With $2018$ points, a network composed of hexagons is formed as a sample the figure: [asy] unitsize(1 cm); int i; path hex = dir(30)--(0,1)--dir(150)--dir(210)--(0,-1)--dir(330)--cycle; draw(hex); draw(shift((sqrt(3),0))*(hex)); draw(shift((2*sqrt(3),0))*(hex)); draw(shift((4*sqrt(3),0))*(hex)); draw(shift((5*sqrt(3),0))*(hex)); dot((3*sqrt(3) - 0.3,0)); dot((3*sqrt(3),0)); dot((3*sqrt(3) + 0.3,0)); dot(dir(150)); dot(dir(210)); for (i = 0; i <= 5; ++i) { if (i != 3) { dot((0,1) + i*(sqrt(3),0)); dot(dir(30) + i*(sqrt(3),0)); dot(dir(330) + i*(sqrt(3),0)); dot((0,-1) + i*(sqrt(3),0)); } } dot(dir(150) + 4*(sqrt(3),0)); dot(dir(210) + 4*(sqrt(3),0)); [/asy] A bee and a fly play the following game: initially the bee chooses one of the $2018$ dots and paints it red, then the fly chooses one of the $2017$ unpainted dots and paint it blue. Then the bee chooses an unpainted point and paints it red and then the fly chooses an unpainted one and paints it blue and so they alternate. If at the end of the game there is an equilateral triangle with red vertices, the bee wins, otherwise Otherwise the fly wins. Determine which of the two insects has a winning strategy.

A table tennis tournament has $101$ contestants, where each pair of contestants will play each other exactly once. In each match, the player who gets $11$ points first is the winner, and the other the loser. At the end of the tournament, it turns out that there exist matches with scores $11$ to $0$ and $11$ to $10$. Show that there exists 3 contestants $A,B,C$ such that the score of the losers in the matches between $A,B$ and $A,C$ are equal, but different from the score of the loser in the match between $B,C$.

Prove that if $ n\geq 4$, $ n\in\mathbb Z$ and $ \left \lfloor \frac {2^n}{n} \right\rfloor$ is a power of 2, then $ n$ is also a power of 2.

Non-zero real numbers $a$, $b$ and $c$ are such that any two of the three equations $ax^{11}+bx^4+c=0$, $bx^{11}+cx^4+a=0$, $cx^{11}+ax^4+b=0$ have a common root. Prove that all three equations have a common root. (Author: I. Bogdanov)

Evaluate the integral $$\int^1_0\left(\sqrt{(x-1)^3+1}+x^{2/3}-(1-x)^{3/2}-\sqrt[3]{1-x^2}\right)dx$$

If $ a,b,c$ are three positive real numbers, prove that $ \frac {a^{2}\plus{}1}{b\plus{}c}\plus{}\frac {b^{2}\plus{}1}{c\plus{}a}\plus{}\frac {c^{2}\plus{}1}{a\plus{}b}\ge 3$

Prove that the following statement is true for two natural nos. $m,n$ if and only $v(m) = v(n)$ where $v(k)$ is the highest power of $2$ dividing $k$. $\exists$ a set $A$ of positive integers such that $(i)$ $x,y \in \mathbb{N}, |x-y| = m \implies x \in A $ or $y \in A$ $(ii)$ $x,y \in \mathbb{N}, |x-y| = n \implies x \not\in A $ or $y \not\in A$

Triangle $ABC$ has side-lengths $AB=12$, $BC=24$, and $AC=18$. The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. What is the perimeter of $\triangle AMN$? $ \textbf{(A)}\ 27 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 42 $

Prove that there exist infinitely many positive integers $d$ such that we can find a polynomial $P\in\mathbb{Z}[x]$ of degree $d$ and $N\in\mathbb{N}$ such that for all integers $x>N$ and any prime $p$, we have $$\nu_p(P(x)^3+3P(x)^2-3)<\frac{d\cdot\log(x)}{2024^{2024}}.$$ [i]Proposed by Marius Cerlat[/i]

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j<k$, $a_j\neq a_k$. (So $1\in S$; $2\in S$ if and only if $a_1\neq a_2$; $3\in S$ if and only if $a_1\neq a_3$ and $a_2\neq a_3$; and so on.) In simplest form, let $\dfrac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.

