On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set. [i]Proposed by Tonći Kokan, Croatia[/i]

The circles $\mathcal C_1$ and $\mathcal C_2$ touch each other externally at $D$, and touch a circle $\omega$ internally at $B$ and $C$, respectively. Let $A$ be an intersection point of $\omega$ and the common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $D$. Lines $AB$ and $AC$ meet $\mathcal C_1$ and $\mathcal C_2$ again at $K$ and $L$, respectively, and the line $BC$ meets $\mathcal C_1$ again at $M$ and $\mathcal C_2$ again at $N$. Prove that the lines $AD$, $KM$, $LN$ are concurrent. [i]Greece[/i]

In convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ meet at the point $P$. We know that $\angle DAC = 90^o$ and $2 \angle ADB = \angle ACB$. If we have $ \angle DBC + 2 \angle ADC = 180^o$ prove that $2AP = BP$. Proposed by Iman Maghsoudi

The figure below is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m}+\sqrt{n},$ where $m$ and $n$ are positive integers. What is $m+n?$ [asy] /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,S); label("D",D,S); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy] $\textbf{(A) }20 \qquad \textbf{(B) }21 \qquad \textbf{(C) }22\qquad \textbf{(D) }23 \qquad \textbf{(E) }24$ Proposed by [b]djmathman[/b]

Find all positive integers $ n$ such $ 20n\plus{}2$ can divide $ 2003n \plus{} 2002.$

Let $ a$ be a positive real number and $ n$ a non-negative integer. Determine $ S\minus{}T$, where $ S\equal{} \sum_{k\equal{}\minus{}2n}^{2n\plus{}1} \frac{(k\minus{}1)^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$ and $ T\equal{} \sum_{k\equal{}\minus{}2n}^{2n\plus{}1} \frac{k^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$

Find unequal integers $m, n$ such that $mn + n$ and $mn + m$ are both squares. Can you find such integers between $988$ and $1991$?

Prove that the product of 4 consecutive natural numbers cannot be a perfect cube.

Emily starts with an empty bucket. Every second, she either adds a stone to the bucket or removes a stone from the bucket, each with probability $\frac{1}{2}$. If she wants to remove a stone from the bucket and the bucket is currently empty, she merely does nothing for that second (still with probability $\hfill \frac{1}{2}$). What is the probability that after $2017$ seconds her bucket contains exactly $1337$ stones?

Do there exist any three relatively prime natural numbers so that the square of each of them is divisible by the sum of the two remaining numbers?

Prove that the solid angle based on a given closed contour is a function of the vertex of the angle that is harmonic outside the contour.

One of Euler's conjectures was disproved in the $1980$s by three American Mathematicians when they showed that there is a positive integer $n$ such that \[n^{5}= 133^{5}+110^{5}+84^{5}+27^{5}.\] Find the value of $n$.

In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"? $ \textbf{(A) } \frac{1}{22,050} \qquad \textbf{(B) } \frac{1}{21,000}\qquad \textbf{(C) } \frac{1}{10,500}\qquad \textbf{(D) } \frac{1}{2,100} \qquad \textbf{(E) } \frac{1}{1,050} $

On every side of a square $ABCD$, we consider three points different (to each other). a) Find the number of line segments defined with endpoints those points , that do not lie on sides of the square. b) If there are no three of the previous line segments passing through the same point, find how many of the intersection points of those segmens line in the interior of the square.

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and yellow when in the sun. Initially the ratio of green to yellow frogs was 3:1. Then 3 green frogs moved to the sunny side and 5 yellow frogs moved to the shady side. Now the ratio is 4:1. What is the difference between the number of green frogs and yellow frogs now? $\textbf{(A) } 10\qquad\textbf{(B) } 12\qquad\textbf{(C) } 16\qquad\textbf{(D) } 20\qquad\textbf{(E) } 24$

$ABCD$ is a regular tetrahedron of side length $4.$ Four congruent spheres are inside $ABCD$ such that each sphere is tangent to exactly three of the faces, the spheres have distinct centers, and the four spheres are concurrent at one point. Let $v$ be the volume of one of the spheres. If $v^2$ can be written as $\tfrac{a}{b}\pi^2,$ where $a$ and $b$ are relatively prime positive integers, find $a+b.$

Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers $a_i$.

Fix a real polynomial $P$ with degree at least $1$, and a real number $c$. Prove that there exist a real number $k$ such that for all reals $a$ and $b$, $$P(a)+P(b)=c \quad \Rightarrow \quad |a+b|<k$$ [i]Proposed by Wong Jer Ren[/i]

Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by $\$2$ from that of each adjacent book. For the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then: $ \textbf{(A)}\ \text{The adjacent book referred to is at the left of the middle book}$ $\qquad\textbf{(B)}\ \text{The middle book sells for \$36 }$ $\qquad\textbf{(C)}\ \text{The cheapest book sells for \$4 }$ $\qquad\textbf{(D)}\ \text{The most expensive book sells for \$64 }$ $\qquad\textbf{(E)}\ \text{None of these is correct } $

In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?

Let $$f(x) = \left|\left|\cdots\left|\left|\left|\left|x\right|-1\right|-2\right|-3\right|-\cdots \right|-10\right|.$$ Compute $f(1)+f(2)+\cdots+f(54)+f(55).$

A pack of MIT students are holding an escape room, where students may compete in teams of 4, 5, or 6. There is \$60 dollars worth of prize money in Amazon gift cards for the winning team. If each gift card can contain any whole number of dollars, what is the minimum number of gift cards required so that the prize money can be distributed evenly among any team? [i]Lightning 4.1[/i]

Let $ABC$ be a triangle, having no right angles, and let $D$ be a point on the side $BC$. Let $E$ and $F$ be the feet of the perpendiculars drawn from the point $D$ to the lines $AB$ and $AC$ respectively. Let $P$ be the point of intersection of the lines $BF$ and $CE$. Prove that the line $AP$ is the altitude of the triangle $ABC$ from the vertex $A$ if and only if the line $AD$ is the angle bisector of the angle $CAB$.

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?