Post your solutions below! :D [b]Also, I think it is beneficial to everyone if you all attempt to comment on each other's solutions.[/b] 5/1/31. Let $n$ be a positive integer. For integers a, b with $0 \leq a b \leq n - 1$, let $r_n(a, b)$ denote the remainder when $ab$ is divided by $n$. If $S_n$ denotes the sum of all $n^2$ remainders $r_n(a, b)$, prove that $\frac{1}{2}-\frac{1}{\sqrt{n}}\leq \frac{S_n}{n^3} \leq \frac{1}{2}$

Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.

A paper equilateral triangle of side length $2$ on a table has vertices labeled $A,B,C.$ Let $M$ be the point on the sheet of paper halfway between $A$ and $C.$ Over time, point $M$ is lifted upwards, folding the triangle along segment $BM,$ while $A,B,$ and $C$ on the table. This continues until $A$ and $C$ touch. Find the maximum volume of tetrahedron $ABCM$ at any time during this process.

This questions comprises two independent parts. (i) Let \(g:\mathbb{R}\to\mathbb{R}\) be continuous and such that \(g(0)=0\) and \(g(x)g(-x)>0\) for any \(x > 0\). Find all solutions \(f : \mathbb{R}\to\mathbb{R}\) to the functional equation \[g(f(x+y))=g(f(x))+g(f(y)),\ x,y\in\mathbb{R}\] (ii) Find all continuously differentiable functions \(\phi : [a, \infty) \to \mathbb{R}\), where \(a > 0\), that satisfies the equation \[(\phi(x))^2=\int_a^x \left(\left|\phi(y)\right|\right)^2+\left(\left|\phi'(y)\right|\right)^2\mathrm{d}y -(x-a)^3,\ \forall x\geq a.\]

Determine all real constants $t$ such that whenever $a$, $b$ and $c$ are the lengths of sides of a triangle, then so are $a^2+bct$, $b^2+cat$, $c^2+abt$.

A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.

Let the sequence $\{a_n\}_{n \geq 1}$ be defined by \[ a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N} \] Prove that \[ a_n^{2025} >n^{2024} \] for all positive integers $n \geq 2$. $\textbf{Proposed by Prajit Adhikari, Nepal.}$

Let $f(x) = x^n+x^{n-1}+\dots+x+1$ for an integer $n\ge 1.$ For which $n$ are there polynomials $g, h$ with real coefficients and degrees smaller than $n$ such that $f(x) = g(h(x)).$

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function . Suppose \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$ and all $t>0$ . Then show that there exists a constant $c$ such that $f(x)=cx\ \forall x$

Initially, a non-constant polynomial $S(x)$ with real coefficients is written down on a board. Whenever the board contains a polynomial $P(x)$, not necessarily alone, one can write down on the board any polynomial of the form $P(C + x)$ or $C + P(x)$ where $C$ is a real constant. Moreover, if the board contains two (not necessarily distinct) polynomials $P(x)$ and $Q(x)$, one can write $P(Q(x))$ and $P(x) + Q(x)$ down on the board. No polynomial is ever erased from the board. Given two sets of real numbers, $A = \{ a_1, a_2, \dots, a_n \}$ and $B = \{ b_1, \dots, b_n \}$, a polynomial $f(x)$ with real coefficients is $(A,B)$-[i]nice[/i] if $f(A) = B$, where $f(A) = \{ f(a_i) : i = 1, 2, \dots, n \}$. Determine all polynomials $S(x)$ that can initially be written down on the board such that, for any two finite sets $A$ and $B$ of real numbers, with $|A| = |B|$, one can produce an $(A,B)$-[i]nice[/i] polynomial in a finite number of steps. [i]Proposed by Navid Safaei, Iran[/i]

How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.)

We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

Find the value of $ k\ (0<k<5)$ such that $ \int_0^{\infty} \frac{x^k}{2\plus{}4x\plus{}3x^2\plus{}5x^3\plus{}3x^4\plus{}4x^5\plus{}2x^6}\ dx$ is minimal.