Find all non-zero real numbers $a, b, c$ such that the following polynomial has four (not necessarily distinct) positive real roots. $$P(x) = ax^4 - 8ax^3 + bx^2 - 32cx + 16c$$

Let's consider the graph of a square trinomial having roots $1$ and $4$. Let's draw two tangents to it from point $O$ (the origin of coordinates), touching it at points $A$ and $B$. What values can the cosine of angle $\angle AOB$ take?

Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any $1011$ of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using [b]only[/b] a straightedge. Can Vikram always complete this challenge? [i]Note.[/i] A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points. [i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]

Find the value(s) of $ x$ such that $ 8xy\minus{}12y\plus{}2x\minus{}3\equal{}0$ is true for all values of $ y$. $ \textbf{(A)}\ \frac{2}{3} \qquad \textbf{(B)}\ \frac{3}{2}\text{ or }\minus{}\frac{1}{4} \qquad \textbf{(C)}\ \minus{}\frac{2}{3}\text{ or }\minus{}\frac{1}{4} \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \minus{}\frac{3}{2}\text{ or }\minus{}\frac{1}{4}$

Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle? [i]Ray Li.[/i]

Let $n$ be a positive integer. On a blackboard, Bobo writes a list of $n$ non-negative integers. He then performs a sequence of moves, each of which is as follows: -for each $i = 1, . . . , n$, he computes the number $a_i$ of integers currently on the board that are at most $i$, -he erases all integers on the board, -he writes on the board the numbers $a_1, a_2,\ldots , a_n$. For instance, if $n = 5$ and the numbers initially on the board are $0, 7, 2, 6, 2$, after the first move the numbers on the board will be $1, 3, 3, 3, 3$, after the second they will be $1, 1, 5, 5, 5$, and so on. (a) Show that, whatever $n$ and whatever the initial configuration, the numbers on the board will eventually not change any more. (b) As a function of $n$, determine the minimum integer $k$ such that, whatever the initial configuration, moves from the $k$-th onwards will not change the numbers written on the board.

A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number $s$ for which Bob can always pick $s$ members of the sequence, and guess all their values correctly. Alice and Bob can discuss a strategy before the game with the aim of maximizing the number of correct guesses of Bob. The only information Bob has is the length of the sequence and the member of the sequence picked by Alice.

An airport contains 25 terminals which are two on two connected by tunnels. There is exactly 50 main tunnels which can be traversed in the two directions, the others are with single direction. A group of four terminals is called [i]good[/i] if of each terminal of the four we can arrive to the 3 others by using only the tunnels connecting them. Find the maximum number of good groups.

Find all positive integers $m$ such that there exists integers $x$ and $y$ that satisfies $$m \mid x^2+11y^2+2022.$$

Let $D$ be a point on the circumcircle of some triangle $ABC$. Let $E, F$ be points on $AC$, $AB$, respectively, such that $A,D,E,F$ are concyclic. Let $M$ be the midpoint of $BC$. Show that if $DM$, $BE$, $CF$ are concurrent, then either $BE \cap CF$ is on the circle $ADEF$, or $EF$ is parallel to $BC$. [i]proposed by USJL[/i]

Let $n \ne 0$ be a natural number and integers $x_1, x_2, ...., x_n, y_1, y_2, ...., y_n$ with the properties: a) $x_1 + x_2 + .... + x_n = y_1 + y_2 + .... + y_n = 0,$ b) $x_1 ^ 2 + y_1 ^ 2 = x_2 ^ 2 + y_2 ^ 2 = .... = x_n ^ 2 + y_n ^ 2$. Show that $n$ is even.

There are integers $a$, $b$, and $c$, each greater than 1, such that $$\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}$$ for all $N > 1$. What is $b$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Simon is playing chess. He wins with probability 1/4, loses with probability 1/4, and draws with probability 1/2. What is the probability that, after Simon has played 5 games, he has won strictly more games than he has lost? [i]2016 CCA Math Bonanza Individual #7[/i]

For a table $n \times 9$ ($n$ rows and $9$ columns), determine the maximum of $n$ that we can write one number in the set $\left\{ {1,2,...,9} \right\}$ in each cell such that these conditions are satisfied: 1. Each row contains enough $9$ numbers of the set $\left\{ {1,2,...,9} \right\}$. 2. Any two rows are distinct. 3. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.

Let $\triangle ABC$ be a triangle with $AB = 10.$ Let $D$ be a point on the opposite side of line $AC$ as $B$ so that $\triangle ACD$ is directly similar to $\triangle ABC$ (i.e. $\angle ACD = \angle ABC,$ etc). Let $M$ be the midpoint of $AD.$ Given that $A$ is the centroid of triangle $\triangle BCM,$ compute $BC^2.$ .

A circle with center $I$ and radius $r$ is inscribed in a triangle $ABC$ with a right angle at $C$. Rays $AI$ and $CI$ meet the opposite sides at $D$ and $E$ respectively. Prove that $\frac{1}{AE}+\frac{1}{BD}=\frac{1}{r}$