A string of letters is called $good$ if it contains a continuous substring $IMONST$ in it. For example, the string $NSIMONSTIM$ is $good$, but the string $IMONNNST$ is not. Find the number of good strings consisting of $12$ letters from $I$, $M$, $O$, $N$, $S$, $T$ only.

Jia Herng has a circle $\omega$ with center $O$, and $P$ is a point outside of $\omega$. Let $PX$ and $PY$ are two lines tangent to $\omega$ at $X$ and $Y$ , and $Q$ is a point on segment $PX$. Let $R$ is a point on the ray $PY$ beyond $Y$ such that $QX = RY$. Help Jia Herng prove that the points $O$, $P$, $Q$, $R$ are concyclic.

Janson wants to find a sequence of positive integers $a_{1}, a_{2}, . . . , a_{2024}$ such that each term is at least $10$, and $a_{i}$ has exactly $a_{i+1}$ divisors for all $1 \leq i \leq 2023$. Can you help him find one such sequence, or is this task impossible?

Pingu is given two positive integers $m$ and $n$ without any common factors greater than $1$. a) Help Pingu find positive integers $p, q$ such that $$\operatorname{gcd}(pm+q, n) \cdot \operatorname{gcd}(m, pn+q) = mn$$b) Prove to Pingu that he can never find positive integers $r, s$ such that $$\operatorname{lcm}(rm+s, n) \cdot \operatorname{lcm}(m, rn+s) = mn$$regardless of the choice of $m$ and $n$.

A duck drew a square $ABCD$, then he reflected $C$ across $B$ to obtain a point $E$. He also drew the center of the square to be $F$. Then, he drew a point $G$ on ray $EF$ beyond $F$ such that $\angle AGC = 135^{\circ}$. Help the Duck prove that $\angle CGD = 135^{\circ}$ as well.

There are $2n$ points on a circle, $n$ are red and $n$ are blue. Janson found a red frog and a blue frog at a red point and a blue point on the circle respectively. Every minute, the red frog moves to the next red point in the clockwise direction and the blue frog moves to the next blue point in the anticlockwise direction. Prove that for any initial position of the two frogs, Janson can draw a line through the circle, such that the two frogs are always on opposite sides of the line.

Suppose $a, b, c, d$ are positive reals such that $a \geq b \geq c \geq d$ and $ab^2c^3d^4 = 1$. Help Janson prove that $a+b+c+d \geq 4$.

A sequence of integers $a_{1}, a_{2}, \cdots$ is called $good$ if: • $a_{1}=1$, and; • $a_{i+1}-a_{i}$ is either $1$ or $2$ for all $i \geq 1$. Find all positive integers n that cannot be written as a sum $n = a_{1} + a_{2} + \cdots + a_{k}$, such that the integers $a_{1} , a_{2} , \cdots , a_{k}$ forms a good sequence.

Ivan claims that for all positive integers $n$, $$\left\lfloor\sqrt[2]{\frac{n}{1^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{2^3}}\right\rfloor + \left\lfloor\sqrt[2]{\frac{n}{3^3}}\right\rfloor + \cdots = \left\lfloor\sqrt[3]{\frac{n}{1^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{2^2}}\right\rfloor + \left\lfloor\sqrt[3]{\frac{n}{3^2}}\right\rfloor + \cdots$$Why is he correct? (Note: $\lfloor x \rfloor$ denotes the floor function.)

For all $n \geq 1$, define $a_{n}$ to be the fraction $\frac{k}{2^n}$ such that $a_{n}$ is the closest to $\frac{1}{3}$ over all integer values of $k$. Prove that the sequence $a_{1}, a_{2}, \cdots $satisfies the equation $2a_{i+2} = a_{i+1} + a_{i}$ for all $i \geq 1$.

Janson found $2025$ dogs on a circle. Janson wants to select some (possibly none) of the dogs to take home, such that no two selected dogs have exactly two dogs (whether selected or not) in between them. Let $S_{1}$ be the number of ways for him to do so. Ivan also found $2025$ cats on a circle. Ivan wants to select some (possibly none) of the cats to take home, such that no two selected cats have exactly five cats (whether selected or not) in between them. Let $S_{2}$ be the number of ways for him to do so. a) Prove that $S_{1}=S_{2}$. b) Prove that $S_{1}$ and $S_{2}$ are both perfect cubes.

Rui Xuen has a circle $\omega$ with center $O$, and a square $ABCJ$ with vertices on $\omega$. Let $M$ be the midpoint of $AB$, and let $\Gamma$ be the circle passing through the points $J$, $O$, $M$. Suppose $\Gamma$ intersect line $AJ$ at a point $P \neq J$, and suppose $\Gamma$ intersect $\omega$ at a point $Q \neq J$. A point $R$ lies on side $BC$ so that $RC = 3RB$. Help Rui Xuen prove that the points $P$, $Q$, $R$ are collinear.