Real numbers $b>a>0$ are given. Find the number $r$ in $[a,b]$ which minimizes the value of $\max\left\{\left|\frac{r-x}x\right||a\le x\le b\right\}$.

Andy, Bella, and Chris are playing in a trivia contest. Andy has $21,200$ points, Bella has $23,600$ points, and Chris has $11,200$ points. They have reached the final round, which works as follows: [list] [*] they are given a hint as to what the only question of the round will be about; [*] then, each of them must bet some amount of their points---this bet must be a nonnegative integer (a player does not know any of the other players' bets, and this bet cannot be changed later on); [*] then, they will be shown the question, where they will have $30$ seconds to individually submit a response (a player does not know any of the other players' answers); [*] finally, once time runs out, their responses and bets are revealed---if a player's response is correct, then the number of points they bet will be added to their score, otherwise, it will be subtracted from their score, and whoever ends up having the most points wins the contest (if there is a tie for the win, then the winner is randomly decided). [/list] Suppose that the contestants are currently deciding their bets based on the hint that the question will be about history. Bella knows that she will likely get the question wrong, but she also knows that Andy, who dislikes history, will definitely get it wrong. Knowing this, Bella wagers an amount that will guarantee her a win. Find the maximum number of points Bella could have ended up with.

For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.

Let $f$ be a function whose domain is $[1, 20]$ and whose range is a subset of $[-100, 100].$ Suppose $\tfrac{f(x)}{y} - \tfrac{f(y)}{x} \leq (x - y)^2$ for all $x$ and $y$ in $[1, 20].$ Compute the largest value of $f(x) - f(y)$ over all such functions $f$ and all $x$ and $y$ in the domain $[1, 20].$

Let $a,b,$ and $c$ be real numbers such that \begin{align*} a+b+c &= 2, \text{ and} \\ a^2+b^2+c^2&= 12 \end{align*} What is the difference between the maximum and minimum possible values of $c$? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ \frac{10}{3}\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ \frac{16}{3}\qquad\textbf{(E)}\ \frac{20}{3} $

The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$

The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$

For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.

$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with $(a)$ maximal area; $(b)$ minimal area?

A set $S$ consists of $k$ sequences of $0,1,2$ of length $n$. For any two sequences $(a_i),(b_i)\in S$ we can construct a new sequence $(c_i)$ such that $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$ and include it in $S$. Assume that after performing finitely many such operations we obtain all the $3n$ sequences of $0,1,2$ of length $n$. Find the least possible value of $k$.

Find the minimum value of \[\max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g)\] subject to the constraints (i) $a, b, c, d, e, f, g \geq 0,$ (ii)$ a + b + c + d + e + f + g = 1.$

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

For the triangles of fixed perimeter, find the maximum value of the product of the radius of the incircle with the radius of the excircle.

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers?

Let $x$ be a variable that can take any positive real value. For certain positive real constants $s$ and $t$, the value of $x^2 + \frac{s}{x}$ is minimized at $x = t$, and the value of $t^2\ln(2 + tx) + \frac{1}{x^2}$ is minimized at $x = s$. Compute the ordered pair $(s, t)$.

Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.

Let $a,b,c$ be reals and \[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\] Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$

For real numbers $x,$ let $T_x$ be the triangle with vertices $(5, 5^3),$ $(8, 8^3),$ and $(x, x^3)$ in $\mathbb{R}^2.$ Over all $x$ in the interval $[5, 8],$ the area of the triangle $T_x$ is maximized at $x = \sqrt{n},$ for some positive integer $n.$ Compute $n.$

Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$