Show that for any real numbers $a$ and $b$ different from $0$, the inequality $$\bigg \lvert \frac{a}{b} + \frac{b}{a}+ab \bigg \lvert \geq \lvert a+b+1 \rvert$$ holds. When is equality achieved?

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

Given an integer $n\ge2$, the graph $G$ is defined by: - Vertices of $G$ are represented by binary strings of length $n$ - Two vertices $a,b$ are connected by an edge if and only if they differ in exactly $2$ places Let $S$ be a subset of the vertices of $G$, and let $S'$ be the set of edges between vertices in $S$ and vertices not in $S$. Show that if $|S|$ (the size of $S$) $\le2^{n-2}$, then $|S'|\ge|S|$.

$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$? $ \textbf{(A)}\ -3 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ 2\sqrt 3 \qquad\textbf{(D)}\ 3\sqrt 3 \qquad\textbf{(E)}\ \text{None of above} $

Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$

Find all positive integers $a$ and $b$ such that number $p=\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is rational number

The measure of angle $ABC$ is $50^\circ $, $\overline{AD}$ bisects angle $BAC$, and $\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is [asy] pair A,B,C,D; A = (0,0); B = (9,10); C = (10,0); D = (6.66,3); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--cycle); draw(A--D--C); label("$A$",A,SW); label("$B$",B,N); label("$C$",C,SE); label("$D$",D,N); label("$50^\circ $",(9.4,8.8),SW); [/asy] $\text{(A)}\ 90^\circ \qquad \text{(B)}\ 100^\circ \qquad \text{(C)}\ 115^\circ \qquad \text{(D)}\ 122.5^\circ \qquad \text{(E)}\ 125^\circ $

Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be random integers chosen independently and uniformly from the set $\{ 0, 1, 2, \dots, 23 \}$. (Note that the integers are not necessarily distinct.) Find the probability that \[ \sum_{k=1}^{5} \operatorname{cis} \Bigl( \frac{a_k \pi}{12} \Bigr) = 0. \] (Here $\operatorname{cis} \theta$ means $\cos \theta + i \sin \theta$.)

Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$.

Given are a closed broken line $A_1A_2\ldots A_n$ and a circle $\omega$ which touches each of lines $A_1A_2,A_2A_3,\ldots,A_nA_1$. Call the link [i]good[/i], if it touches $\omega$, and [i]bad[/i] otherwise (i.e. if the extension of this link touches $\omega$). Prove that the number of bad links is even.

Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$. A sequence of polynomials is defined as $$P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$ Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that $$P_{2n}(x)=P_n(x^2+c).$$ [i]Proposed by Navid Safaei[/i]

In the acute triangle $ABC$ the point $M$ is on the side $BC$. The inscribed circle of triangle $ABM$ touches the sides $BM$, $MA$ and $AB$ in points $D$, $E$ and $F$, and the inscribed circle of triangle $ACM$ touches the sides $CM$, $MA$ and $AC$ in points $X$, $Y$ and $Z$. The lines $FD$ and $ZX$ intersect in point $H$. Prove that lines $AH$, $XY$ and $DE$ are concurrent.

In the cells of the $4\times N$ table, integers are written, modulo no more than $2024$ (i.e. numbers from the set $\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}$) so that in each of the four lines there are no two equal numbers. At what maximum $N$ could it turn out that in each column the sum of the numbers is equal to $23$? [i]G.M.Sharafetdinova[/i]

Let $ABC$ be an obtuse triangle, and let $H$ denote its orthocenter. Let $\omega_A$ denote the circle with center $A$ and radius $AH$. Let $\omega_B$ and $\omega_C$ be defined in a similar way. For all points $X$ in the plane of triangle $ABC$ let circle $\Omega(X)$ be defined in the following way (if possible): take the polars of point $X$ with respect to circles $\omega_A$, $\omega_B$ and $\omega_C$, and let $\Omega(X)$ be the circumcircle of the triangle defined by these three lines. With a possible exception of finitely many points find the locus of points $X$ for which point $X$ lies on circle $\Omega(X)$. [i]Proposed by Vilmos Molnár-Szabó, Budapest[/i]

A sequence $(a_n)_0^N$ of real numbers is called concave if $2a_n\ge a_{n-1} + a_{n+1}$ for all integers $n, 1 \le n \le N - 1$. $(a)$ Prove that there exists a constant $C >0$ such that \[\left(\displaystyle\sum_{n=0}^{N}a_n\right)^2\ge C(N - 1)\displaystyle\sum_{n=0}^{N}a_n^2\:\:\:\:\:(1)\] for all concave positive sequences $(a_n)^N_0$ $(b)$ Prove that $(1)$ holds with $C = \frac{3}{4}$ and that this constant is best possible.

