A set \( A \) of integers is said to be \textit{admissible} if it satisfies the property: \[ \text{If } x, y \in A, \text{ then } x^2 + kxy + y^2 \in A \text{ for all } k \in \mathbb{Z}. \] Determine all pairs \( (m, n) \) of nonzero integers such that the only admissible set containing both \( m \) and \( n \) is the set of all integers.

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$. Find the locus of the centers of the circles circumscribed around the triangles $PXY$.

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

Paul is in the desert and has a pile of gypsum crystals. No matter how he divides the pile into two nonempty piles, at least one of the resulting piles has a number of crystals that, when written in base $10,$ has a sum of digits at least $7.$ Given that Paul’s initial pile has at least two crystals, compute the smallest possible number of crystals in the initial pile.

Let $s(n)$ be the number of positive integer divisors of $n$. Find the all positive values of $k$ that is providing $k=s(a)=s(b)=s(2a+3b)$.

For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

Let $\vartriangle ABC$ be an isosceles triangle with $\angle BAC = 100^o$. Let $D, E$ be points on ray $\overrightarrow{AB}$ so that $BC = AD = BE$. Show that $BC \cdot DE = BD \cdot CE$

Given $2012$ distinct points $A_1,A_2,\dots,A_{2012}$ on the Cartesian plane. For any permutation $B_1,B_2,\dots,B_{2012}$ of $A_1,A_2,\dots,A_{2012}$ define the [i]shadow[/i] of a point $P$ as follows: [i]Point $P$ is rotated by $180^{\circ}$ around $B_1$ resulting $P_1$, point $P_1$ is rotated by $180^{\circ}$ around $B_2$ resulting $P_2$, ..., point $P_{2011}$ is rotated by $180^{\circ}$ around $B_{2012}$ resulting $P_{2012}$. Then, $P_{2012}$ is called the shadow of $P$ with respect to the permutation $B_1,B_2,\dots,B_{2012}$.[/i] Let $N$ be the number of different shadows of $P$ up to all permutations of $A_1,A_2,\dots,A_{2012}$. Determine the maximum value of $N$. [i]Proposer: Hendrata Dharmawan[/i]

Let the inscribed circle $\omega$ of the triangle $ABC$ have a center $I{}$ and touch the sides $BC, CA$ and $AB$ at points $D, E$ and $F{}$ respectively. Let $M{}$ and $N{}$ be points on the straight line $EF$ such that $BM \parallel AC$ and $CN \parallel AB$. Let $P{}$ and $Q{}$ be points on the segments $DM{}$ and $DN{}$, respectively, such that $BP \parallel CQ$. Prove that the intersection point of the lines $PF$ and $QE$ lies on $\omega$. [i]Proposed by Don Luu (Vietnam)[/i]

You have $128$ teams in a single elimination tournament. The Engineers and the Crimson are two of these teams. Each of the $128$ teams in the tournament is equally strong, so during each match, each team has an equal probability of winning. Now, the $128$ teams are randomly put into the bracket. What is the probability that the Engineers play the Crimson sometime during the tournament?

If \[ \sum_{k=1}^{1000}\left( \frac{k+1}{k}+\frac{k}{k+1}\right)=\frac{m}{n} \] for relatively prime positive integers $m,n$, compute $m+n$. [i]2020 CCA Math Bonanza Lightning Round #2.4[/i]

Let the maximum and minimum value of $f(x)=\cos \left(x^{2018}\right) \sin x$ are $M$ and $m$ respectively where $x \in[-2 \pi, 2 \pi] .$ Then $$ M+m= $$ [list=1] [*] $\frac{1}{2}$ [*] $-\frac{1}{\sqrt{2}}$ [*] $\frac{1}{2018}$ [*] Does not exists [/list]

The values of $ k$ for which the equation $ 2x^2\minus{}kx\plus{}x\plus{}8\equal{}0$ will have real and equal roots are: $ \textbf{(A)}\ 9 \text{ and }\minus{}7 \qquad \textbf{(B)}\ \text{only }\minus{}7 \qquad \textbf{(C)}\ \text{9 and 7} \\ \textbf{(D)}\ \minus{}9 \text{ and }\minus{}7 \qquad \textbf{(E)}\ \text{only 9}$

We have six positive numbers $a_1, a_2, \ldots , a_6$ such that $a_1a_2\ldots a_6 =1$. Prove that: $$ \frac{1}{a_1(a_2 + 1)} + \frac{1}{a_2(a_3 + 1)} + \ldots + \frac{1}{a_6(a_1 + 1)} \geq 3.$$

The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\rm nine}$. What is the remainder when $N$ is divided by $5?$ $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$