For any given positive integer $n$, $n^6+3a$ is a perfect cube, where $a$ is a positive integer. Then $\text{(A)}$There is no such $a$. $\text{(B)}$There are infinitely many such $a$. $\text{(C)}$There is finitely many such $a$. $\text{(D)}$None of $\text{(A)(B)(C)}$ is correct.

In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.

For $n$ a positive integer, define $f_1(n)=n$ and then for $i$ a positive integer, define $f_{i+1}(n)=f_i(n)^{f_i(n)}$. Determine $f_{100}(75)\pmod{17}$. Justify your answer.

Let $ ABC$ be an isosceles triangle with $ \left|AB\right| \equal{} \left|AC\right| \equal{} 10$ and $ \left|BC\right| \equal{} 12$. $ P$ and $ R$ are points on $ \left[BC\right]$ such that $ \left|BP\right| \equal{} \left|RC\right| \equal{} 3$. $ S$ and $ T$ are midpoints of $ \left[AB\right]$ and $ \left[AC\right]$, respectively. If $ M$ and $ N$ are the foot of perpendiculars from $ S$ and $ R$ to $ PT$, then find $ \left|MN\right|$. $\textbf{(A)}\ \frac {9\sqrt {13} }{26} \qquad\textbf{(B)}\ \frac {12 \minus{} 2\sqrt {13} }{13} \qquad\textbf{(C)}\ \frac {5\sqrt {13} \plus{} 20}{13} \qquad\textbf{(D)}\ 15\sqrt {3} \qquad\textbf{(E)}\ \frac {10\sqrt {13} }{13}$

Find the least integer $n > 60$ so that when $3n$ is divided by $4$, the remainder is $2$ and when $4n$ is divided by $5$, the remainder is $1$.

The numbers $S_1=2^2, S_2=2^4,\ldots, S_n=2^{2n}$ are given. A rectangle $OABC$ is constructed on the Cartesian plane according to these numbers. For this, starting from the point $O$ the points $A_1,A_2,\ldots,A_n$ are consistently marked along the axis $Ox$, and the points $C_1,C_2,\ldots,C_n$ are consistently marked along the axis $Oy$ in such a way that for all $k$ from $1$ to $n$ the lengths of the segments $A_{k-1}A_k=x_k$ and $C_{k-1}C_k=y_k$ are positive integers (let $A_0=C_0=O$, $A_n=A$, and $C_n=C$) and $x_k\cdot y_k=S_k$. [b]a)[/b] Find the maximal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this maximal area is achieved. [b]b)[/b] Find the minimal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this minimal area is achieved. [i](E. Manzhulina, B. Rublyov)[/i]

A point object of mass $m$ is connected to a cylinder of radius $R$ via a massless rope. At time $t = 0$ the object is moving with an initial velocity $v_0$ perpendicular to the rope, the rope has a length $L_0$, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds $T_{max}$. Express your answers in terms of $T_{max}$, $m$, $L_0$, $R$, and $v_0$. [asy] size(200); real L=6; filldraw(CR((0,0),1),gray(0.7),black); path P=nullpath; for(int t=0;t<370;++t) { pair X=dir(180-t)+(L-t/180)*dir(90-t); if(X.y>L) X=(X.x,L); P=P--X; } draw(P,dashed,EndArrow(size=7)); draw((-1,0)--(-1,L)--(2,L),EndArrow(size=7)); filldraw(CR((-1,L),0.25),gray(0.7),black);[/asy]What is the length (not yet wound) of the rope? $ \textbf{(A)}\ L_0 - \pi R $ $ \textbf{(B)}\ L_0 - 2 \pi R$ $ \textbf{(C)}\ L_0 - \sqrt{18} \pi R $ $ \textbf{(D)}\ \frac{mv_0^2}{T_{max}} $ $ \textbf{(E)}\ \text{none of the above} $

Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by $n.$ [b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$

Given $2017$ positive real numbers $a_1,a_2,\dots ,a_{2017}$. For each $n>2017$, set $$a_n=\max\{ a_{i_1}a_{i_2}a_{i_3}|i_1+i_2+i_3=n, 1\leq i_1\leq i_2\leq i_3\leq n-1\}.$$ Prove that there exists a positive integer $m\leq 2017$ and a positive integer $N>4m$ such that $a_na_{n-4m}=a_{n-2m}^2$ for every $n>N$.

Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

Compute the positive integer $n$ such that $\log_3 n < \log_2 3 < \log_3 (n + 1)$.

Let $a$ and $b$ be positive integers satisfying \[ \frac a{a-2} = \frac{b+2021}{b+2008} \] Find the maximum value $\dfrac ab$ can attain.

Let $M$ and $N$ be two points on different sides of the square $ABCD$. Suppose that segment $MN$ divides the square into two tangential polygons. If $R$ and $r$ are radii of the circles inscribed in these polygons ($R> r$), calculate the length of the segment $MN$ in terms of $R$ and $r$. (Moldova)

Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]

A class has $24$ students. Each group consisting of three of the students meet, and choose one of the other $21$ students, A, to make him a gift. In this case, A considers each member of the group that offered him a gift as being his friend. Prove that there is a student that has at least $10$ friends.

Let $\{g_i\}_{i=0}^{\infty}$ be a sequence of positive integers such that $g_0=g_1=1$ and the following recursions hold for every positive integer $n$: \begin{align*} g_{2n+1} &= g_{2n-1}^2+g_{2n-2}^2 \\ g_{2n} &= 2g_{2n-1}g_{2n-2}-g_{2n-2}^2 \end{align*} Compute the remainder when $g_{2011}$ is divided by $216$.

What is the last two digits of the number $(11 + 12 + 13 + ... + 2006)^2$?

Let the number of ways for a rook to return to its original square on a $4\times 4$ chessboard in 8 moves if it starts on a corner be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path. [i]Proposed by Bradley Guo[/i]

In a one-player game, you have three cards. At the beginning, a nonnegative integer is written on each of the cards, and the sum of these three integers is $2006$. At each step, you can select two of the three chards, subtract $1$ from the integer written on each of these two cards - as long as the resulting integers are still nonnegative -, and add $1$ to the integer written on the third card. You play this game until you can’t perform a step anymore because two of the cards have $0$’s written on them. Assume that, at this moment, the third card has a $1$ written on it. Prove that I can tell you which card contains the $1$ without knowing how exactly you proceeded in your game, but only knowing the starting configuration (i. e., the numbers written on the cards at the beginning of the game) and the fact that at the end, you were left with two $0$’s and a $1$.

Alice and Bob play a game on a sphere which is initially marked with a finite number of points. Alice and Bob then take turns making moves, with Alice going first: - On Alice’s move, she counts the number of marked points on the sphere, \(n\). She then marks another \(n + 1\) points on the sphere. - On Bob’s move, he chooses one hemisphere and removes all marked points on that hemisphere, including any marked points on the boundary of the hemisphere. Can Bob always guarantee that after a finite number of moves, the sphere contains no marked points? (A [i]hemisphere[/i] is the region on a sphere that lies completely on one side of any plane passing through the centre of the sphere.) [i]proposed by the ICMC Problem Committee[/i]

An array $n \times n$ is given, consisting of $n^2$ unit squares. A pawn is placed arbitrarily on a unit square. A [i]move[/i] of the pawn means a jump from a square of the $k$th column to any square of the $k$th row. Show that there exists a sequence of $n^2$ moves of the pawn so that all the unit squares of the array are visited once and, in the end, the pawn returns to the original position.

Determine all positive integers $n$ with the property that $n = (d(n))^2$. Here $d(n)$ denotes the number of positive divisors of $n$.

In an acute triangle $ABC$, altitudes at vertices $A$ and $B$ and bisector line at angle $C$ intersect the circumcircle again at points $A_1, B_1$ and $C_0$. Using the straightedge and compass, reconstruct the triangle by points $A_1, B_1$ and $C_0$.

A classroom has $30$ students, each of whom is either male or female. For every student $S$, we define his or her [i]ratio[/i] to be the number of students of the opposite gender as $S$ divided by the number of students of the same gender as $S$ (including $S$). Let $\Sigma$ denote the sum of the ratios of all $30$ students. Find the number of possible values of $\Sigma$. [i]Proposed by Evan Chen[/i]

Let $f,g:[a,b]\to [0,\infty)$ be two continuous and non-decreasing functions such that each $x\in [a,b]$ we have \[ \int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt. \] Prove that \[ \int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt. \]