Bob the bomb-defuser has stumbled upon an active bomb. He opens it up, and finds the red and green wires conveniently located for him to cut. Being a seasoned member of the bomb-squad, Bob quickly determines that it is the green wire that he should cut, and puts his wirecutters on the green wire. But just before he starts to cut, the bomb starts to count down, ticking every second. Each time the bomb ticks, starting at time $t=15$ seconds, Bob panics and has a certain chance to move his wirecutters to the other wire. However, he is a rational man even when panicking, and has a $\frac{1}{2t^2}$ chance of switching wires at time t, regardless of which wire he is about to cut. When the bomb ticks at $t=1$, Bob cuts whatever wire his wirecutters are on, without switching wires. What is the probability that Bob cuts the green wire?

What is the number formed by the last three digits of $1201^{1202}$?

Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let \[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]

The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights. [i]Warut Suksompong, Thailand[/i]

Let $n$ be a positive integer such that $n(n+3)$ is a perfect square of an integer, prove that $n$ is not a multiple of $3$.

Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?

In a rectangle $ABCD$ two not intersecting circles $\omega_1$ and $\omega_2$ are drawn such that $\omega_1$ is tangent to $AB$ and $AD$ at points $P$ and $S$ respectively, and $\omega_2$ is tangent to $CB$ and $CD$ at $T$ and $Q$ respectively. It is known that $PQ=11, ST=10, BD=14$. Find the distance between centers of circles $\omega_1$ and $\omega_2$. [i]I. Voronovich[/i]

Let $I,H,O$ be the incenter, centroid, and circumcenter of the nonisosceles triangle $ABC$. Prove that $AI \parallel HO$ if and only if $\angle BAC =120^{\circ}$.

Let $P_1$, $P_2$, $\ldots$, $P_{1993} = P_0$ be distinct points in the $xy$-plane with the following properties: (i) both coordinates of $P_i$ are integers, for $i = 1, 2, \ldots, 1993$; (ii) there is no point other than $P_i$ and $P_{i+1}$ on the line segment joining $P_i$ with $P_{i+1}$ whose coordinates are both integers, for $i = 0, 1, \ldots, 1992$. Prove that for some $i$, $0 \leq i \leq 1992$, there exists a point $Q$ with coordinates $(q_x, q_y)$ on the line segment joining $P_i$ with $P_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.

Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.

In $\triangle{ABC}$, $cos\angle{A} =\frac{2}{3}$, $cos\angle{B} =\frac{1}{9}$, and $BC = 24$. Find the length $AC$.

Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$, where the sum is taken over all positive divisors of $n$. What is $f(2023)$? $\textbf{(A)}~-1536\qquad\textbf{(B)}~96\qquad\textbf{(C)}~108\qquad\textbf{(D)}~116\qquad\textbf{(E)}~144$

Nine numbers $a, b, c, \dots$ are arranged around a circle. All numbers of the form $a+b^c, \dots$ are prime. What is the largest possible number of different numbers among $a, b, c, \dots$?

Let $ABC$ be a triangle. Points $D, E, F$ are on segments $BC$, $CA$, $AB$, respectively. Suppose that $AF = 10$, $F B = 10$, $BD = 12$, $DC = 17$, $CE = 11$, and $EA = 10$. Suppose that the circumcircles of $\vartriangle BFD$ and $\vartriangle CED$ intersect again at $X$. Find the circumradius of $\vartriangle EXF$.

The sequence $a_n$ is given as $$a_1=1, a_2=2 \;\;\; \text{and} \;\;\;\; a_{n+2}=a_n(a_{n+1}+1) \quad \forall n\geq 1$$ Prove that $a_{a_n}$ is divisible by $(a_n)^n$ for $n\geq 100$.

A trapezoid is divided into seven strips of equal width as shown. What fraction of the trapezoid’s area is shaded?