| + ... + |a_{50} - b_{50}|$. PS. You should use hide for answers.

a) Prove that, whatever the real number x would be, the following inequality takes place ${{x}^{4}}-{{x}^{3}}-x+1\ge 0.$ b) Solve the following system in the set of real numbers: ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$. The Mathematical Gazette

A boy buys oranges at $ 3$ for $ 10$ cents. He will sell them at $ 5$ for $ 20$ cents. In order to make a profit of $ \$ 1.00$, he must sell: $ \textbf{(A)}\ 67 \text{ oranges} \qquad\textbf{(B)}\ 150 \text{ oranges} \qquad\textbf{(C)}\ 200 \text{ oranges} \\ \textbf{(D)}\ \text{an infinite number of oranges} \qquad\textbf{(E)}\ \text{none of these}$

For which value of $m$, there is no integer pair $(x,y)$ satisfying the equation $3x^2-10xy-8y^2=m^{19}$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3$

Let $A$ be a finite set of positive real numbers satisfying the property: [i]For any real numbers a > 0, the sets $\{x \in A | x > a\}$ and $\{x \in A | x < \frac{1}{a}\}$ have the cardinals of the same parity.[/i] Show that the product of all elements in $A$ is equal to $1$.

Investigate the convergence of the integral \[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dxdy}{1+x^4y^4}\]

What is the largest prime $p$ that makes $\sqrt{17p+625}$ an integer? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 67 \qquad\textbf{(C)}\ 101 \qquad\textbf{(D)}\ 151 \qquad\textbf{(E)}\ 211 $

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]D1.[/b] Triangle $ABC$ has $AB = 3$, $BC = 4$, and $\angle B = 90^o$. Find the area of triangle $ABC$. [b]D2 / L1.[/b] Let $ABCDEF$ be a regular hexagon. Given that $AD = 5$, find $AB$. [b]D3.[/b] Caroline glues two pentagonal pyramids to the top and bottom of a pentagonal prism so that the pentagonal faces coincide. How many edges does Caroline’s figure have? [b]D4 / L3.[/b] The hour hand of a clock is $6$ inches long, and the minute hand is $10$ inches long. Find the area of the region swept out by the hands from $8:45$ AM to $9:15$ AM of a single day, in square inches. [b]D5 / L2.[/b] Circles $A$, $B$, and $C$ are all externally tangent, with radii $1$, $10$, and $100$, respectively. What is the radius of the smallest circle entirely containing all three circles? [b]D6.[/b] Four parallel lines are drawn such that they are equally spaced and pass through the four vertices of a unit square. Find the distance between any two consecutive lines. [b]D7 / L4.[/b] In rectangle $ABCD$, $AB = 2$ and $AD > AB$. Two quarter circles are drawn inside of $ABCD$ with centers at $A$ and $C$ that pass through $B$ and $D$, respectively. If these two quarter circles are tangent, find the area inside of $ABCD$ that is outside both of the quarter circles. [b]D8 / L6.[/b] Triangle $ABC$ is equilateral. A circle passes through $A$ and is tangent to side $BC$. It intersects sides AB and $AC$ again at $E$ and $F$, respectively. If $AE = 10$ and $AF = 11$, find $AB$. [b]L5.[/b] Find the area of a triangle with side lengths $\sqrt{2}$, $\sqrt{58}$, and $2\sqrt{17}$. [b]L7.[/b] Triangle $ABC$ has area $80$. Point $D$ is in the interior of $\vartriangle ABC$ such that $AD =6$, $BD = 4$, $CD = 16$, and the area of $\vartriangle ADC = 48$. Determine the area of $\vartriangle ADB$. [b]L8. [/b]Given two points $A$ and $B$ in the plane with $AB = 1$, define $f(C)$ to be the circumcenter of triangle $ABC$, if it exists. Find the number of points $X$ so that $f^{2019}(X) = X$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].