Fix a positive integer $n\geq 3$. Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n\rbrace$, such that $\gcd (a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n)=1$, $\lbrace a_i\rbrace _{i=1}^n$, $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$?

The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$. [i]Proposed by Eugene Chen [/i]

Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty).

Let $n$ be an odd integer. Placing no more than one $X$ in each cell of a $n \times n$ grid, what is the greatest number of $X$ 's that can be put on the grid without getting $n$ $X$'s together vertically, horizontally or diagonally? [list=1] [*] $2{{n}\choose {2}}$ [*] ${{n}\choose {2}}$ [*] $2n $ [*] $2{{n}\choose {2}}-1$ [/list]

A cube has a volume of $125 ^3$ cm . What is the area of one face of the cube? $\textbf{(A)}\ 20^2 \qquad \textbf{(B)}\ 25^2 \qquad \textbf{(C)}\ 41\frac{2}{3}^2 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 75$

A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$ $ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$

A uniform circular disk of radius $R$ begins with a mass $M$; about an axis through the center of the disk and perpendicular to the plane of the disk the moment of inertia is $I_\text{0}=\frac{1}{2}MR^2$. A hole is cut in the disk as shown in the diagram. In terms of the radius $R$ and the mass $M$ of the original disk, what is the moment of inertia of the resulting object about the axis shown? [asy] size(14cm); pair O=origin; pair A=O, B=(3,0), C=(6,0); real r_1=1, r_2=.5; pen my_fill_pen_1=gray(.8); pen my_fill_pen_2=white; pen my_fill_pen_3=gray(.7); pen my_circleline_draw_pen=black+1.5bp; //fill(); filldraw(circle(A,r_1),my_fill_pen_1,my_circleline_draw_pen); filldraw(circle(B,r_1),my_fill_pen_1,my_circleline_draw_pen); // Ellipse filldraw(yscale(.2)*circle(C,r_1),my_fill_pen_1,my_circleline_draw_pen); draw((C.x,C.y-.75)--(C.x,C.y-.2), dashed); draw(C--(C.x,C.y+1),dashed); label("axis of rotation",(C.x,C.y-.75),3*S); // small ellipse pair center_small_ellipse; center_small_ellipse=midpoint(C--(C.x+r_1,C.y)); //dot(center_small_ellipse); filldraw(yscale(.15)*circle(center_small_ellipse,r_1/2),white); pair center_elliptic_arc_arrow; real gr=(sqrt(5)-1)/2; center_elliptic_arc_arrow=(C.x,C.y+gr); //dot(center_elliptic_arc_arrow); draw(//shift((0*center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y-.2))* ( yscale(.2)* ( arc((center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y+2.4), .4,120,360+60)) ),Arrow); //dot(center_elliptic_arc_arrow); // lower_Half-Ellipse real downshift=1; pair C_prime=(C.x,C.y-downshift); path lower_Half_Ellipse=yscale(.2)*arc(C_prime,r_1,180,360); path upper_Half_Ellipse=yscale(.2)*arc(C,r_1,180,360); draw(lower_Half_Ellipse,my_circleline_draw_pen); //draw(upper_Half_Ellipse,red); // Why here ".2*downshift" instead of downshift seems to be not absolutely clean. filldraw(upper_Half_Ellipse--(C.x+r_1,C.y-.2*downshift)--reverse(lower_Half_Ellipse)--cycle,gray); //filldraw(shift(C-.1)*(circle((B+.5),.5)),my_fill_pen_2);// filldraw(circle((B+.5),.5),my_fill_pen_2);//shift(C-.1)* /* filldraw(//shift((C.x,C.y-.45))* yscale(.2)*circle((C.x,C.y-1),r_1),my_fill_pen_3,my_circleline_draw_pen); */ draw("$R$",A--dir(240),Arrow); draw("$R$",B--shift(B)*dir(240),Arrow); draw(scale(1)*"$\scriptstyle R/2$",(B+.5)--(B+1),.5*LeftSide,Arrow); draw(scale(1)*"$\scriptstyle R/2$",(B+.5)--(B+1),.5*LeftSide,Arrow); [/asy] (A) $\text{(15/32)}MR^2$ (B) $\text{(13/32)}MR^2$ (C) $\text{(3/8)}MR^2$ (D) $\text{(9/32)}MR^2$ (E) $\text{(15/16)}MR^2$

Consider two finite sequences of real numbers \( a_1, a_2, \dots, a_n \) and \( b_1, b_2, \dots, b_n \). Let \( \alpha(x) = \#\{i | a_i = x \} \) and \( \beta(x) = \#\{i | b_i = -x \} \). Prove that there exists a permutation \( \sigma \in S_n \) (the symmetric group of \( n \) elements) such that \( a_{\sigma(i)} + b_i \neq 0 \) for all \( i = 1, \dots, n \) if and only if \( \alpha(x) + \beta(x) \leq n \) for all \( x \in \mathbb{R} \).

