If $n$ is an integer, then find all values of $n$ for which $\sqrt{n}+\sqrt{n+2005}$ is an integer as well.

Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Let $(O_1), (O_2), (O_3), (O_4)$ be the circles going through $(A,B), (B,C),(C,D),(D,A)$. Let $X, Y,Z, T$ be the second intersection of the pairs of the circles: $(O_1)$ and $(O_2), (O_2)$ and $(O_3), (O_3)$ and $(O_4), (O_4)$ and $(O_1)$. (a) Prove that $X, Y,Z, T$ are on the same circle of radius $I$. (b) Prove that the midpoints of the line segments $O_1O_3,O_2O_4,OI$ are collinear. Nguyễn Văn Linh

How many integer values satisfy $|x|<3\pi$? $\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20$

In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? [asy] unitsize(150); pair A,B,C,D,E,F,G,H,J,K; A=(1,0); B=(0,0); C=(0,1); D=(1,1); draw(A--B--C--D--A); E=(2-sqrt(3),0); F=(2-sqrt(3),1); draw(E--F); G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1); K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0); draw(G--H--K--J--G); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$E$",E,S); label("$F$",F,N); label("$G$",G,E); label("$H$",H,N); label("$K$",K,W); label("$J$",J,S); [/asy] $ \textbf{(A) }\dfrac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\dfrac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\dfrac{\sqrt{3}}{6}\qquad\textbf{(E) }1-\dfrac{\sqrt{2}}{2} $

[color=darkblue]Let $ X$ be a group of people, where any two people are friends or enemies. Each pair of friends from $ X$ doesn't have any common friends, and any two enemies have exactly two common friends. Prove that each person from $ X$ has the same number of friends as others.[/color]

Let $S$ be the set of $10$-tuples of non-negative integers that have sum $2019$. For any tuple in $S$, if one of the numbers in the tuple is $\geq 9$, then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely many operations, then denote $A\rightarrow B$. (1) Find the smallest integer $k$, such that if the minimum number in $A,B\in S$ respectively are both $\geq k$, then $A\rightarrow B$ implies $B\rightarrow A$. (2) For the $k$ obtained in (1), how many tuples can we pick from $S$, such that any two of these tuples $A,B$ that are distinct, $A\not\rightarrow B$.

Let $S$ be the set of integer triplets $(a, b, c)$ with $1 \le a \le b \le c$ that satisfy $a + b + c = 77$ and: \[\frac{1}{a} +\frac{1}{b}+\frac{1}{c}= \frac{1}{5}.\]What is the value of the sum $\sum_{a,b,c \in S} a\cdot b \cdot c$?

Carlos took $70\%$ of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left? $\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ 30\%\qquad\textbf{(E)}\ 35\%$

One container of paint is exactly enough to cover the inside of an old rectangle which is three times as long as it is wide. If we make a new rectangle by shortening the old rectangle by $18$ feet and widening it by $8$ feet as shown below, one container of paint is also exactly enough to cover the inside of the new rectangle. Find the length in feet of the perimeter of the new rectangle. [asy] size(250); defaultpen(linewidth(0.8)); draw((-2,0)--(-2,5)--(13,5)--(13,0)--cycle^^(16,-1)--(16,6)--(27,6)--(27,-1)--cycle^^(9,5)--(9,0)^^(16,4)--(27,4)); path rect1=(13,5)--(13,0)--(9,0)--(9,5)--cycle,rect2=(16,6)--(16,4)--(27,4)--(27,6)--cycle; fill(rect1,lightgray); fill(rect2,lightgray); draw(rect1^^rect2); [/asy]

Call a set of integers [i]unpredictable[/i] if no four elements in the set form an arithmetic sequence. How many unordered [i]unpredictable[/i] sets of five distinct positive integers $\{a, b, c, d, e\}$ exist such that all elements are strictly less than $12$? [i]Proposed by Anthony Yang[/i]

Minivan and Megavan play a game. For a positive integer $n$, Minivan selects a sequence of integers $a_1,a_2,\ldots,a_n$. An operation on $a_1,a_2,\ldots,a_n$ means selecting an $a_i$ and increasing it by $1$. Minivan and Megavan take turns, with Minivan going first. On Minivan's turn, he performs at most $2025$ operations, and he may choose the same integer repeatedly. On Megavan's turn, he performs exactly $1$ operation instead. Megavan wins if at any point in the game, including in the middle of Minivan's operations, two numbers in the sequence are equal. [i](Proposed by Ho Janson)[/i]

For a finite sequence $A = (a_1, a_2,\ldots,a_n)$ of numbers, the [i]Cesaro sum[/i] of $A$ is defined to be \[\frac{S_1 + S_2 + \cdots + S_n}{n}\] where $S_k = a_1 + a_2 + \cdots + a_k\ \ \ \ (1 \le k \le n)$. If the Cesaro sum of the 99-term sequence $(a_1, a_2, \ldots, a_{99})$ is $1000$, what is the Cesaro sum of the 100-term sequence $(1,a_1,a_2,\ldots,a_{99})$? $ \textbf{(A)}\ 991\qquad\textbf{(B)}\ 999\qquad\textbf{(C)}\ 1000\qquad\textbf{(D)}\ 1001\qquad\textbf{(E)}\ 1009 $

What is the maximum number of rational points that can lie on a circle in $ \mathbb{R}^2$ whose center is not a rational point? (A [i]rational point[/i] is a point both of whose coordinates are rational numbers.)

Let $ABCD$ be a convex quadrilateral, and $M$, $N$, and $P$ be the midpoints of diagonals $AC$ and $BD$, and side $AD$, respectively. Also, suppose that $\angle{ABC} + \angle{DCB} = 90$ and that $AB = 6$, $CD = 8$. Calculate the perimeter of triangle $MNP$.

The notation $\phi (n)$ is the number of positive integers smaller than $n$ and coprime with $n$, $\pi (n)$ is the number of primes that do not exceed $n$. Prove that for any natural number $n > 1$, we have $$\phi (n) \ge \frac{\pi (n)}{2}$$

Let $f$ and $g$ be two quadratic polynomials with real coefficients such that the equation $f(g(x)) = 0$ has four distinct real solutions: $112, 131, 146,$ and $a.$ Compute the sum of all possible values of $a.$