In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

Balls numbered $1,2,3,\ldots$ are deposited in $5$ bins, labeled $A,B,C,D,$ and $E$, using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposted in $B$. The next three balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, $22,23,\ldots,28$ are despoited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited? $\textbf{(A) }A\qquad\textbf{(B) }B\qquad\textbf{(C) }C\qquad\textbf{(D) }D\qquad\textbf{(E) }E$

Consider the following game for a positive integer $n$. Initially, the numbers $1, 2, \ldots, n$ are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers $3,6,11$ and $17$ are on the board, then $3$ can be erased as $6 - 3=3$, or $6$ as $17 - 11=6$, or $11$ as $17 - 6=11$.) For which values of $n$ is it possible to end with only one number remaining on the board? [i](Michael Reitmeir)[/i]

The elevator buttons in Harvard's Science Center form a $3\times 2$ grid of identical buttons, and each button lights up when pressed. One day, a student is in the elevator when all the other lights in the elevator malfunction, so that only the buttons which are lit can be seen, but one cannot see which floors they correspond to. Given that at least one of the buttons is lit, how many distinct arrangements can the student observe? (For example, if only one button is lit, then the student will observe the same arrangement regardless of which button it is.)

Given a permutation $(a_{1}, a_{1},...,a_{n})$ of the numbers $1, 2,...,n$ one may interchange any two consecutive "blocks" - that is, one may transform ($a_{1}, a_{2},...,a_{i}$,$\underbrace {a_{i+1},... a_{i+p},}_{A} $ $ \underbrace{a_{i+p+1},...,a_{i+q},}_{B}...,a_{n}) $ into $ (a_{1}, a_{2},...,a_{i},$ $ \underbrace {a_{i+p+1},...,a_{i+q},}_{B} $ $ \underbrace {a_{i+1},... a_{i+p}}_{A}$$,...,a_{n}) $ by interchanging the "blocks" $A$ and $B$. Find the least number of such changes which are needed to transform $(n, n-1,...,1)$ into $(1,2,...,n)$

In an $6 \times 6$ grid of lattice points, how many ways are there to choose $ 4$ points that are vertices of a nondegenerate quadrilateral with at least one pair of opposite sides parallel to the sides of the grid?

For reals $x\ge3$, let $f(x)$ denote the function \[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^2} .\] [i]Ray Li.[/i]

Consider in a square $[ABCD]$ a point $E$ on the side $AB$, different from $A$ and $B$. On the side $BC$ consider the point $F$ such that $\angle AED = \angle DEF$ . Prove that $EF = AE + FC$.

We have $n>2$ non-zero integers such that each one of them is divisible by the sum of the other $n-1$ numbers. Prove that the sum of all the given numbers is zero.

[list] [*] A test was conducted in class. It is known that at least $\frac{2}{3}$ of the problems were hard. Each such problems were not solved by at least $\frac{2}{3}$ of the students. It is also known that at least $\frac{2}{3}$ of the students passed the test. Each such student solved at least $\frac{2}{3}$ of the suggested problems. Is this possible? [*] Previous problem with $\frac{2}{3}$ replaced by $\frac{3}{4}$. [*] Previous problem with $\frac{2}{3}$ replaced by $\frac{7}{10}$.[/list]

The entries on an $ n$ × $ n$ board are colored black and white like it is usually done in a chessboard, and the upper left hand corner is black. We color the entries on the chess board black according to the following rule: In each step we choose an arbitrary $ 2$×$ 3$ or $ 3$× $ 2$ rectangle that still contains $ 3$ white entries, and we color these three entries black. For which values of $ n$ can the whole board be colored black in a finite number of steps

In triangle $\triangle ABC$, the points $A', B', C'$ are on sides $BC, AC, AB$ respectively. Also, $AA', BB', CC'$ intersect at the point $O$(they are concurrent at $O$). Also, $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$. Find the value of $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$.

A jar was filled with jelly beans so that $54\%$ of the beans were red, $30\%$ of the beans were green, and $16\%$ of the beans were blue. Alan then removed the same number of red jelly beans and green jelly beans from the jar so that now $20\%$ of the jelly beans in the jar are blue. What percent of the jelly beans in the jar are now red?

In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point of side $CD$ such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.

