Suppose $a$ and $b$ are real numbers such that $0 < a < b < 1$. Let $$x= \frac{1}{\sqrt{b}} - \frac{1}{\sqrt{b+a}},\hspace{1cm} y= \frac{1}{b-a} - \frac{1}{b}\hspace{0.5cm}\textrm{and}\hspace{0.5cm} z= \frac{1}{\sqrt{b-a}} - \frac{1}{\sqrt{b}}.$$ Show that $x$, $y$, $z$ are always ordered from smallest to largest in the same way, regardless of the choice of $a$ and $b$. Find this order among $x$, $y$, $z$.

Let $n$ be a positive integer, and $a,b\ge 1$, $c>0$ arbitrary real numbers. Prove that \[\frac{(ab+c)^n-c}{(b+c)^n-c}\le a^n.\]

Let $p$ a prime number and $r$ an integer such that $p|r^7-1$. Prove that if there exist integers $a, b$ such that $p|r+1-a^2$ and $p|r^2+1-b^2$, then there exist an integer $c$ such that $p|r^3+1-c^2$.

Let $\alpha$, $\beta$ and $\gamma$ be the interior angles of an acute triangle. Prove that if $\alpha < \beta < \gamma$ then $$\sin 2\alpha >\ sin 2 \beta > \sin 2\gamma.$$

A sequence of integers $a_0, a_1 …$ is called [i]kawaii[/i] if $a_0 =0, a_1=1,$ and $$(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0$$ for all integers $n \geq 0$. An integer is called [i]kawaii[/i] if it belongs to some kawaii sequence. Suppose that two consecutive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that $m$ is divisible by $3,$ and that $m/3$ is also kawaii.

Prove that there exists a program in standard Pascal which prints out its own ASCII code. No disk operations are permitted.

Find the largest $k$ for which there exists a permutation $(a_1, a_2, \ldots, a_{2022})$ of integers from $1$ to $2022$ such that for at least $k$ distinct $i$ with $1 \le i \le 2022$ the number $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$ is an integer larger than $1$. [i](Proposed by Oleksii Masalitin)[/i]

Let be $a,b$ real numbers such that $|a|\neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6$ . Calculate: $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}$

How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$? (Note that $i=\sqrt{-1}$) $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$

\[\int \frac{1}{(1+x)\sqrt{x}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Calculate: $\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{e}^{{{x}^{n}}}}dx}$

Each point of a plane is colored in one of three colors: red, black and blue. Prove that there exists a rectangle in this plane whose vertices all have the same color.

A polynomial whose roots are all equal to each other is called a [i]unicorn[/i]. Compute the number of distinct ordered triples $(M, P, G)$, where $M$, $P$, $G$ are complex numbers, such that the polynomials $z^3 + M z^2 + Pz + G$ and $z^3 + G z^2 + Pz + M$ are both unicorns.

Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.

Test whether equation $$\frac{1}{x - a} + \frac{1}{x - b} + \frac{1}{x - c} = 0,$$ where $ a $, $ b $, $ c $ denote the given real numbers, has real roots.

Let $\triangle ABC$ be equilateral. Let $D$ be the midpoint of side $AC,$ and let $DEFG$ be a square such that $D, F, B$ are collinear and $E,G$ lie on $AB,CB$ respectively. What fraction of the area of $\triangle ABC$ is covered by square $DEFG?$ [i]Proposed by Lohith Tummala[/i]

In making Euclidean constructions in geometry it is permitted to use a ruler and a pair of compasses. In the constructions considered in this question no compasses are permitted, but the ruler is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the rule. Then the distance between the parallel lines is equal to the breadth of the ruler. Carry through the following constructions with such a ruler. Construct: [b]a)[/b] The bisector of a given angle. [b]b)[/b] The midpoint of a given rectilinear line segment. [b]c)[/b] The center of a circle through three given non-collinear points. [b]d)[/b] A line through a given point parallel to a given line.

Determine the largest integer $n$ such that there exist monic quadratic polynomials $p_1(x)$, $p_2(x)$, $p_3(x)$ with integer coefficients so that for all integers $ i \in [1, n]$ there exists some $j \in [1, 3]$ and $m \in Z$ such that $p_j (m) = i$.

Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$. If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$. [i]Proposed by Andy Xu[/i]

Let $a_1, a_2, \ldots , a_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = 1$. Prove that $$\dfrac{a_1}{\sqrt{1-a_1}}+\cdots+\dfrac{a_n}{\sqrt{1-a_n}} \geq \dfrac{1}{\sqrt{n-1}}(\sqrt{a_1}+\cdots+\sqrt{a_n}).$$

Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

Given $30$ students such that each student has at most $5$ friends and for every $5$ students there is a pair of students that are not friends, determine the maximum $k$ such that for all such possible configurations, there exists $k$ students who are all not friends.

Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$. [i]Dan Schwartz[/i]

At a tourist camp, each person has at least $50$ and at most $100$ friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) $1331$ different colors, and any person has $20$ friends whose t-shirts all have pairwisely different colors.

Let $X$ be a regular topological space and let $S$ be a countably compact dense subspace in $X$. (The countably compact property means that every infinite subset of $S$ has an accumulation point in $S$.) Show that $S$ is also $G_\delta$-dense in $X$, i.e., $S$ intersects all nonempty $G_\delta$ sets.