Consider the triangle $ABC$ with circumcenter $O$. Let $D$ be the intersection of the angle bisector of $\angle{A}$ with $BC$. Show that $OA$, the perpendicular bisector of $AD$ and the perpendicular to $BC$ passing through $D$ are concurrent.

Sheldon was really annoying Leonard. So to keep him quiet, Leonard decided to do something. He gave Sheldon the following grid $\begin{tabular}{|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 1 & 1 & 0\\ \hline 1 & 1 & 1 & 1 & 0 & 0\\ \hline 1 & 1 & 1 & 0 & 0 & 0\\ \hline 1 & 1 & 0 & 0 & 0 & 1\\ \hline 1 & 0 & 0 & 0 & 1 & 0\\ \hline 0 & 0 & 0 & 1 & 0 & 0\\ \hline \end{tabular}$ and asked him to transform it to the new grid below $\begin{tabular}{|c|c|c|c|c|c|} \hline 1 & 2 & 18 &24 &28 &30\\ \hline 21 & 3 & 4 &16 &22 &26\\ \hline 23 &19 & 5 & 6 &14 &20\\ \hline 32 &25 &17 & 7 & 8 &12\\ \hline 33 &34 &27 &15 & 9 &10\\ \hline 35 &31 &36 &29 &13 &11\\ \hline \end{tabular}$ by only applying the following algorithm: $\bullet$ At each step, Sheldon must choose either two rows or two columns. $\bullet$ For two columns $c_1, c_2$, if $a,b$ are entries in $c_1, c_2$ respectively, then we say that $a$ and $b$ are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them. Leonard hoped this would keep Sheldon occupied for some time, but Sheldon immediately said, "But this is impossible!". Was Sheldon right? Justify.

Let function $f : \mathbb Z_{\geq 0}^0 \to \mathbb N$ satisfy the following conditions: (i) $ f(0, 0, 0) = 1;$ (ii) $f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);$ (iii) when applying above relation iteratively, if any of $x', y', z$' is negative, then $f(x', y', z') = 0.$ Prove that if $x, y, z$ are the side lengths of a triangle, then $\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)}$ is not an integer for any integers $k, m > 1.$

Prove that for any positive real $x$ and $y$, holds the inequality $$\frac{1}{(x+y)^2}+\frac{1}{x^2}+\frac{1}{y^2} \ge \frac{9}{4xy}$$

Let $G$ be a graph on $n$ vertices $V_1,V_2,\dots, V_n$ and let $P_1,P_2, \dots, P_n$ be points in the plane. Suppose that, whenever $V_i$ and $V_j$ are connected by an edge, $P_iP_j$ has length $1$; in this situation, we say that the $P_i$ form an [i]embedding[/i] of $G$ in the plane. Consider a set $S\subseteq \{1,2,\dots, n\}$ and a configuration of points $Q_i$ for each $i\in S$. If the number of embeddings of $G$ such that $P_i=Q_i$ for each $i\in S$ is finite and nonzero, we say that $S$ is a [i]tasty[/i] set. Out of all tasty sets $S$, we define a function $f(G)$ to be the smallest size of a tasty set. Let $T$ be the set of all connected graphs on $n$ vertices with $n-1$ edges. Choosing $G$ uniformly and at random from $T$, let $a_n$ be the expected value of $\frac{f(G)^2}{n^2}$. Compute $\left\lfloor 2019 \displaystyle\lim_{n\to \infty} a_n \right\rfloor$. [i]Proposed by Vincent Huang[/i]

$a_1,a_2,a_3,a_4,a_5$ are non-zero complex numbers, satisfying: $\displaystyle\begin{cases} \displaystyle\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\frac{a_5}{a_4}\\ \displaystyle a_1+a_2+a_3+a_4+a_5=4\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+\frac{1}{a_5}\right)=S \end{cases}$ Where $S$ is a real number that $|S|\leq2$ Prove that points that $a_1,a_2,a_3,a_4,a_5$ refers to in the complex plane are concyclic.

Points $M$ and $N$ lie on the sides $BC$ and $CD$ of the square $ABCD,$ respectively, and $\angle MAN = 45^{\circ}$. The circle through $A,B,C,D$ intersects $AM$ and $AN$ again at $P$ and $Q$, respectively. Prove that $MN || PQ.$

Find all real solutions of the equation: $ [x]\plus{}[x\plus{}\dfrac{1}{2}]\plus{}[x\plus{}\dfrac{2}{3}]\equal{}2002$

Al, Betty, and Clara are in the same class of 50 students total, but are not friends with each other. Al is friends with 24 students, Betty is friends with 39, and Clara is friends with 20. What is the greatest number of students that could be friends with all 3 of them?

Prove that, abc≤1 and a,b,c>0 \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq 1+ \frac{6}{a+b+c} \]