I give you a function $\textbf{rand}$ that returns a number chosen uniformly at random from $[0,T]$ for some number $T$ that you don't know. Your task is to approximate $T$. You do this by calling $\textbf{rand}$ $100$ times, recording the results as $X_1,X_2,\dots,X_{100}$, and guessing \[\hat{T}=\alpha\cdot\max\{X_1,X_2,\dots,X_{100}\}\] for some $\alpha$. Which value of $\alpha$ ensures that $\mathbb{E}[\hat{T}]=T$?

How many three-digit positive integers $\underline{a}$ $\underline{b}$ $\underline{c}$ are there whose nonzero digits $a$, $b$, and $c$ satisfy $$0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$$ (The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ in the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$) $\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

For each positive integer $n$ we denote by $p(n)$ the greatest square less than or equal to $n$. [list=a] [*]Find all pairs of positive integers $( m,n)$, with $m\leq n$, for which \[ p( 2m+1) \cdot p( 2n+1) =400 \] [*]Determine the set $\mathcal{P}=\{ n\in\mathbb{N}^{\ast}\vert n\leq100\text{ and }\dfrac{p(n+1)}{p(n)}\notin\mathbb{N}^{\ast}\}$[/list]

Let $ABC$ an acute triangle and $H$ its orthocenter. Let $E$ and $F$ be the intersection of lines $BH$ and $CH$ with $AC$ and $AB$ respectively, and let $D$ be the intersection of lines $EF$ and $BC$. Let $\Gamma_1$ be the circumcircle of $AEF$, and $\Gamma_2$ the circumcircle of $BHC$. The line $AD$ intersects $\Gamma_1$ at point $I \neq A$. Let $J$ be the feet of the internal bisector of $\angle{BHC}$ and $M$ the midpoint of the arc $\stackrel{\frown}{BC}$ from $\Gamma_2$ that contains the point $H$. The line $MJ$ intersects $\Gamma_2$ at point $N \neq M$. Show that the triangles $EIF$ and $CNB$ are similar.

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

Triangle $JHT$ has side lengths $JH = 14$, $HT = 10$, and $TJ = 16$. Points $I$ and $U$ lie on $\overline{JH}$ and $\overline{JT},$ respectively, so that $HI = TU = 1.$ Let $M$ and $N$ be the midpoints of $\overline{HT}$ and $\overline{IU},$ respectively. Line $MN$ intersects another side of $\triangle JHT$ at a point $P$ other than $M.$ Compute $MP^2.$

Let G be an abelian group, $0\leq\varepsilon<1$ and $f : G\to\Bbb R^n$ a function that satisfies the inequality. $$||f(x+y)-f(x)-f(y)|| \leq \varepsilon ||f (y)|| \qquad (x, y)\in G^2$$ Prove that there is an additive function $A : G\to \Bbb R^n$ and a continuous function $\varphi : A (G) \to\Bbb R^n$ such that $f = \varphi\circ A$.

The sum $$\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}$$ $ $ \\ can be expressed as a rational number $N$. Find the last 3 digits of $2021! \cdot N$.

A list of positive integers has the following properties: - The sum of the items in the list is $30$. - The unique mode of the list is $9$. - The median of the list is a positive integer that does not appear in the list itself. Find the sum of the squares of all the items in the list.

It is given a convex hexagon $A_1A_2 \cdots A_6$ such that all its interior angles are same valued (congruent). Denote by $a_1= \overline{A_1A_2},\ \ a_2=\overline{A_2A_3},\ \cdots , a_6=\overline{A_6A_1}.$ $a)$ Prove that holds: $ a_1-a_4=a_2-a_5=a_3-a_6 $ $b)$ Prove that if $a_1,a_2,a_3,...,a_6$ satisfy the above equation, we can construct a convex hexagon with its same-valued (congruent) interior angles.

Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\tfrac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. A player wins the game if their score is higher than the other player's score. The probability Dukejukem defeats the Trollner to win WASS equals $\tfrac{m}{n}$ where $m$ and $n$ are coprime positive integers. Computer $m+n$.

In the parliament of Nekurnekadzeme, all activities take place in commissions, which consist of exactly three members. The constitution stipulates that any three commissions must have at least five members. We will call a family of commissions a [i]clique[/i] if every two of them have exactly two members in common, but if any other commission is added to this family, this condition is no longer fulfilled. Prove that two different cliques cannot have more than one commission in common.

Determine, with proof, if there exists an odd prime $p$ such that the following equation holds: $$\sum_{n = 1}^{\frac{p-1}{2}} \cot\left(\frac{\pi n^2}{p}\right) = 69\sqrt{p}$$ [i]Proposed by Chris Bao[/i]

Let $S = \{1,..., 999\}$. Determine the smallest integer $m$. for which there exist $m$ two-sided cards $C_1$,..., $C_m$ with the following properties: $\bullet$ Every card $C_i$ has an integer from $S$ on one side and another integer from $S$ on the other side. $\bullet$ For all $x,y \in S$ with $x\ne y$, it is possible to select a card $C_i$ that shows $x$ on one of its sides and another card $C_j$ (with $i \ne j$) that shows $y$ on one of its sides.

A game consists of a grid of $4\times 4$ and tiles of two colors (Yellow and White). A player chooses a type of token and gives it to the second player who places it where he wants, then the second player chooses a type of token and gives it to the first who places it where he wants, They continue in this way and the one who manages to form a line with three tiles of the same color wins (horizontal, vertical or diagonal and regardless of whether it is the tile you started with or not). Before starting the game, two yellow and two white pieces are already placed as shows the figure below. [img]https://cdn.artofproblemsolving.com/attachments/b/5/ba11377252c278c4154a8c3257faf363430ef7.png[/img] Yolanda and Xinia play a game. If Yolanda starts (choosing the token and giving it to Xinia for this to place) indicate if there is a winning strategy for either of the two players and, if any, describe the strategy.

Let $ABC$ be a triangle, with $A'$, $B'$, and $C'$ the points of tangency of the incircle with $BC$, $CA$, and $AB$ respectively. Let $X$ be the intersection of the excircle with respect to $A$ with $AB$, and $M$ the midpoint of $BC$. Let $D$ be the intersection of $XM$ with $B'C'$. Show that $\angle C'A'D' = 90^o$.