Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. For $x\in \mathbb{R}$ we say that $f$ is [i]increasing in $x$[/i] if there exists $\epsilon_x > 0$ such that $f(x)\geq{f(a)}$, $\forall a\in (x-\epsilon_x,x)$ and $f(x)\leq f(b)$, $\forall b\in (x,x+\epsilon_x)$. $\textbf{(a)}$ Prove that if $f$ is increasing in $x$, $\forall x\in \mathbb{R}$ then $f$ is increasing over $\mathbb{R}$. $\textbf{(b)}$ We say that $f$ is [i]increasing to the left[/i] in $x$ if there exists $\epsilon_x > 0$ such that $f(x)\geq f(a) $, $ \forall a \in (x-\epsilon_x,x)$. Provide an example of a function $f: [0,1]\rightarrow \mathbb{R}$ for which there exists an infinite set $M \subset (0,1)$ such that $f$ is increasing to the left in every point of $M$, yet $f$ is increasing over no proper subinterval of $[0,1]$.

In a trapezoid $ABCD$, the legs $AB$ and $CD$ meet in $M$ and the diagonals $AC$ and $BD$ meet in $N.$ Let $AC=a$ and $BC=b.$ Find the area of triangles $AMD$ and $AND$ in terms of $a$ and $b.$

Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$ and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.

We have four charged batteries, four uncharged batteries and a radio which needs two charged batteries to work. Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batteries in the radio and check if the radio works or not.

Let $BE$ and $CF$ be the altitudes of an acute triangle $ABC$ with $E$ on $AC$ and $F$ on $AB$. Let $O$ be the point of intersection of $BE$ and $CF$. Take any line $KL$ through $O$ with $K$ on $AB$ and $L$ on $AC$. Suppose $M$ and $N$ are located on $BE$ and $CF$ respectively. such that $KM$ is perpendicular to $BE$ and $LN$ is perpendicular to $CF$. Prove that $FM$ is parallel to $EN$.

Four friends, Anna, Bob, Celia, and David, exchanged some money. For any two of these friends, exactly one gave money to the other. For example, Celia could have given some money to David but then David would not have given money to Celia. In the end, each person broke even (meaning that no one made or lost any money). (a) Is it possible that the amounts of money given were $10$, $20$, $30$, $40$, $50$, $60$? (b) Is it possible that the amounts of money given were $20$, $30$, $40$, $50$, $60$, $70$? For each part, if your answer is yes, show that the situation is possible by describing who could have given what amounts to whom. If your answer is no, prove that the situation is not possible.

Each of $n$ boys and $n$ girls chooses a random number from the set $\{ 1, 2, 3, 4, 5 \}$, uniformly and independently. Let $p_n$ be the probability that every boy chooses a different number than every girl. As $n$ approaches infinity, what value does $\sqrt[n]{p_n}$ approach?

Find all functions $f: \mathbb R^+ \to \mathbb R^+$ such that \[ f \left( x+\frac{f(xy)}{x} \right) = f(xy) f \left( y + \frac 1y \right) \] holds for all $x,y\in\mathbb R^+.$

Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?

Real numbers $x$, $y$, and $z$ satisfy the inequalities $$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$ Which of the following numbers is nessecarily positive? $\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\ \textbf{(E) } y+z$

\[\int_0^{2022} \{x\lfloor x \rfloor\}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$

Let $D,E,F$ be the points of tangency of the incircle of a given triangle $ABC$ with sides $BC, CA, AB,$ respectively. Denote by $I$ the incenter of $ABC$, by $M$ the midpoint of $BC$ and by $G$ the foot of the perpendicular from $M$ to line $EF$. Prove that the line $ID$ is tangent to the circumcircle of the triangle $MGI$.

Prove that if $n$ is an odd natural number, then $1^n +2^n +\cdots +n^n$ is divisible by $n^2$.

The roots of the equation $ax^2 - 4bx + 4c = 0$ with $ a > 0$ belong to interval $[2, 3]$. Prove that: a) $a \le b \le c < a + b.$ b) $\frac{a}{a+c} + \frac{b}{b+a} > \frac{c}{b+c} .$

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which are differentiable in $0$ and satisfy the following inequality for all real numbers $x,y$ \[f(x+y)+f(xy)\geq f(x)+f(y).\][i]Dan Ștefan Marinescu and Mihai Piticari[/i]

Let $\tfrac{a}{b}$ be a fraction such that $a$ and $b$ are positive integers and the first three digits of its decimal expansion are $527$. What is the smallest possible value of $a+b?$

Given is an acute triangle $ABC$. Point $P$ lies inside this triangle and lies on the bisector of angle $\angle BAC$. Suppose that the point of intersection of the altitudes $H$ of triangle $ABP$ lies inside triangle $ABC$. Let $Q$ be the intersection of the line $AP$ and the line perpendicular to $AC$ passing through $H$. Prove that $Q$ is the point symmetrical to $P$ wrt the line $BH$.

If $\sum_{i=1}^{\infty} u_{i}^{2}$ and $\sum_{i=1}^{\infty} v_{i}^{2}$ are convergent series of real numbers, prove that $$\sum_{i=1}^{\infty}(u_{i}-v_{i})^{p}$$ is convergent, where $p\geq 2$ is an integer.

Let the AP of the form $4$, $9$, $\ldots$ be $\mathbf{A}$, and the AP of the form $16$, $25$, $\ldots$ be $\mathbf{B}$. Find the number of integers from $1$ to $2024$ inclusive, that appear in only one of the AP's. For clarification, the AP's $\mathbf{A}$ and $\mathbf{B}$ start from 4 and 16 respectively. [hide=Answer]584[/hide]

Katie Ledecky and Michael Phelps each participate in $7$ swimming events in the Olympics (and there is no event that they both participate in). Ledecky receives $g_L$ gold, $s_L$ silver, and $b_L$ bronze medals, and Phelps receives $g_P$ gold, $s_P$ silver, and $b_P$ bronze medals. Ledecky notices that she performed objectively better than Phelps: for all positive real numbers $w_b<w_s<w_g$, we have $$w_gg_l+w_ss_L+w_bb_L>w_gg_P+w_ss_P+w_bb_P.$$ Compute the number of possible $6$-tuples $(g_L,s_L,b_L,g_P,s_P,b_P).$

$n(>1)$ lotus leaves are arranged in a circle. A frog jumps from a particular leaf from another under the following rule: [list] [*]It always moves clockwise. [*]From starting it skips one leaf and then jumps to the next. After that it skips two leaves and jumps to the following. And the process continues. (Remember the frog might come back on a leaf twice or more.)[/list] Given that it reaches all leaves at least once. Show $n$ cannot be odd.

A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.

Let $ABC$ be a triangle with $AB=5$, $BC=6$, $CA=7$. Let $D$ be a point on ray $AB$ beyond $B$ such that $BD=7$, $E$ be a point on ray $BC$ beyond $C$ such that $CE=5$, and $F$ be a point on ray $CA$ beyond $A$ such that $AF=6$. Compute the area of the circumcircle of $DEF$. [i]Proposed by James Lin.[/i]

A triangle $ABC$ has its orthocentre $H$ distinct from its vertices and from the circumcenter $O$ of $\triangle ABC$. Denote by $M, N$ and $P$ respectively the circumcenters of triangles $HBC, HCA$ and $HAB$. Show that the lines $AM, BN, CP$ and $OH$ are concurrent.