Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawig the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies on between the parallel lines (excluding the intersection points on the lines) ? $ \textbf{(A)}\ 5600 \qquad\textbf{(B)}\ 5650 \qquad\textbf{(C)}\ 6500 \qquad\textbf{(D)}\ 6560 \qquad\textbf{(E)}\ 6600 $

Find all pairs $(p,q)$ such that the equation $$x^4+2px^2+qx+p^2-36=0$$ has exactly $4$ integer roots(counting multiplicity).

Find all integers $a$ for which $x^3 -x+a$ has three integer roots.

Source: 1976 Euclid Part A Problem 8 ----- Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is $\textbf{(A) } 1 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } -7 \qquad \textbf{(E) } \text{none of these}$

Acute scalene triangle $ABC$ has $G$ as its centroid and $O$ as its circumcenter. Let $H_a,\, H_b,\, H_c$ be the projections of $A,\, B,\, C$ on respective opposite sides and $D,\, E,\, F$ be the midpoints of $BC,\, CA,\, AB$ in that order. $\overrightarrow{GH_a},\, \overrightarrow{GH_b},\, \overrightarrow{GH_c}$ intersect $(O)$ at $X,\,Y,\,Z$ respectively. a. Prove that the circle $(XCE)$ pass through the midpoint of $BH_a$ b. Let $M,\, N,\, P$ be the midpoints of $AX,\, BY,\, CZ$ respectively. Prove that $\overleftrightarrow{DM},\, \overleftrightarrow{EN},\,\overleftrightarrow{FP}$ are concurrent.

Compute the remainder when $98!$ is divided by $101$.

Suppose that $m$ and $n$ are integers, such that both the quadratic equations $x^2+mx-n=0$ and $x^2-mx+n=0$ have integer roots. Prove that $n$ is divisible by $6$.

A league consists of $2024$ players. A [i]round[/i] involves splitting the players into two different teams and having every member of one team play with every member of the other team. A round is called [i]balanced[/i] if both teams have an equal number of players. A tournament consists of several rounds at the end of which any two players have played each other. The committee organised a tournament last year which consisted of $N$ rounds. Prove that the committee can organise a tournament this year with $N$ balanced rounds. [i]Proposed by Anant Mudgal and Navilarekallu Tejaswi[/i]

Let $p_1, p_2, ..., p_n$ be positive real numbers, such that $p_1 + p_2 +... + p_n = 1$. Let $x \in [0,1]$ and let $y_1, y_2, ..., y_n$ be such that $y^2_1 + y^2_2 +...+ y^2_n= x$. Prove that $$\left( \sum_{nx\le k \le n }y_k \sqrt{p_k} \right)^2 \le \sum_{k=1}^{n}\frac{k}{n} p_k$$

On a closed track, clockwise, there are five boxes $A, B, C, D$ and $E$, and the length of the track section between boxes $A$ and $B$ is $1$ km, between $B$ and $C$ - $5$ km, between $C$ and $D$ - $2$ km, between $D$ and $E$ - $10$ km, and between $E$ and $A$ - $3$ km. On the track, they drive in a clockwise direction, the race always begins and ends in the box. What box did you start from if the length of the race was exactly $1998$ km?

The product \[\prod^{63}_{k=4} \frac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = \frac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot \frac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot \frac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots \frac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\] is equal to $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$

Jerry's favorite number is $97$. He knows all kinds of interesting facts about $97$: [list][*]$97$ is the largest two-digit prime. [*]Reversing the order of its digits results in another prime. [*]There is only one way in which $97$ can be written as a difference of two perfect squares. [*]There is only one way in which $97$ can be written as a sum of two perfect squares. [*]$\tfrac1{97}$ has exactly $96$ digits in the [smallest] repeating block of its decimal expansion. [*]Jerry blames the sock gnomes for the theft of exactly $97$ of his socks.[/list] A repunit is a natural number whose digits are all $1$. For instance, \begin{align*}&1,\\&11,\\&111,\\&1111,\\&\vdots\end{align*} are the four smallest repunits. How many digits are there in the smallest repunit that is divisible by $97?$

Find all natural $n{}$ such that for every natural $a{}$ that is mutually prime with $n{}$, the number $a^n - 1$ is divisible by $2n^2$.

Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$. Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.

Find all polynomials $f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n}$ with the following properties (a) all the coefficients $a_{1}, a_{2}, ..., a_{n}$ belong to the set $\{ -1, 1 \}$; and (b) all the roots of the equation $f(x)=0$ are real.

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $cn\log(n)$ pairwise coprime $n$th powers. [i]Canada[/i]

Find all positive integers $x, y$ and primes $p$, such that $x^5+y^4=pxy$.

Prove that exprssion $A=\frac{25}{2}(n+2-\sqrt{2n+3})$, $(n\in\mathbb{N})$ is a perfect square of an integer if exprssion $A$ is an integer .

Given vertex $A$ and $A$-excircle $\omega_A$ . Construct all possible triangles such that circumcenter of $\triangle ABC$ coincide with centroid of the triangle formed by tangent points of $\omega_A$ and triangle sides. [i]Proposed by Seyed Reza Hosseini Dolatabadi, Pooya Esmaeil Akhondy[/i] [b]Rated 4[/b]

Write $2013$ as a sum of $m$ prime numbers. The smallest value of $m$ is: (A): $2$, (B): $3$, (C): $4$, (D): $1$, (E): None of the above.

