Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored? $\mathrm{(A) \ } 400 \qquad \mathrm{(B) \ } 402 \qquad \mathrm {(C) \ } 597 \qquad \mathrm{(D) \ } 600 \qquad \mathrm{(E) \ } 602$

The circumcenter and orthocenter of $ABC$ are $O$ and $H$, respectively. Let $XACH$ be a parallelogram. Show that if $OH$ is parallel to $BC$, then $OX$ and $AB$ intersect at some point on the perpendicular bisector of $AH$. [i]proposed by USJL[/i]

Samuel Tsui and Jason Yang each chose a different integer between $1$ and $60$, inclusive. They don’t know each others’ numbers, but they both know that the other person’s number is between $1$ and $60$ and distinct from their own. They have the following conversation: Samuel Tsui: Do our numbers have any common factors greater than $1$? Jason Yang: Definitely not. However their least common multiple must be less than$ 2023$. Samuel Tsui: Ok, thismeans that the sumof the factors of our two numbers are equal. What is the sumof Samuel Tsui’s and Jason Yang’s numbers? [i]Proposed by Samuel Tsui[/i]

Prove that for any integer $n$ one can find integers $a$ and $b$ such that \[n=\left[ a\sqrt{2}\right]+\left[ b\sqrt{3}\right] \]

Given a natural number $ n\ge 2 $ and an $ n\times n $ matrix with integer entries, consider the multiplicative monoid $$ M=\{ M_k=I+kA| k\in \mathbb{Z} \} . $$ [b]a)[/b] Prove that $ M $ is a commutative group if the [url=https://en.wikipedia.org/wiki/Nilpotent_matrix]index[/url] of $ A $ is $ 2. $ [b]b)[/b] Prove that all elements of $ M $ are units if $ M_1,M_2,\ldots M_{2n} $ are all units.

Pick two consecutive positive integers whose sum is less than $100$. Square both of those integers and then find the difference of the squares. Which of the following could be the difference? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 79 \qquad \textbf{(D)}\ 96 \qquad \textbf{(E)}\ 131$

What is $ (20\minus{}(2010\minus{}201)) \plus{} (2010\minus{}(201\minus{}20))$? $ \textbf{(A)}\ \minus{}4020\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 401\qquad \textbf{(E)}\ 4020$

Let $ ABCD$ be an isosceles trapezium with $ AB \parallel{} CD$ and $ \bar{BC} \equal{} \bar{AD}.$ The parallel to $ AD$ through $ B$ meets the perpendicular to $ AD$ through $ D$ in point $ X.$ The line through $ A$ drawn which is parallel to $ BD$ meets the perpendicular to $ BD$ through $ D$ in point $ Y.$ Prove that points $ C,X,D$ and $ Y$ lie on a common circle.

Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which \[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\] where $ x_0, x_1, \ldots , x_n$ are real variables.

A triangle with side lengths in the ratio $ 3: 4: 5$ is inscribed in a circle of radius $ 3.$ What is the area of the triangle? $ \textbf{(A)}\ 8.64 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 5\pi \qquad \textbf{(D)}\ 17.28 \qquad \textbf{(E)}\ 18$

There are $n$ vertices and $m > n$ edges in a graph. Each edge is colored either red or blue. In each year, we are allowed to choose a vertex and flip the color of all edges incident to it. Prove that there is a way to color the edges (initially) so that they will never all have the same color

Each person stands on a whole number on the number line from $0$ to $2022$ . In each turn, two people are selected by a distance of at least $2$. These go towards each other by $1$. When no more such moves are possible, the process ends. Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.) [i](Birgit Vera Schmidt)[/i] [hide=original wording]Bei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person. In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang. Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)[/hide]

Determine if there are triplets ($x,y,z)$ of real numbers such that $$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$ If the answer is affirmative, find the minimum and maximum values ​​of $z$ in such a triplet.

[b]a.)[/b] Let $m>1$ be a positive integer. Prove there exist finite number of positive integers $n$ such that $m+n|mn+1$. [b]b.)[/b] For positive integers $m,n>2$, prove that there exists a sequence $a_0,a_1,\cdots,a_k$ from positive integers greater than $2$ that $a_0=m$, $a_k=n$ and $a_i+a_{i+1}|a_ia_{i+1}+1$ for $i=0,1,\cdots,k-1$.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

Let $\Gamma$ be the circumcircle of a triangle $ABC,$ and let $D$ and $E$ be two points different from the vertices on the sides $AB$ and $AC,$ respectively. Let $A'$ be the second point where $\Gamma$ intersects the bisector of the angle $BAC,$ and let $P$ and $Q$ be the second points where $\Gamma$ intersects the lines $A'D$ and $A'E,$ respectively. Let $R$ and $S$ be the second points of intersection of the lines $AA'$ and the circumcircles of the triangles $APD$ and $AQE,$ respectively. Show that the lines $DS, \: ER$ and the tangent line to $\Gamma$ through $A$ are concurrent.

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: \[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\] Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

Find the number of ordered integer triplets $ x, y, z $ with absolute value less than or equal to 100 such that $ 2x^2 + 3y^2 + 3z^2 + 2xy + 2xz - 4yz < 5 $.

Given three points $A, B$ and $C$ (not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.

Given two 2's, "plus" can be changed to "times" without changing the result: 2+2=2·2. The solution with three numbers is easy too: 1+2+3=1·2·3. There are three answers for the five-number case. Which five numbers with this property has the largest sum?

Let $A_1A_2 . . .A_{100}$ be the vertices of a regular $100$-gon. Let $\pi$ be a randomly chosen permutation of the numbers from $1$ through $100$. The segments $A_{\pi (1)}A_{\pi (2)}$, $A_{\pi (2)}A_{\pi (3)}$, $...$ ,$A_{\pi (99)}A_{\pi (100)}, A_{\pi (100)}A_{\pi (1)}$ are drawn. Find the expected number of pairs of line segments that intersect at a point in the interior of the $100$-gon.

A group of $25$ friends were discussing a large positive integer. "It can be divided by $1$," said the first friend. "It can be divided by $2$," said the second friend. "And by $3$," said the third friend. "And by $4$," added the fourth friend. This continued until everyone had made such a comment. If exactly $2$ friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?

Determine whether there exist an odd positive integer $n$ and $n\times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions: [list=1] [*]$\det (B)=1$;[/*] [*]$AB=BA$;[/*] [*]$A^4+4A^2B^2+16B^4=2019I$.[/*] [/list] (Here $I$ denotes the $n\times n$ identity matrix.) [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]

Compute the sum of the real solutions to $\lfloor x \rfloor \{x\} = 2020x$. Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$, and$ \{x\} = x -\lfloor x \rfloor$.