Let $R$ be a ring of characteristic zero. Let $e,f,g\in R$ be idempotent elements (an element $x$ is called idempotent if $x^2=x$) satisfying $e+f+g=0$. Show that $e=f=g=0$.

A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$.

Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, ..., 30\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1.$

A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.

The three distinct points$ B, C, D$ are collinear with C between B and D. Another point A not on the line BD is such that $|AB| = |AC| = |CD|.$ Prove that ∠$BAC = 36$ if and only if $1/|CD|-1/|BD|=1/(|CD| + |BD|)$ .

An urn initially contains one red ball and one blue ball. In each step, we choose a uniform random ball from the urn. If it is red, then another red ball and another blue ball are placed in the urn. And when we choose a blue ball for the $k$-th time, we put a blue ball and $2k + 1$ red balls in the urn. (The chosen balls are not removed; they remain in the urn.) Let $G_n$ denote the number of red balls in the urn after $n$ steps. Prove that there exist constants $0 < c, \alpha < \infty$ such that $\frac{G_n}{n^\alpha} \to c$ almost surely.

Positive integers $a, b, c$ satisfy $\frac1a +\frac1b +\frac1c<1$. Prove that $\frac1a +\frac1b +\frac1c\le \frac{41}{42}$. Also prove that equality in fact holds in the second inequality.

A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning. Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).

Let $P$ be a point in the interior of a triangle $ABC$ and let the rays $\overrightarrow{AP}, \overrightarrow{BP}$ and $\overrightarrow{CP}$ intersect the sides $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$, respectively. Let $D$ be the foot of the perpendicular from $A_1$ to $B_1C_1$. Show that \[\frac{CD}{BD}=\frac{B_1C}{BC_1} \cdot \frac{C_1A}{AB_1}.\]

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]

Is it possible to divide a $6 \times 6$ chessboard into $18$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint? (A Shapovalov)

Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$

Let a, b be positive integers. a, b and a.b are not perfect squares. Prove that at most one of following equations $ ax^2 \minus{} by^2 \equal{} 1$ and $ ax^2 \minus{} by^2 \equal{} \minus{} 1$ has solutions in positive integers.

Two quadratic polynomials $ f_{1},f_{2}$ satisfy $ f_{1}'(x)f_{2}'(x)\geq |f_{1}(x)|\plus{}|f_{2}(x)|\forall x\in\mathbb{R}$ . Prove that $ f_{1}\cdot f_{2}\equal{} g^{2}$ for some $ g\in\mathbb{R}[x]$.

A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$. Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.

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]