Laura and Daniel play with quadratic polynomials. First Laura says a nonzero real number $r$. Then Daniel says a nonzero real number $s$, and then again Laura says another nonzero real number $t$. Finally. Daniel writes the polynomial $P(x) = ax^2 + bx + c$ where $a,b$, and $c$ are $r,s$, and $t$ in some order Daniel chooses. Laura wins if the equation $P(x) = 0$ has two different real solutions, and Daniel wins otherwise. Determine who has a winning strategy and describe that strategy.

Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$. [i]A. Golovanov[/i]

If $p,q\in\mathbb{R}[x]$ satisfy $p(p(x))=q(x)^2$, does it follow that $p(x)=r(x)^2$ for some $r\in\mathbb{R}[x]$?

Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set $A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$ $B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$ Prove that $2|A|\geq |B|$.

[i]Round 5[/i] [b]p13.[/b] Kelvin Amphibian, a not-frog who lives on the coordinate plane, likes jumping around. Each step, he jumps either to the spot that is $1$ unit to the right and 2 units up, or the spot that is $2$ units to the right and $1$ unit up, from his current location. He chooses randomly among these two choices with equal probability. He starts at the origin and jumps for a long time. What is the probability that he lands on $(10, 8)$ at some time in his journey? [b]p14.[/b] Points $A, B, C$, and $D$ are randomly chosen on the circumference of a unit circle. What is the probability that line segments $AB$ and $CD$ intersect inside the circle? [b]p15.[/b] Let $P(x)$ be a quadratic polynomial with two consecutive integer roots. If it is also known that $\frac{P(2017)} {P(2016)} = \frac{2016}{2017}$ , find the larger root of $P(x)$. [u]Round 6[/u] [b]p16.[/b] Let $S_n$ be the sum of reciprocals of the integers between $1$ and $n$ inclusive. Find a triple $(a, b, c)$ of positive integers such that $S_{2017} \cdot S_{2017} - S_{2016} \cdot S_{2018} = \frac{S_a+S_b}{c}$ . [b]p17.[/b] Suppose that $m$ and $n$ are both positive integers. Alec has $m$ standard $6$-sided dice, each labelled $1$ to $6$ inclusive on the sides, while James has $n$ standard $12$-sided dice, each labelled $1$ to $12$ inclusive on the sides. They decide to play a game with their dice. They each toss all their dice simultaneously and then compute the sum of the numbers that come up on their dice. Whoever has a higher sum wins (if the sums are equal, they tie). Given that both players have an equal chance of winning, determine the minimum possible value of mn. [b]p18.[/b] Overlapping rectangles $ABCD$ and $BEDF$ are congruent to each other and both have area $1$. Given that $A,C,E, F$ are the vertices of a square, find the area of the square. [u]Round 7[/u] [b]p19.[/b] Find the number of solutions to the equation $$||| ... |||||x| + 1| - 2| + 3| - 4| +... - 98| + 99| - 100| = 0$$ [b]p20.[/b] A split of a positive integer in base $10$ is the separation of the integer into two nonnegative integers, allowing leading zeroes. For example, $2017$ can be split into $2$ and $017$ (or $17$), $20$ and $17$, or $201$ and $7$. A split is called squarish if both integers are nonzero perfect squares. $49$ and $169$ are the two smallest perfect squares that have a squarish split ($4$ and $9$, $16$ and $9$ respectively). Determine all other perfect squares less than $2017$ with at least one squarish split. [b]p21.[/b] Polynomial $f(x) = 2x^3 + 7x^2 - 3x + 5$ has zeroes $a, b$ and $c$. Cubic polynomial $g(x)$ with $x^3$-coefficient $1$ has zeroes $a^2$, $b^2$ and $c2$. Find the sum of coefficients of $g(x)$. [u]Round 8[/u] [b]p22.[/b] Two congruent circles, $\omega_1$ and $\omega_2$, intersect at points $A$ and $B$. The centers of $\omega_1$ and $\omega_2$ are $O_1$ and $O_2$ respectively. The arc $AB$ of $\omega_1$ that lies inside $\omega_2$ is trisected by points $P$ and $Q$, with the points lying in the order $A, P, Q,B$. Similarly, the arc $AB$ of $\omega_2$ that lies inside $\omega_1$ is trisected by points $R$ and $S$, with the points lying in the order $A,R, S,B$. Given that $PQ = 1$ and $PR =\sqrt2$, find the measure of $\angle AO_1B$ in degrees. [b]p23.[/b] How many ordered triples of $(a, b, c)$ of integers between $-10$ and $10$ inclusive satisfy the equation $-abc = (a + b)(b + c)(c + a)$? [b]p24.[/b] For positive integers $n$ and $b$ where $b > 1$, define $s_b(n)$ as the sum of digits in the base-$b$ representation of $n$. A positive integer $p$ is said to dominate another positive integer $q$ if for all positive integers $n$, $s_p(n)$ is greater than or equal to $s_q(n)$. Find the number of ordered pairs $(p, q)$ of distinct positive integers between $2$ and $100$ inclusive such that $p$ dominates $q$. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2936487p26278546]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$ Prove that there exists a polynomial $F(t)$ in one variable such that $$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$

Two polynomials $f(x)$ and $g(x)$ with real coefficients are called similar if there exist nonzero real number a such that $f(x) = q \cdot g(x)$ for all $x \in R$. [b]I.[/b] Show that there exists a polynomial $P(x)$ of degree 1999 with real coefficients which satisfies the condition: $(P(x))^2 - 4$ and $(P'(x))^2 \cdot (x^2-4)$ are similar. [b]II.[/b] How many polynomials of degree 1999 are there which have above mentioned property.

Prove that there exists a polynomial $P(x)$ with real coefficients and degree greater than 1 such that both of the following conditions are true $\cdot$ $P(a)+P(b)+P(c)\ge 2021$ holds for all nonnegative real numbers $a,b,c$ such that $a+b+c=3$ $\cdot$ There are infinitely many triples $(a,b,c)$ of nonnegative real numbers such that $a+b+c=3$ and $P(a)+P(b)+P(c)= 2021$

Suppose that $\omega$ is a primitive $2007^{\text{th}}$ root of unity. Find $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

Call a polynomial with real roots [i]n-local[/i] if the greatest difference between any pair of its roots is $n$. Let $f(x)=x^2+ax+b$ be a 1-[i]local[/i] polynomial with distinct roots such that $a$ and $b$ are non-zero integers. If $f(f(x))$ is a 23-[i]local[/i] polynomial, find the sum of the roots of $f(x)$. [i]Proposed by Anthony Yang[/i]

Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k \plus{} 1\}$, then $ F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1)$.

Fix an integer $n \geq 2$ and let $a_1, \ldots, a_n$ be integers, where $a_1 = 1$. Let $$ f(x) = \sum_{m=1}^n a_mm^x. $$ Suppose that $f(x) = 0$ for some $K$ consecutive positive integer values of $x$. In terms of $n$, determine the maximum possible value of $K$.

Prove the following inequality: \[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1} x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in \mathbb{N}.$

Determine all positive integers $n$ for which there exists a polynomial $P_n(x)$ of degree $n$ with integer coefficients that is equal to $n$ at $n$ different integer points and that equals zero at zero.