Let $n$ be a positive integer, and let $p$ be a prime number. Prove that if $p^p | n!$, then $p^{p+1} | n!$.

Consider the following modified algorithm for binary search, which we will call $\textit{weighted binary search}$: \begin{tabular}{l} 01: \textbf{FUNCTION} SEARCH($L$, value) \\ 02:$\qquad$ hi $\leftarrow$ $\operatorname{len}(L) - 1$ \\ 03:$\qquad$ lo $\leftarrow$ 0 \\ 04:$\qquad$ \textbf{WHILE} hi $\geq$ lo \\ 05:$\qquad\qquad$ guess $\leftarrow$ $\lfloor w \cdot\text{lo} + (1-w) \cdot \text{hi}\rfloor$ \\ 06:$\qquad\qquad$ mid $\leftarrow$ $L[\text{guess}]$ \\ 07:$\qquad\qquad$ \textbf{IF} mid $> \text{value}$ \\ 08: $\qquad\qquad\qquad$ hi $\leftarrow$ $\text{guess} - 1$ \\ 09: $\qquad\qquad$ \textbf{ELSE IF} mid $< \text{value}$ \\ 10: $\qquad\qquad\qquad$ lo $\leftarrow$ $\text{guess} + 1$ \\ 11: $\qquad\qquad$ \textbf{ELSE} \\ 12: $\qquad\qquad\qquad$ \textbf{RETURN} guess \\ 13:$\qquad$ \textbf{RETURN} -1 (not found) \end{tabular}\\ Assume $L$ is a list of the integers $\{1,2,\ldots,100\}$, in that order. Further assume that accessing the $k$th index of $L$ costs $k+1$ tokens (e.g. $L[0]$ costs $1$ token). Let $S$ be the set of all $w\in[\tfrac12,1)$ which minimize the average cost when $\texttt{value}$ is an integer selected at random in the range $[1,50]$. Given that $S=\left(x,\tfrac {74}{99}\right]$, determine $x$.

Let $ABC$ be a triangle. Consider three circles, centered at $A, B, C$, with respective radii $$\sqrt{AB \cdot AC},\sqrt{BC \cdot BA},\sqrt{CA \cdot CB}.$$ Given that there are six distinct pairwise intersections between these three circles, show that they lie on two concentric circles. [i](Two circles are concentric if they have the same center.)[/i]

A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?

There are $11$ empty boxes and a pile of stones. Two players play the following game by alternating moves: In one move a player takes $10$ stones from the pile and places them into boxes, taking care to place no more than one stone in any box. The winner is the player after whose move there appear $21$ stones in one of the boxes for the first time. If a player wants to guarantee that they win the game, should they go first or second? Explain your reasoning.

Let $n$ points, where there are not $3$ of them on a line, and consider the segments that are formed by connecting any $2$ of the points. There are enough colors available to paint the points and the segments, coloring them with the following two rules: a) All the segments that reach the same point are painted of different colors. b) Each point is painted a different color to all the segments that reach it. Find the minimum number of colors needed to make such a coloring.

Andrew and Barry play the following game: there are two heaps with $a$ and $b$ pebbles, respectively. In the first round Barry chooses a positive integer $k,$ and Andrew takes away $k$ pebbles from one of the two heaps (if $k$ is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round, the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one of the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game. Which player has a winning strategy? [i]Submitted by András Imolay, Budapest[/i]

There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies. [i]Proposed by Tejaswi Navilarekallu, India[/i]

Set $u_0 = \frac{1}{4},$ and for $k \geq 0$ let $u_{k+1}$ be determined by the recurrence $u_{k+1} = 2u_k - 2u_k^2.$ This sequence tends to a limit, call it $L.$ What is the least value of $k$ such that $$|u_k - L| \leq \frac{1}{2^{1000}}?$$ $\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 253 \qquad\textbf{(D)}\ 329 \qquad\textbf{(E)}\ 401$

Given that the value of \[\sum_{k=1}^{2021} \frac{1}{1^2+2^2+3^2+\cdots+k^2}+\sum_{k=1}^{1010} \frac{6}{2k^2-k}+\sum_{k=1011}^{2021} \frac{24}{2k+1}\] can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Aidan Duncan[/i]

Let $ABC$ be a triangle with $\angle B=120^{o}$. Let $D$ be point on the $B$-angle bisector, such that $\angle ADB=2\angle ACB$. Point $E$ lies on the segment $AB$, so that $AE=AD$. Show that $EC=ED$.

Let $P$ be an interior point of acute triangle $\Delta ABC$, which is different from the orthocenter. Let $D$ and $E$ be the feet of altitudes from $A$ to $BP$ and $CP$, and let $F$ and $G$ be the feet of the altitudes from $P$ to sides $AB$ and $AC$. Denote by $X$ the midpoint of $[AP]$, and let the second intersection of the circumcircles of triangles $\Delta DFX$ and $\Delta EGX$ lie on $BC$. Prove that $AP$ is perpendicular to $BC$ or $\angle PBA = \angle PCA$.

The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$, respectively. Let $S$ be the intersection point of lines passing through points $B$ and $C$ and parallel to $A_1C_1$ and $A_1B_1$ respectively, $A_0$ be the foot of the perpendicular drawn from point $A_1$ on $B_1C_1$, $G_1$ be the centroid of triangle $A_1B_1C_1$, $P$ be the intersection point of the ray $G_1A_0$ with $\omega$. Prove that points $S, A_1$, and $P$ lie on a straight line.

Charlyn walks completely around the boundary of a square whose sides are each $5$ km long. From any point on her path she can see exactly $1$ km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number? $\text{(A)}\ 24 \qquad\text{(B)}\ 27\qquad\text{(C)}\ 39\qquad\text{(D)}\ 40 \qquad\text{(E)}\ 42$

Around the triangle $ABC$ the circle is circumscribed, and at the vertex ${C}$ tangent ${t}$ to this circle is drawn. The line ${p}$, which is parallel to this tangent intersects the lines ${BC}$ and ${AC}$ at the points ${D}$ and ${E}$, respectively. Prove that the points $A,B,D,E$ belong to the same circle. (Montenegro)

Find all $7$-digit numbers which are multiples of $21$ and which have each digit $3$ or $7$.

Find all real number $p$, such that the three roots of the equation $5x^3-5(p+1)x^2+(71p-1)x+1=66p$ are all positive integers.