Given $ \triangle{ABC}$, where $ A$ is at $ (0,0)$, $ B$ is at $ (20,0)$, and $ C$ is on the positive $ y$-axis. Cone $ M$ is formed when $ \triangle{ABC}$ is rotated about the $ x$-axis, and cone $ N$ is formed when $ \triangle{ABC}$ is rotated about the $ y$-axis. If the volume of cone $ M$ minus the volume of cone $ N$ is $ 140\pi$, find the length of $ \overline{BC}$.

A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?

For which real values of $m$ does the equation $x^2-\frac{m^2+1}{m -1}x+2m+2=0$ has root $x=-1$?

Let $ABC$ be an acute scalene triangle with incenter $I$. Show that the circumcircle of $BIC$ intersects the Euler line of $ABC$ in two distinct points. (Recall that the [i]Euler line[/i] of a scalene triangle is the line that passes through its circumcenter, centroid, orthocenter, and the nine-point center.) [i]Andrew Gu[/i]

Given that $x, y$, and $z$ are a combination of positive integers such that $xyz = 2(x + y + z)$, compute the sum of all possible values of $x + y + z$.

The lines $a$ and $b$ are parallel and the point $A$ lies on $a$. One chooses one circle $\gamma$ through A tangent to $b$ at $B$. $a$ intersects $\gamma$ for the second time at $T$. The tangent line at $T$ of $\gamma$ is called $t$. Prove that independently of the choice of $\gamma$, there is a fixed point $P$ such that $BT$ passes through $P$. Prove that independently of the choice of $\gamma$, there is a fixed circle $\delta$ such that $t$ is tangent to $\delta$.

Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.) [i]David Yang.[/i]

Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \le 1\}$.

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

We inscribe a circle $\omega$ in equilateral triangle $ABC$ with radius $1$. What is the area of the region inside the triangle but outside the circle?

Consider the complex numbers $a_1, a_2,\cdots , a_n$, $n \geq 2$. Which have the following properties: [list] [*] $|a_i|=1$ $\forall$ $i=1,2,\cdots , n$ [*] $\sum \limits_{k=1}^n arg(a_k)\leq \pi$ [/list] Show that the inequality$$\left(n^2\cot \left(\frac{\pi}{2n}\right)\right)^{-1}\left |\sum \limits_{k=0}^n(-1)^k\left[3n^2-(8k+5)n+4k(k+1)\sigma_k\right]\right |\geq \sqrt{\left(1+\frac{1}{n}\right)^2\cot^2 \left(\frac{\pi}{2n}\right)}+16\left |\sum \limits_{k=0}^n(-1)^k\sigma_k\right |$$where $\sigma_0=1$, $\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}$, $\forall$ $k=1,2,\cdots , n$

Alf, the alien from the $1980$s TV show, has a big appetite for the mineral apatite. However, he’s currently on a diet, so for each integer $k \ge 1,$ he can eat exactly $k$ pieces of apatite on day $k.$ Additionally, if he eats apatite on day $k,$ he cannot eat on any of days $k + 1, k + 2, \ldots, 2k - 1.$ Compute the maximum total number of pieces of apatite Alf could eat over days $1,2, \ldots,99,100.$

What is the maximal number of solutions can the equation have $$\max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0$$ where $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$ are real numbers, all $a_i$ not equal to $0$.

Consider a $3 \times 3$ grid of squares. A circle is inscribed in the lower left corner, the middle square of the top row, and the rightmost square of the middle row, and a circle $O$ with radius $r$ is drawn such that $O$ is externally tangent to each of the three inscribed circles. If the side length of each square is 1, compute $r$.

There are $2^n$ words of length $n$ over the alphabet $\{0, 1\}$. Prove that the following algorithm generates the sequence $w_0, w_1, \ldots, w_{2^n-1}$ of all these words such that any two consecutive words differ in exactly one digit. (1) $w_0 = 00 \ldots 0$ ($n$ zeros). (2) Suppose $w_{m-1} = a_1a_2 \ldots a_n,\quad a_i \in \{0, 1\}$. Let $e(m)$ be the exponent of $2$ in the representation of $n$ as a product of primes, and let $j = 1 + e(m)$. Replace the digit $a_j$ in the word $w_{m-1}$ by $1 - a_j$. The obtained word is $w_m$.