What is the smallest positive integer value of $x$ for which $x \equiv 4$ (mod $9$) and $x \equiv 7$ (mod $8$)?

Find all positive integers $n$ that can be represented in the form $$n=\mathrm{lcm}(a,b)+\mathrm{lcm}(b,c)+\mathrm{lcm}(c,a)$$ where $a,b,c$ are positive integers.

Solve the system of equations: $$\begin{cases} xy+zu=14 \\ xz+yu=11 \\ xu+yz=10 \\ x+y+z+u=10 \end{cases}$$

The Wonder Island is inhabited by Hedgehogs. Each Hedgehog consists of three segments of unit length having a common endpoint, with all three angles between them $120^{\circ}$. Given that all Hedgehogs are lying flat on the island and no two of them touch each other, prove that there is a finite number of Hedgehogs on Wonder Island.

Evaluate the sum $\frac{3!+4!}{2(1!+2!)}+\frac{4!+5!}{3(2!+3!)}+\cdots+\frac{12!+13!}{11(10!+11!)}$

A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$? [asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy] $\textbf{(A) }2+2\sqrt 7\qquad \textbf{(B) }3+2\sqrt 5\qquad \textbf{(C) }4+2\sqrt 7\qquad \textbf{(D) }4\sqrt 5\qquad \textbf{(E) }4\sqrt 7\qquad$

A rectangular wooden block has a square top and bottom, its volume is $576$, and the surface area of its vertical sides is $384$. Find the sum of the lengths of all twelve of the edges of the block.

$ n\geq 2$ cars are participating in a rally. The cars leave the start line at different times and arrive at the finish line at different times. During the entire rally each car takes over any other car at most once , the number of cars taken over by each car is different and each car is taken over by the same number of cars. Find all possible values of $ n$

The lengths of two line segments are $ a$ units and $ b$ units respectively. Then the correct relation between them is: $ \textbf{(A)}\ \frac{a\plus{}b}{2} > \sqrt{ab} \qquad\textbf{(B)}\ \frac{a\plus{}b}{2} < \sqrt{ab} \qquad\textbf{(C)}\ \frac{a\plus{}b}{2} \equal{} \sqrt{ab}\\ \textbf{(D)}\ \frac{a\plus{}b}{2} \leq \sqrt{ab} \qquad\textbf{(E)}\ \frac{a\plus{}b}{2} \geq \sqrt{ab}$

The diagram below shows twelve $30-60-90$ triangles placed in a circle so that the hypotenuse of each triangle coincides with the longer leg of the next triangle. The fourth and last triangle in this diagram are shaded. The ratio of the perimeters of these two triangles can be written as $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(200); defaultpen(linewidth(0.8)); pair point=(-sqrt(3),0); pair past,unit; path line; for(int i=0;i<=12;++i) { past = point; line=past--origin; unit=waypoint(line,1/200); point=extension(past,rotate(90,past)*unit,origin,dir(180-30*i)); if (i == 4) { filldraw(origin--past--point--cycle,gray(0.7)); } else if (i==12) { filldraw(origin--past--point--cycle,gray(0.7)); } else { draw(origin--past--point); } } draw(origin--point); [/asy]

Prove that there exist infinitely pairs $(m,n)$ such that $m+n$ divides $(m!)^n+(n!)^m+1$

Let $ a,b,c$ be positive integers. If $ 30|a\plus{}b\plus{}c$, prove that $ 30|a^5\plus{}b^5\plus{}c^5$.

Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes? $\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$

Let $ p_i $ denote the $ i $-th prime number. Let $ n = \lfloor\alpha^{2015}\rfloor $, where $ \alpha $ is a positive real number such that $ 2 < \alpha < 2.7 $. Prove that $$ \displaystyle\sum_{2 \le p_i \le p_j \le n}\frac{1}{p_ip_j} < 2017 $$

Show that every triangle can be dissected into nine convex nondegenrate pentagons.

The decimal expansion of $37^9$ is $129A617B979C077$ for digits $A, B,$ and $C$. Find the three digit number $ABC$. [i]Individual #14[/i]

Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.

Rishabh has $2024$ pairs of socks in a drawer. He draws socks from the drawer uniformly at random, without replacement, until he has drawn a pair of identical socks. Compute the expected number of unpaired socks he has drawn when he stops.

[b](a)[/b] The integers $x$, $x^2$ and $x^3$ begin with the same digit. Does it imply that this digit is $1$? [i]($2$ points) [/i] [b](b)[/b] The same question for the integers $x, x^2, x^3, \cdots, x^{2015}$ [i]($3$ points)[/i] .

Let $A,B\in\mathcal{M}_n(\mathbb{Z})$ and $p$ a prime number. Prove that $$\text{Tr}((A+B)^p)\equiv\text{Tr}(A^p+B^p)\pmod p.$$

The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have different remainders when divided by $n + 1$?

Let $ABC$ be a triangle. The circle $\omega_A$ through $A$ is tangent to line $BC$ at $B$. The circle $\omega_C$ through $C$ is tangent to line $AB$ at $B$. Let $\omega_A$ and $\omega_C$ meet again at $D$. Let $M$ be the midpoint of line segment $[BC]$, and let $E$ be the intersection of lines $MD$ and $AC$. Show that $E$ lies on $\omega_A$.

Find the product of all positive real solutions to the equation $x^{-x} + x^{\frac{1}{x}} = \frac{2021}{2020}.$ [i]Proposed by Gabriel Wu[/i]

The equation $ x^3 - x^2 + ax - 2^n = 0$ has three integer roots. Determine $a$ and $n$.

An equiangular octagon has four sides of length $ 1$ and four sides of length $ \frac{\sqrt{2}}{2}$, arranged so that no two consecutive sides have the same length. What is the area of the octagon? $ \textbf{(A)}\ \frac{7}{2}\qquad \textbf{(B)}\ \frac{7\sqrt{2}}{2}\qquad \textbf{(C)}\ \frac{5 \plus{} 4\sqrt{2}}{2}\qquad \textbf{(D)}\ \frac{4 \plus{} 5\sqrt{2}}{2}\qquad \textbf{(E)}\ 7$