Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( 0 < f'(t) \leq 1 \) for all \( t \in [0, 1] \). Show that: \[ \left( \int_0^1 f(t) \, dt \right)^2 \geq \int_0^1 f(t)^3 \, dt. \]

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that $$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$ holds for all real numbers $x; y$

Let $f : (0,+\infty) \to \mathbb{R}$ a convex function and $\alpha, \beta, \gamma > 0$. Then $$\frac{1}{6\alpha}\int \limits_0^{6\alpha}f(x)dx\ +\ \frac{1}{6\beta}\int \limits_0^{6\beta}f(x)dx\ +\ \frac{1}{6\gamma}\int \limits_0^{6\gamma}f(x)dx$$ $$\geq \frac{1}{3\alpha +2\beta +\gamma}\int \limits_0^{3\alpha +2\beta +\gamma}f(x)dx\ +\ \frac{1}{\alpha +3\beta +2\gamma}\int \limits_0^{\alpha +3\beta +2\gamma}f(x)dx\ $$ $$+\ \frac{1}{2\alpha +\beta +3\gamma}\int \limits_0^{2\alpha +\beta +3\gamma}f(x)dx$$

Let $ f : \mathbf{R} \to \mathbf{R} $ be a continuous function with $ \displaystyle\int_{0}^{1} f(x) f'(x) \, \mathrm{d}x = 0 $ and $ \displaystyle\int_{0}^{1} f(x)^2 f'(x) \, \mathrm{d}x = 18 $. What is $ \displaystyle\int_{0}^{1} f(x)^4 f'(x) \, \mathrm{d} x $?

Let $\tau(x)$ be the number of positive divisors of $x$ (including $1$ and $x$). Find \[\tau\left( \tau\left( \dots \tau\left(2024^{2024^{2024}}\right) \right)\right),\] where there are $4202^{4202^{4202}}$ $\tau$'s. [i]Lightning 2.1[/i]

Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i\minus{}P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.

A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k\geq 1$, the circles in $\textstyle\bigcup_{j=0}^{k-1} L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\textstyle\bigcup_{j=0}^6 L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is \[\sum_{C\in S}\dfrac1{\sqrt{r(C)}}?\] [asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.78)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.78)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.78)); }[/asy] $\textbf{(A) }\dfrac{286}{35}\qquad\textbf{(B) }\dfrac{583}{70}\qquad\textbf{(C) }\dfrac{715}{73}\qquad\textbf{(D) }\dfrac{143}{14}\qquad\textbf{(E) }\dfrac{1573}{146}$

If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that \[\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.\]

Let $f : N \to N$ be a function which satisfies $f(x)+ f(x+2) \le 2 f(x+1)$ for any $x \in N$. Prove that there exists a line in the coordinate plane containing infinitely many points of the form $(n, f(n)), n \in N$.

Let $f_n(x)=\sin^n x + \cos^n x$. For how many $x$ in $[0,\pi]$ is it true that \[6f_4(x)-4f_6(x)=2f_2(x)?\] $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{more than 8}$

Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.

Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy $$f(\sin{x})+f(\cos{x})=2020$$ for any real number $x.$

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

Given a positive integer $ n$, find all functions $ F: \mathbb{N} \rightarrow \mathbb{R}$ such that $ F(x\plus{}y)\equal{}F(xy\minus{}n)$ whenever $ x,y \in \mathbb{N}$ satisfy $ xy>n$.

A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?

A function $\psi \colon {\mathbb Z} \to {\mathbb Z}$ is said to be [i]zero-requiem[/i] if for any positive integer $n$ and any integers $a_1$, $\ldots$, $a_n$ (not necessarily distinct), the sums $a_1 + a_2 + \dots + a_n$ and $\psi(a_1) + \psi(a_2) + \dots + \psi(a_n)$ are not both zero. Let $f$ and $g$ be two zero-requiem functions for which $f \circ g$ and $g \circ f$ are both the identity function (that is, $f$ and $g$ are mutually inverse bijections). Given that $f+g$ is [i]not[/i] a zero-requiem function, prove that $f \circ f$ and $g \circ g$ are both zero-requiem. [i]Sutanay Bhattacharya[/i]

Define a sequence $\{a_n\}$ as: $\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.$ Prove that this sequence has a finite limit as $n\to+\infty$ . Also determine the limit.

Find all functions $f: N_0 \to N_0$, (where $N_0$ is the set of all non-negative integers) such that $f(f(n))=f(n)+1$ for all $n \in N_0$ and the minimum of the set $\{ f(0), f(1), f(2) \cdots \}$ is $1$.

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$ for all real numbers $x$, $y$ and $z$ with $x+y+z=0$.

A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

In each of (a) to (d) you have to find a strictly increasing surjective function from A to B or prove that there doesn't exist any. (a) $A=\{x|x\in \mathbb{Q},x\leq \sqrt{2}\}$ and $B=\{x|x\in \mathbb{Q},x\leq \sqrt{3}\}$ (b) $A=\mathbb{Q}$ and $B=\mathbb{Q}\cup \{\pi \} $ In (c) and (d) we say $(x,y)>(z,t)$ where $x,y,z,t \in \mathbb{R}$ , whenever $x>z$ or $x=z$ and $y>t$. (c) $A=\mathbb{R}$ and $B=\mathbb{R}^2$ (d) $X=\{2^{-x}| x\in \mathbb{N}\}$ , then $A=X \times (X\cup \{0\})$ and $B=(X \cup \{ 0 \}) \times X$ (e) If $A,B \subset \mathbb{R}$ , such that there exists a surjective non-decreasing function from $A$ to $B$ and a surjective non-decreasing function from $B$ to $A$ , does there exist a surjective strictly increasing function from $B$ to $A$? Time allowed for this problem was 2 hours.

The function $ F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $ n \geq 0,$ (i) $ F(4n) \equal{} F(2n) \plus{} F(n),$ (ii) $ F(4n \plus{} 2) \equal{} F(4n) \plus{} 1,$ (iii) $ F(2n \plus{} 1) \equal{} F(2n) \plus{} 1.$ Prove that for each positive integer $ m,$ the number of integers $ n$ with $ 0 \leq n < 2^m$ and $ F(4n) \equal{} F(3n)$ is $ F(2^{m \plus{} 1}).$

Let $ f(x) \equal{} 1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{100}$. Find $ f'(1)$.

The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A) }19\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad$