The product of three consecutive positive integers is $ 8$ times their sum. What is the sum of their squares? $ \textbf{(A)}\ 50 \qquad \textbf{(B)}\ 77 \qquad \textbf{(C)}\ 110 \qquad \textbf{(D)}\ 149 \qquad \textbf{(E)}\ 194$

In a circle with a radius of $1$, $64$ mutually different points are selected. Prove that $10$ mutually different points can be selected from them, which lie in a circle with a radius $\frac12$.

Let $ABCD$ be an arbitrary quadrilateral. Let squares $ABB_1A_2, BCC_1B_2, CDD_1C_2, DAA_1D_2$ be constructed in the exterior of the quadrilateral. Furthermore, let $AA_1PA_2$ and $CC_1QC_2$ be parallelograms. For any arbitrary point $P$ in the interior of $ABCD$, parallelograms $RASC$ and $RPTQ$ are constructed. Prove that these two parallelograms have two vertices in common.

Let $a$, $b$, and $c$ be positive real numbers such that $a^2+b^2+c^2=989$ and $(a+b)^2+(b+c)^2+(c+a)^2=2013$. Find $a+b+c$.

A and B are friends at a summer school. When B asks A for his address, he answers: "My house is on XYZ street, and my house number is a 3-digit number with distinct digits, and if you permute its digits, you will have other 5 numbers. The interesting thing is that the sum of these 5 numbers is exactly 2017. That's all.". After a while, B can determine A's house number. And you, can you find his house number?

Let $n$ be a positive integer. Alice writes $n$ real numbers $a_1, a_2,\dots, a_n$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ($a_n$ is not a neighbor of $a_1$, nor vice versa). A number [i]changes sign[/i] if it changes from being nonnegative to negative or vice versa. In terms of $n$, determine the maximum number of times that $a_1$ can change sign, across all possible values of $a_1,a_2,\dots, a_n$ and all possible sequences of moves Alice may make.

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

In triangle $ABC$, $BC=3, CA=4$, and $AB=5$. For any point $P$ in the same plane as $ABC$, define $f(P)$ as the sum of the distances from $P$ to lines $AB, BC$, and $CA$. The area of the locus of $P$ where $f(P)\leq 12$ is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Yannick Yao[/i]

Evaluate $\displaystyle\sum_{n=1}^\infty \dfrac{1}{n^2+2n}$.

Find all prime numbers $p$ such that there exists a positive integer $n$ where $2^n p^2 + 1$ is a square number.

Given two circles $k$ and $l$ which intersect at two points. One of their common tangents touches $k$ at point $K$, while the other common tangent touches $l$ at $L.$ Let $A$ and $B$ be the intersections of the line $KL$ with the circles $k$ and $l$, respectively. Prove that $\overline{AK} = \overline{BL}.$

We call a positive integer $n$ [i]happy [/i] if there exist integers $a,b$ such that $a^2+b^2 = n$. If $t$ is happy, show that (a) $2t$ is [i]happy[/i], (b) $3t$ is not [i]happy[/i]

Find solution in positive integers : $$n^5+n^4=7^m-1$$

Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? $\text{(A)}\ \text{1 hr} \qquad \text{(B)}\ \text{1 hr 10 min} \qquad \text{(C)}\ \text{1 hr 20 min} \qquad \text{(D)}\ \text{1 hr 40 min} \qquad \text{(E)}\ \text{2 hr}$

If $AG_a,BG_b$, and $CG_c$ are symmedians in $\Delta ABC$ ($G_a\in BC,G_b\in AC,G_c\in AB$), is it possible for $\Delta G_a G_b G_c$ to be equilateral when $\Delta ABC$ is not equilateral?

Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?

Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.

Let [i]Revolution[/i]$(x) = x^3 +Ux^2 +Sx + A$, where $U$, $S$, and $A$ are all integers and $U +S + A +1 = 1773$. Given that [i]Revolution[/i] has exactly two distinct nonzero integer roots $G$ and $B$, find the minimum value of $|GB|$. [i]Proposed by Jacob Xu[/i] [hide=Solution] [i]Solution.[/i] $\boxed{392}$ Notice that $U + S + A + 1$ is just [i]Revolution[/i]$(1)$ so [i]Revolution[/i]$(1) = 1773$. Since $G$ and $B$ are integer roots we write [i]Revolution[/i]$(X) = (X-G)^2(X-B)$ without loss of generality. So Revolution$(1) = (1-G)^2(1-B) = 1773$. $1773$ can be factored as $32 \cdot 197$, so to minimize $|GB|$ we set $1-G = 3$ and $1-B = 197$. We get that $G = -2$ and $B = -196$ so $|GB| = \boxed{392}$. [/hide]

Let $ABC$ be a triangle and $D$ a point on the side $BC$. Point $E$ is the symmetric of $D$ with respect to $AB$. Point $F$ is the symmetric of $E$ with respect to $AC$. Point $P$ is the intersection of line $DF$ with line $AC$. Prove that the quadrilateral $AEDP$ is cyclic. (Malik Talbi)

How many triangles appear in the diagram below? [asy] import graph; size(4.4cm); real labelscalefactor = 0.5; pen dotstyle = black; draw((-2,5)--(-2,1)); draw((-2,5)--(2,5)); draw((2,5)--(2,1)); draw((-2,1)--(2,1)); draw((0,5)--(0,1)); draw((-2,3)--(2,3)); draw((-1,5)--(-1,1)); draw((1,5)--(1,1)); draw((-2,2)--(2,2)); draw((-2,4)--(2,4)); draw((1,5)--(-2,2)); draw((-2,2)--(-1,1)); draw((-1,1)--(2,4)); draw((2,4)--(1,5)); draw((-1,5)--(-2,4)); draw((-2,4)--(1,1)); draw((1,1)--(2,2)); draw((2,2)--(-1,5)); [/asy]

Find the maximal number of points, such that there exist a configuration of $2023$ lines on the plane, with each lines pass at least $2$ points.

Let $ABC$ be an acute triangle, and its altitudes $AX,BY,CZ$ concurrent at $H$. Construct circles $(K_a), (K_b), (K_c)$ circumscribing the triangles $AY Z, BZX, CXY$ . Construct a circle $(K)$ that is internally tangent to all the three circles $(Ka), (K_b), (K_c)$. Prove that $(K)$ is tangent to the circumcircle $(O)$ of the triangle $ABC$. Đỗ Thanh Sơn

Given triangle $ ABC$ with sidelengths $ a,b,c$. Tangents to incircle of $ ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ ABC$ is equal to $ \frac{\pi (a^{2}\plus{}b^{2}\plus{}c^{2})(b\plus{}c\minus{}a)(c\plus{}a\minus{}b)(a\plus{}b\minus{}c)}{(a\plus{}b\plus{}c)^{3}}$ (hmm,, looks familiar, isn't it? :wink: )

Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$. [i]Ravi Jagadeesan.[/i]