Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$, $b\ge 2$, $c\ge 2$. Adam and Boris play the following game. They alternately take turns with Adam first. On Adam’s turn, he can choose one of the polynomial’s coefficients and replace it with the sum of the other two coefficients. On Boris’s turn, he can choose one of the polynomial’s coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$, $b$ and $c$, determine who has a winning strategy.

It is known that at least one coordinate of the center $(x_0, y_0)$ of the circle $(x -x_0)^2+ (y -y_0)^2 = R^2$ is irrational. Prove that on the circle itself there are at most two points with rational coordinates.

Find all real values of $a$ such that $x^4 - 2ax^2 + x + a^2 -a = 0$ has all its roots real.

Let $ABCD$ be an inscribed quadrilateral in a circumference with center $O$ such that it lies inside $ABCD$ and $\angle{BAC} = \angle{ODA}$. Let $E$ be the intersection of $AC$ with $BD$. Lines $r$ and $s$ are drawn through $E$ such that $r$ is perpendicular to $BC$, and $s$ is perpendicular to $AD$. Let $P$ be the intersection of $r$ with $AD$, and $M$ the intersection of $s$ with $BC$. Let $N$ be the midpoint of $EO$. Prove that $M$, $N$, and $P$ lie on a line.

When making a batch of $N \ge 5$ coins, a worker mistakenly made two coins from a different material (all coins look the same). The boss knows that there are exactly two such coins, that they weigh the same, but differ in weight from the others. The employee knows what coins these are and that they are lighter than others. He needs, after carrying out two weighings on cup scales without weights, to convince his boss that the coins are counterfeit easier than real ones, and in which coins are counterfeit. Can he do it?

In a paralleogram $ABCD$ , a point $P$ on the segment $AB$ is taken such that $\frac{AP}{AB}=\frac{61}{2022}$\\ and a point $Q$ on the segment $AD$ is taken such that $\frac{AQ}{AD}=\frac{61}{2065}$.If $PQ$ intersects $AC$ at $T$, find $\frac{AC}{AT}$ to the nearest integer

Let $P$ be a non-degenerate polygon with $n$ sides, where $n > 4$. Prove that there exist three distinct vertices $A, B, C$ of $P$ with the following property:If $\ell_1,\ell_2,\ell_3$ are the lengths of the three polygonal chains into which $A, B, C$ break the perimeter of $P$, then there is a triangle with side lengths $\ell_1,\ell_2$ and $\ell_3$. [size=150]Remark[/size]: By a non-degenerate polygon we mean a polygon in which every two sides are disjoint, apart from consecutive ones, which share only the common endpoint.(Poland)

A point object of mass $M$ hangs from the ceiling of a car from a massless string of length $L$. It is observed to make an angle $\theta$ from the vertical as the car accelerates uniformly from rest. Find the acceleration of the car in terms of $\theta$, $M$, $L$, and $g$. [asy] size(250); import graph; // Left draw((-3,0)--(-23,0),linewidth(1.5)); draw((-13,0)--(-13,-14)); filldraw(circle((-13,-15),2),gray); draw((-13,-15)--(-21,-15),dashed); draw((-21,-14)--(-21,-1),EndArrow(size=5)); draw((-21,-1)--(-21,-14),EndArrow(size=5)); label(scale(1.5)*"$L$",(-21,-7.5),2*E); // Right draw((3,0)--(23,0),linewidth(1.5)); draw((13,0)--(13,-19),dashed); draw((13,0)--(5,-12)); filldraw(circle((3.89,-13.66),2),gray); label(scale(1.5)*"$\theta$",(12,-9),1.5*W); real f(real x){ return 5x^2/12-95x/12+25; } draw(graph(f,12,7),Arrows); [/asy] (A) $Mg \sin \theta$ (B) $MgL \tan \theta$ (C) $g \tan \theta$ (D) $g \cot \theta$ (E) $Mg \tan \theta$

$n$ is called [i]diamond 2005[/i] if $n=\overline{...ab999...99999cd...}$, e.g. $2005 \times 9$. Let $\{a_n\}:a_n< C\cdot n,\{a_n\}$ is increasing. Prove that $\{a_n\}$ contain infinite [i]diamond 2005[/i]. Compare with [url=http://www.mathlinks.ro/Forum/topic-15091.html]this problem.[/url]

Each of $n$ boys and $n$ girls chooses a random number from the set $\{ 1, 2, 3, 4, 5 \}$, uniformly and independently. Let $p_n$ be the probability that every boy chooses a different number than every girl. As $n$ approaches infinity, what value does $\sqrt[n]{p_n}$ approach?

For each $ x$ in $ [0,1]$, define \[ f(x)=\begin{cases}2x, &\text { if } 0 \leq x \leq \frac {1}{2}; \\ 2 - 2x, &\text { if } \frac {1}{2} < x \leq 1. \end{cases} \]Let $ f^{[2]}(x) = f(f(x))$, and $ f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $ n \geq 2$. For how many values of $ x$ in $ [0,1]$ is $ f^{[2005]}(x) = \frac {1}{2}$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2005 \qquad \textbf{(C)}\ 4010 \qquad \textbf{(D)}\ 2005^2 \qquad \textbf{(E)}\ 2^{2005}$

A game works as follows: the player pays $2$ tokens to enter the game. Then, a fair coin is flipped. If the coin lands on heads, they receive $3$ tokens; if the coin lands on tails, they receive nothing. A player starts with $2$ tokens and keeps playing this game until they do not have enough tokens to play again. What is the expected value of the number of tokens they have left at the end? [i]2020 CCA Math Bonanza Team Round #9[/i]

Let $ n$ be a positive integer. Find all $ x_1,x_2,\ldots,x_n$ that satisfy the relation: \[ \sqrt{x_1\minus{}1}\plus{}2\cdot \sqrt{x_2\minus{}4}\plus{}3\cdot \sqrt{x_3\minus{}9}\plus{}\cdots\plus{}n\cdot\sqrt{x_n\minus{}n^2}\equal{}\frac{1}{2}(x_1\plus{}x_2\plus{}x_3\plus{}\cdots\plus{}x_n).\]

Let $n\in N$ $n\geq2$ and the set $X$ with $n+1$ elements. The ordered sequences $(a_{1}, a_{2},\ldots,a_{n})$ and $(b_{1},b_{2},\ldots b_{n})$ of distinct elements of $X$ are said to be $\textit{separated}$ if there exists $i\neq j$ such that $a_{i}=b_{j}$. Determine the maximal number of ordered sequences of $n$ elements from $X$ such that any two of them are $\textit{separated}$. Note: ordered means that, for example $(1,2,3)\neq(2,3,1)$.

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circle $S$ through $A,B$, and $O$ intersects $AC$ and $BC$ again at points $P$ and $Q$ respectively. Prove that $CO \perp PQ$.