Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > f(x)>0$ for all real numbers $x$. Show that $f(8) > 2022f(0)$. [i]Proposed by Ethan Tan[/i]

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

Find integral solutions to the equation \[(m^{2}-n^{2})^{2}=16n+1.\]

Is there a circle which passes through five points with integer co-ordinates?

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?

A square is called [i]proper[/i] if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$. [i]Proposed by Lewis Chen[/i]

A function $f(x)$ satisfies $f(x)=f\left(\frac{c}{x}\right)$ for some real number $c(>1)$ and all positive number $x$. If $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$, evaluate $\int_1^c \frac{f(x)}{x} dx$

Let $ x_0 \equal{} 1$ and for $ n\ge0,$ let $ x_{n \plus{} 1} \equal{} 3x_n \plus{} \left\lfloor x_n\sqrt {5}\right\rfloor.$ In particular, $ x_1 \equal{} 5,\ x_2 \equal{} 26,\ x_3 \equal{} 136,\ x_4 \equal{} 712.$ Find a closed-form expression for $ x_{2007}.$ ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

Let $E$ be the set of all continuously differentiable real valued functions $f$ on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. Define $$J(f)=\int^1_0(1+x^2)f'(x)^2\text dx.$$ a) Show that $J$ achieves its minimum value at some element of $E$. b) Calculate $\min_{f\in E}J(f)$.

$$\int_0^1 x\ln(x^2)\ln(1+x)\,dx$$ [i]Proposed by Connor Gordon[/i]

$f$ is a continuous real-valued function such that $f(x+y)=f(x)f(y)$ for all real $x$, $y$. If $f(2)=5$, find $f(5)$.

If $x$ and $y$ are real numbers with $(x+y)^4=x-y$, what is the maximum possible value of $y$?

Find the real number $\alpha$ such that the curve $f(x)=e^x$ is tangent to the curve $g(x)=\alpha x^2$.

Find all functions $ f:[0,1]\longrightarrow \mathbb{R} $ that are continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow [0,1] , $ the following equality holds. $$ \int_0^1 f\left( g(x) \right) dx =\int_0^1 g(x) dx $$

Let $n\geq 3$ be integer. Given a regular $n$-polygon $P$ with side length 4 on the plane $z=0$ in the $xyz$-space.Llet $G$ be a circumcenter of $P$. When the center of the sphere $B$ with radius 1 travels round along the sides of $P$, denote by $K_n$ the solid swept by $B$. Answer the following questions. (1) Take two adjacent vertices $P_1,\ P_2$ of $P$. Let $Q$ be the intersection point between the perpendicular dawn from $G$ to $P_1P_2$, prove that $GQ>1$. (2) (i) Express the area of cross section $S(t)$ in terms of $t,\ n$ when $K_n$ is cut by the plane $z=t\ (-1\leq t\leq 1)$. (ii) Express the volume $V(n)$ of $K_n$ in terms of $n$. (3) Denote by $l$ the line which passes through $G$ and perpendicular to the plane $z=0$. Express the volume $W(n)$ of the solid by generated by a rotation of $K_n$ around $l$ in terms of $n$. (4) Find $\lim_{n\to\infty} \frac{V(n)}{W(n)} .$

All the points with integer coordinates in the $ xy$-Plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point $ (0,0)$ is red and the point $ (0,1)$ is blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.

When $ x^9\minus{}x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is: $ \textbf{(A)}\ \text{more than 5} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 2$

Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$

If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\] then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.

\[\int_0^{\pi/2} \frac{\sin(x)}{2-\sin(x)\cos(x)}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

The figure on the right shows the ellipse $\frac{(x-19)^2}{19}+\frac{(x-98)^2}{98}=1998$. Let $R_1,R_2,R_3,$ and $R_4$ denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of $R_1-R_2+R_3-R_4$. [asy] defaultpen(linewidth(0.7)); pair c=(19,98); real dist = 30; real a = sqrt(1998*19),b=sqrt(1998*98); xaxis("x",c.x-a-dist,c.x+a+3*dist,EndArrow); yaxis("y",c.y-b-dist*2,c.y+b+3*dist,EndArrow); draw(ellipse(c,a,b)); label("$R_1$",(100,200)); label("$R_2$",(-80,200)); label("$R_3$",(-60,-150)); label("$R_4$",(70,-150));[/asy]

Let $ a,\ b,\ c$ be postive constants. Evaluate $ \int_0^1 \frac{2a\plus{}3bx\plus{}4cx^2}{2\sqrt{a\plus{}bx\plus{}cx^2}}\ dx$.

Evaluate $ \int_1^e \frac{1\minus{}x(e^x\minus{}1)}{x(1\plus{}xe^x\ln x)}\ dx$.

Define the sequence $\{x_{n}\}$ by $x_{0}=a\in (0,1)$ and $x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...)$. Show that the sequence converges and find its limit.

Alice and Bob play the following number-guessing game. Alice writes down a list of positive integers $x_{1}$, $\cdots$, $x_{n}$, but does not reveal them to Bob, who will try to determine the numbers by asking Alice questions. Bob chooses a list of positive integers $a_{1}$, $\cdots$, $a_{n}$ and asks Alice to tell him the value of $a_{1}x_{1}+\cdots+a_{n}x_{n}$. Then Bob chooses another list of positive integers $b_{1}$, $\cdots$, $b_{n}$ and asks Alice for $b_{1}x_{1}+\cdots+b_{n}x_{n}$. Play continues in this way until Bob is able to determine Alice's numbers. How many rounds will Bob need in order to determine Alice's numbers?