The bisectors of the angles $\angle{A},\angle{B},\angle{C}$ of a triangle $\triangle{ABC}$ intersect with the circumcircle $c_1(O,R)$ of $\triangle{ABC}$ at $A_2,B_2,C_2$ respectively.The tangents of $c_1$ at $A_2,B_2,C_2$ intersect each other at $A_3,B_3,C_3$ (the points $A_3,A$ lie on the same side of $BC$,the points $B_3,B$ on the same side of $CA$,and $C_3,C$ on the same side of $AB$).The incircle $c_2(I,r)$ of $\triangle{ABC}$ is tangent to $BC,CA,AB$ at $A_1,B_1,C_1$ respectively.Prove that $A_1A_2,B_1B_2,C_1C_2,AA_3,BB_3,CC_3$ are concurrent. [hide=Diagram][asy]import graph; size(11cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -9.26871978147865, xmax = 19.467150423463277, ymin = -6.150626456647122, ymax = 10.10782642246474; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((1.0409487561836381,4.30054785243355)--(0.,0.)--(6.,0.)--cycle, aqaqaq); /* draw figures */ draw((1.0409487561836381,4.30054785243355)--(0.,0.), uququq); draw((0.,0.)--(6.,0.), uququq); draw((6.,0.)--(1.0409487561836381,4.30054785243355), uququq); draw(circle((3.,1.550104087253063), 3.376806580383107)); draw(circle((1.9303371951242874,1.5188413314630436), 1.5188413314630436)); draw((1.0226422135625703,7.734611112525813)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((-1.2916762981259242,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((-0.2820306621765219,2.344520485530311)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((1.0559139088339535,1.4932847901569466)--(5.212367857300808,4.101231513568902), linetype("2 2")); draw((1.0559139088339535,1.4932847901569466)--(3.,-1.8267024931300442), linetype("2 2")); draw((12.047991949367804,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((1.0226422135625703,7.734611112525813)--(-1.2916762981259242,-1.8267024931300444)); draw((-1.2916762981259242,-1.8267024931300444)--(12.047991949367804,-1.8267024931300444)); draw((12.047991949367804,-1.8267024931300444)--(1.0226422135625703,7.734611112525813)); /* dots and labels */ dot((1.0409487561836381,4.30054785243355),linewidth(3.pt) + dotstyle); label("$A$", (0.5889800538632699,4.463280489351154), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.5723380089304358,-0.10096957139619551), NE * labelscalefactor); dot((6.,0.),linewidth(3.pt) + dotstyle); label("$C$", (6.233525986976863,0.06107480945873997), NE * labelscalefactor); label("$c_1$", (1.9663572911302232,5.111458012770896), NE * labelscalefactor); dot((3.,-1.8267024931300442),linewidth(3.pt) + dotstyle); label("$A_2$", (2.9386235762598374,-2.3155761097469805), NE * labelscalefactor); dot((5.212367857300808,4.101231513568902),linewidth(3.pt) + dotstyle); label("$B_2$", (5.315274495465561,4.274228711687063), NE * labelscalefactor); dot((-0.2820306621765219,2.344520485530311),linewidth(3.pt) + dotstyle); label("$C_2$", (-0.9234341674494632,2.6807922999468636), NE * labelscalefactor); dot((1.0226422135625703,7.734611112525813),linewidth(3.pt) + dotstyle); label("$A_3$", (1.1291279900463889,7.893219884113956), NE * labelscalefactor); dot((-1.2916762981259242,-1.8267024931300444),linewidth(3.pt) + dotstyle); label("$B_3$", (-1.8146782621516093,-1.4783468086631473), NE * labelscalefactor); dot((12.047991949367804,-1.8267024931300444),linewidth(3.pt) + dotstyle); label("$C_3$", (12.148145888182015,-1.6673985863272387), NE * labelscalefactor); dot((1.9303371951242874,1.5188413314630436),linewidth(3.pt) + dotstyle); label("$I$", (2.047379481557691,1.681518618008095), NE * labelscalefactor); dot((1.9303371951242878,0.),linewidth(3.pt) + dotstyle); label("$A_1$", (1.4532167517562602,-0.5600953171518461), NE * labelscalefactor); label("$c_2$", (1.5072315453745722,3.247947632939138), NE * labelscalefactor); dot((2.9254299438737803,2.666303492733126),linewidth(3.pt) + dotstyle); label("$B_1$", (2.8576013858323694,3.1129106488933584), NE * labelscalefactor); dot((0.45412477306806903,1.8761589424582812),linewidth(3.pt) + dotstyle); label("$C_1$", (0,2.3296961414278368), NE * labelscalefactor); dot((1.0559139088339535,1.4932847901569466),linewidth(3.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/hide]

Let $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$ for real number $ p > 0$. Answer the following questions. (1) Find the value of $ x \equal{} s_p$ for which $ f(x)$ is minimal and draw the graph of $ y \equal{} f(x)$. (2) Let $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$. Find the value of $ t \equal{} t_p$ for which $ g(t)$ is minimal. (3) Use the fact $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$ to find the limit $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$.

Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.

A point $A$ is outside a circle $\mathcal K$ with center $O$. Line $AO$ intersects the circle at $B$ and $C$, and a tangent through $A$ touches the circle in $D$. Let $E$ be an arbitrary point on the line $BD$ such that $D$ lies between $B$ and $E$. The circumcircle of the triangle $DCE$ meets line $AO$ at $C$ and $F$ and line $AD$ at $D$ and $G$. Prove that the lines $BD$ and $FG$ are parallel.

Let $ABC$ be a scalene triangle with circumcenter $O$ and incenter $I$. The $A$-excircle, $B$-excircle, and $C$-excircle of triangle $ABC$ touch $BC$, $CA$, and $AB$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $P$ be the orthocenter of $AB_1C_1$ and $H$ be the orthocenter of $ABC$. Show that if $M$ is the midpoint of $PA_1$, then lines $HM$ and $OI$ are parallel. [i]Michael Ren[/i]

Alice starts with an empty string and randomly appends one of the digits $2$, $0$, $1$, or $8$ until the string ends with $2018$. What is the probability Alice appends less than $9$ digits before stopping?

Find all triples $m$; $n$; $l$ of natural numbers such that $m + n = gcd(m; n)^2$; $m + l = gcd(m; l)^2$; $n + l = gcd(n; l)^2$: [i]S. Tokarev[/i]

Bob chooses a $4$-digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob’s string is $0000$ and the stream of bits begins $101000001 \dots$, then $N = 7$.

A polynomial $p(x)$ with real coefficients is said to be [i]almeriense[/i] if it is of the form: $$ p(x) = x^3+ax^2+bx+a $$ And its three roots are positive real numbers in arithmetic progression. Find all [i]almeriense[/i] polynomials such that $p\left(\frac{7}{4}\right) = 0$

Compute the smallest positive integer $M$ such that there exists a positive integer $n$ such that [list] [*] $M$ is the sum of the squares of some $n$ consecutive positive integers, and [*] $2M$ is the sum of the squares of some $2n$ consecutive positive integers. [/list] [i]Proposed by Jaedon Whyte[/i]

The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$, that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$.