If $a$ and $b$ are distinct real numbers, solve the systems (a) $\begin{cases} x+y = 1 \\ (ax+by)^2 \le a^2x+b^2y \end{cases}$ and (b) $\begin{cases} x+y = 1 \\ (ax+by)^4 \le a^4x+b^4y \end{cases}$

Consider real numbers $A$, $B$, \dots, $Z$ such that \[ EVIL = \frac{5}{31}, \; LOVE = \frac{6}{29}, \text{ and } IMO = \frac{7}{3}. \] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$. [i]Proposed by Evan Chen[/i]

Let $ABCD$ be a parallelogram with area 160. Let diagonals $AC$ and $BD$ intersect at $E$. Point $P$ is on $\overline{AE}$ such that $EC = 4EP$. If line $DP$ intersects $AB$ at $F$, find the area of $BFPC$. [i]Proposed by Andy Xu[/i]

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

Eleven pirates find a treasure chest. When they split up the coins in it, they find that there are 5 coins left. They throw one pirate overboard and split the coins again, only to find that there are 3 coins left over. So, they throw another pirate over and try again. This time, the coins split evenly. What is the least number of coins there could have been?

Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class. Each year at the college one third of the freshman class drops out of school, $40$ students in the sophomore class drop out of school, and one tenth of the junior class drops out of school. Given that the college only admits new freshman students, and that it wants to begin each school year with $3400$ students enrolled, how many students does it need to admit into the freshman class each year?

Let $A$ and $B$ be unit hexagons that share a center. Then, let $\mathcal{P}$ be the set of points contained in at least one of the hexagons. If the maximum possible area of $\mathcal{P}$ is $X$ and the minimum possible area of $\mathcal{P}$ is $Y,$ then the value of $Y-X$ can be expressed as $\tfrac{a\sqrt{b}-c}{d},$ where $a,b,c,d$ are positive integers such that $b$ is square-free and $\gcd(a,c,d)=1.$ Find $a+b+c+d.$

How many positive integers $n$ are there such that $3 \leq n \leq 100$ and $x^{2^{n}} + x + 1$ is divisible by $x^2 + x + 1$?

When $1999{}^2{}^0{}^0{}^0$ is divided by $5$ , the remainder is $ \text{(A)}\ 4\qquad\text{(B)}\ 1\qquad\text{(C)}\ 2\qquad\text{(D)}\ 3\qquad\text{(E)}\ 0 $

In $\triangle ABC$, the interior sides of which are mirrors, a laser is placed at point $A_1$ on side $BC$. A laser beam exits the point $A_1$, hits side $AC$ at point $B_1$, and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side $AB$ at point $C_1$, then side $BC$ at point $A_2$, then side $AC$ again at point $B_2$, then side $AB$ again at point $C_2$, then side $BC$ again at point $A_3$, and finally, side $AC$ again at point $B_3$. (a) Prove that $\angle B_3A_3C = \angle B_1A_1C$. (b) Prove that such a laser exists if and only if all the angles in $\triangle ABC$ are less than $90^{\circ}$.

Rama was asked by her teacher to subtract $3$ from a certain number and then divide the result by $9$. Instead, she subtracted $9$ and then divided the result by $3$. She got $43$ as the answer. What should have been her answer if she had solved the problem correctly?

Let $ABC$ be a triangle and let $P$ be a point on one of the sides of $ABC$. Construct a line passing through $P$ that divides triangle $ABC$ into two parts of equal area.

Let $A$ and $B$ be two given points, distance $1 $ apart. Determine a point $P$ on the line $AB$ such that $$\frac{1}{1 + AP}+\frac{1}{1 + BP}$$ is a maximum.

Let $p$ be a (positive) prime number. Suppose that real numbers $a_1, a_2, . . ., a_{p+1}$ have the property that, whenever one of the numbers is deleted, the remaining numbers can be partitioned into two classes with the same arithmetic mean. Show that these numbers must be equal.

Prove that \[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]

Let $ ABCD $ be a rectangle that has $ M $ on its $ BD $ diagonal. If $ N,P $ are the projections of $ M $ on $ AB, $ respectively, $ AD, $ what's the locus of the intersection between $ CP $ and $ DN? $

There is a polynomial $P(x)$ with integer coefficients such that $$P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$$ holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$