A $20 \times 20$ rectangular grid has been given. It is known that one of the grid's unit squares contains a hidden treasure. To find the treasure, we have been given an opportunity to order several scientific studies at the same time, results of which will be known only after some time. For each study we must choose one $1 \times 4$ rectangle, and the study will tell whether the rectangle contains the treasure. The $1 \times 4$ rectangle can be either horizontal or vertical, and it can extend over a side of the $20 \times 20$ grid, coming back in at the opposite side (you can think of the $20 \times 20$ grid as a torus - the opposite sides are connected). What is the minimal amount of studies that have to ordered for us to precisely determine the unit square containing the treasure?

Let $\gamma_1$ and $\gamma_2$ be circles centered at $O$ and $ P$ respectively, and externally tangent to each other at point $Q$. Draw point $D$ on $\gamma_1$ and point $E$ on $\gamma_2$ such that line $DE$ is tangent to both circles. If the length $OQ = 1$ and the area of the quadrilateral $ODEP$ is $520$, then what is the value of length $PQ$?

A permutation $ \pi$ is selected randomly through all $ n$-permutations. a) if \[ C_a(\pi)\equal{}\mbox{the number of cycles of length }a\mbox{ in }\pi\] then prove that $ E(C_a(\pi))\equal{}\frac1a$ b) Prove that if $ \{a_1,a_2,\dots,a_k\}\subset\{1,2,\dots,n\}$ the probability that $ \pi$ does not have any cycle with lengths $ a_1,\dots,a_k$ is at most $ \frac1{\sum_{i\equal{}1}^ka_i}$

The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

A number is [i]capicua [/i] if it is read equally from left to right as it is from right to the left. For example, $23432$ and $111111$ are capicua numbers. (a) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2022$ equal digits? (b) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2021$ equal digits?

We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there? [i]Proposed by Nathan Xiong[/i]

A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n$$

Five distinct points $A,B,C,D$ and $E$ lie on a line with $|AB|=|BC|=|CD|=|DE|$. The point $F$ lies outside the line. Let $G$ be the circumcentre of the triangle $ADF$ and $H$ the circumcentre of the triangle $BEF$. Show that the lines $GH$ and $FC$ are perpendicular.

Let $k$ be a nonnegative integer. Evaluate \[ \sum_{j=0}^k 2^{k-j} \binom{k+j}{j}. \]

Find the highest point (largest possible $y$-coordinate) on the parabola \[y=-2x^2+28x+418.\]

In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is $m/n$ square miles, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Point $G$ is where the medians of the triangle $ABC$ intersect and point $D$ is the midpoint of side $BC$. The triangle $BDG$ is equilateral with side length 1. Determine the lengths, $AB$, $BC$, and $CA$, of the sides of triangle $ABC$. [asy] size(200); defaultpen(fontsize(10)); real r=100.8933946; pair A=sqrt(7)*dir(r), B=origin, C=(2,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C); draw(A--B--C--A--D^^B--E^^C--F); pair point=G; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$G$", G, dir(20)); label("1", B--G, dir(150)); label("1", D--G, dir(30)); label("1", B--D, dir(270));[/asy]

At the 2013 Winnebago County Fair a vendor is offering a ``fair special" on sandals. If you buy one pair of sandals at the regular price of \$50, you get a second pair at a 40\% discount, and a third pair at half the regular price. Javier took advantage of the ``fair special" to buy three pairs of sandals. What percentage of the \$150 regular price did he save? $\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qquad \textbf{(D)}\ 40 \qquad \textbf{(E)}\ 45$

$27$ students are given a number from $1$ to $27.$ How many ways are there to divide $27$ students into $9$ groups of $3$ with the following condition? (i) The sum of students number in each group is $1\pmod{3}$ (ii) There are no such two students where their numbering differs by $3.$

What is the maximum number of integers that can be chosen from $1,2,\dots,99$ so that the chosen integers can be arranged in a circle with the property that the product of every pair of neighbouring integers is 3-digit number?

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]

Let's call several numbers written in a row a 'combination of numbers'. In the country of Robotland, some combinations of numbers have been declared prohibited. It is known that there are a finite number of forbidden combinations and there is an infinite decimal fraction that does not contain forbidden combinations. Prove that there is an infinite periodic decimal fraction that does not contain prohibited combinations.

The chord $MN$ on the circle is fixed. For every diameter $AB$ of the circle consider the intersection point $C$ of the lines $AM$ and $BN$ and construct the line $\ell$ passing through $C$ perpendicularly to $AB$. Prove that all the lines $\ell$ pass through a fixed point. (E. Kulanin, Moscow)

Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$. Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.

The coefficient of $x^7$ in the expansion of $(\frac{x^2}{2}-\frac{2}{x})^8$ is: $ \textbf{(A)}\ 56\qquad\textbf{(B)}\ -56\qquad\textbf{(C)}\ 14\qquad\textbf{(D)}\ -14\qquad\textbf{(E)}\ 0 $

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that \[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \] [i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$. Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$. [i]Proposed by Dirk Laurie, South Africa[/i]

Let $a,b,c,d,e,f$ be six real numbers with sum 10, such that \[ (a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6. \] Find the maximum possible value of $f$. [i]Cyprus[/i]

Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds: \[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\] where $S=\sum_{i=1}^n x_i$.

Two beasts, Rosencrans and Gildenstern, play a game. They have a circle with $n$ points ($n \ge 5$) on it. On their turn, each beast (starting with Rosencrans) draws a chord between a pair of points in such a way that any two chords have a shared point. (The chords either intersect or have a common endpoint.) For example, two potential legal moves for the second player are drawn below with dotted lines. [asy] unitsize(0.7cm); draw(circle((0,0),1)); dot((0,-1)); pair A = (-1/2,-(sqrt(3))/2); dot(A); pair B = ((sqrt(2))/2,-(sqrt(2))/2); dot(B); pair C = ((sqrt(3))/2,1/2); dot(C); draw(A--C); pair D = (-(sqrt(0.05)),sqrt(0.95)); dot(D); pair E = (-(sqrt(0.2)),sqrt(0.8)); dot(E); draw(B--E,dotted); draw(C--D,dotted); [/asy] The game ends when a player cannot draw a chord. The last beast to draw a chord wins. For which $n$ does Rosencrans win?