[u]General Round[/u] [b]p1.[/b] Alice can bake a pie in $5$ minutes. Bob can bake a pie in $6$ minutes. Compute how many more pies Alice can bake than Bob in $60$ minutes. [b]p2.[/b] Ben likes long bike rides. On one ride, he goes biking for six hours. For the first hour, he bikes at a speed of $15$ miles per hour. For the next two hours, he bikes at a speed of $12$ miles per hour. He remembers biking $90$ miles over the six hours. Compute the average speed, in miles per hour, Ben biked during the last three hours of his trip. [b]p3.[/b] Compute the perimeter of a square with area $36$. [b]p4.[/b] Two ants are standing side-by-side. One ant, which is $4$ inches tall, casts a shadow that is $10$ inches long. The other ant is $6$ inches tall. Compute, in inches, the length of the shadow that the taller ant casts. [b]p5.[/b] Compute the number of distinct line segments that can be drawn inside a square such that the endpoints of the segment are on the square and the segment divides the square into two congruent triangles. [b]p6.[/b] Emily has a cylindrical water bottle that can hold $1000\pi$ cubic centimeters of water. Right now, the bottle is holding $100\pi$ cubic centimeters of water, and the height of the water is $1$ centimeter. Compute the radius of the water bottle. [b]p7.[/b] Given that $x$ and $y$ are nonnegative integers, compute the number of pairs $(x, y)$ such that $5x + y = 20$. [b]p8.[/b] A sequence an is recursively defined where $a_n = 3(a_{n-1}-1000)$ for $n > 0$. Compute the smallest integer $x$ such that when $a_0 = x$, $a_n > a_0$ for all $n > 0$. [b]p9.[/b] Compute the probability that two random integers, independently chosen and both taking on an integer value between $1$ and $10$ with equal probability, have a prime product. [b]p10.[/b] If $x$ and $y$ are nonnegative integers, both less than or equal to $2$, then we say that $(x, y)$ is a friendly point. Compute the number of unordered triples of friendly points that form triangles with positive area. [b]p11.[/b] Cindy is thinking of a number which is $4$ less than the square of a positive integer. The number has the property that it has two $2$-digit prime factors. What is the smallest possible value of Cindy's number? [b]p12.[/b] Winona can purchase a pencil and two pens for $250$ cents, or two pencils and three pens for $425$ cents. If the cost of a pencil and the cost of a pen does not change, compute the cost in cents of five pencils and six pens. [b]p13.[/b] Colin has an app on his phone that generates a random integer betwen $1$ and $10$. He generates $10$ random numbers and computes the sum. Compute the number of distinct possible sums Colin can end up with. [b]p14.[/b] A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle. [b]p15.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result. [b]p16.[/b] A unit square is subdivided into a grid composed of $9$ squares each with sidelength $\frac13$ . A circle is drawn through the centers of the $4$ squares in the outermost corners of the grid. Compute the area of this circle. [b]p17.[/b] There exists exactly one positive value of $k$ such that the line $y = kx$ intersects the parabola $y = x^2 + x + 4$ at exactly one point. Compute the intersection point. [b]p18.[/b] Given an integer $x$, let $f(x)$ be the sum of the digits of $x$. Compute the number of positive integers less than $1000$ where $f(x) = 2$. [b]p19.[/b] Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let $BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$? [b]p20.[/b] Compute all real solutions to $16^x + 4^{x+1} - 96 = 0$. [b]p21.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red? [b]p22.[/b] $ABCDEFGH$ is an equiangular octagon with side lengths $2$, $4\sqrt2$, $1$, $3\sqrt2$, $2$, $3\sqrt2$, $3$, and $2\sqrt2$,in that order. Compute the area of the octagon. [b]p23.[/b] The cubic $f(x) = x^3 +bx^2 +cx+d$ satisfies $f(1) = 3$, $f(2) = 6$, and $f(4) = 12$. Compute $f(3)$. [b]p24.[/b] Given a unit square, two points are chosen uniformly at random within the square. Compute the probability that the line segment connecting those two points touches both diagonals of the square. [b]p25.[/b] Compute the remainder when: $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$. [u]General Tiebreaker [/u] [b]Tie 1.[/b] Trapezoid $ABCD$ has $AB$ parallel to $CD$, with $\angle ADC = 90^o$. Given that $AD = 5$, $BC = 13$ and $DC = 18$, compute the area of the trapezoid. [b]Tie 2.[/b] The cubic $f(x) = x^3- 7x - 6$ has three distinct roots, $a$, $b$, and $c$. Compute $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ . [b]Tie 3.[/b] Ben flips a fair coin repeatedly. Given that Ben's first coin flip is heads, compute the probability Ben flips two heads in a row before Ben flips two tails in a row. PS. You should use hide for answers.

Let $\triangle ABC$ be an acute scalene triangle. Denote by $k_A$ the circle with diameter $BC$, and let $B_A,C_A$ be the contact points of the tangents from $A$ to $k_A$, chosen so that $B$ and $B_A$ lie on opposite sides of $AC$ and $C$ and $C_A$ lie on opposite sides of $AB$. Similarly, let $k_B$ be the circle with diameter $CA$, with tangents from $B$ touching at $C_B,A_B$, and $k_C$ the circle with diameter $AB$, with tangents from $C$ touching at $A_C,B_C$. Prove that the lines $B_AC_A, C_BA_B, A_CB_C$ are concurrent.

Evaluate $ \int_0^1 \frac{2e^{2x}\plus{}xe^x\plus{}3e^x\plus{}1}{(e^x\plus{}1)^2(e^x\plus{}x\plus{}1)^2}\ dx$.