Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.

An acute triangle $ABC$ has $AB$ as one of its longest sides. The incircle of $ABC$ has center $I$ and radius $r$. Line $CI$ meets the circumcircle of $ABC$ at $D$. Let $E$ be a point on the minor arc $BC$ of the circumcircle of $ABC$ with $\angle ABE > \angle BAD$ and $E\notin \{B,C\}$. Line $AB$ meets $DE$ at $F$ and line $AD$ meets $BE$ at $G$. Let $P$ be a point inside triangle $AGE$ with $\angle APE=\angle AFE$ and $P\neq F$. Let $X$ be a point on side $AE$ with $XP\parallel EG$ and let $S$ be a point on side $EG$ with $PS\parallel AE$. Suppose $XS$ and $GP$ meet on the circumcircle of $AGE$. Determine the possible positions of $E$ as well as the minimum value of $\frac{BE}{r}$.

When $10^{93}-93$ is expressed as a single whole number, the sum of the digits is $\text{(A)}\ 10 \qquad \text{(B)}\ 93 \qquad \text{(C)}\ 819 \qquad \text{(D)}\ 826 \qquad \text{(E)}\ 833$

Triangle $A'B'C'$ is located inside triangle $ABC$ such that $AB \parallel A'B' $, $BC \parallel B'C'$ and $CA \parallel C'A'$ , and all three sides of these parallel sides are at distance $d$ at each case. Let $O$ and $O'$ be the centers of the inscribed circles of the triangles $ABC$ and $A'B'C'$ and $K$ and $K'$ are the the centers of their circumcircles. Prove that points $O, O', K$ and $K'$ lie on a straight line.

What is the minimum $k{}$ for which among any three nonzero real numbers there are two numbers $a{}$ and $b{}$ such that either $|a-b|\leqslant k$ or $|1/a-1/b|\leqslant k$? [i]Maxim Didin[/i]

In a tetrahedron of volume $V$ the sum of the squares of the lengths of its edges equals $S$. Prove that $$V \le \frac{S\sqrt{S}}{72\sqrt{3}}$$

Let $d\geq 1$ and $n\geq 0$ be integers. Find the number of ways to write down a nonnegative integer in each square of a $d\times d$ grid such that the numbers in any set of $d$ squares, no two in the same row or column, sum to $n$.

On every side of a square $ABCD$, we consider three points different (to each other). a) Find the number of line segments defined with endpoints those points , that do not lie on sides of the square. b) If there are no three of the previous line segments passing through the same point, find how many of the intersection points of those segmens line in the interior of the square.

Let $n$ be a positive integer. For a permutation $a_1, a_2, \dots, a_n$ of the numbers $1, 2, \dots, n$ we define $$b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j$$ We say that the permutation $a_1, a_2, \dots, a_n$ is [i]guadiana[/i] if the sequence $b_1, b_2, \dots, b_n$ does not contain two consecutive equal terms. How many guadiana permutations exist?

Compute the number of triangles of different sizes which contain the gray triangle in the figure below. [asy] size(5cm); real n = 4; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } filldraw((1.5,0.866)--(2,2*0.866)--(2.5,0.866)--cycle, gray); [/asy] [i]Proposed by Nathan Xiong[/i]

$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then the circumference of the track in yards is $\textbf{(A) }400\qquad\textbf{(B) }440\qquad\textbf{(C) }480\qquad\textbf{(D) }560\qquad \textbf{(E) }880$

A lightly damaged rook moves around on a $m \times n$ chessboard by taking turns moves to a horizontal or vertical field. For which $m$ and $n$, is it possible for him to have visited each field exactly once? The starting field counts as visited, squares skipped during a move, however, are not.

Find the sum of all positive integers $n$ with the following properties: [list] [*] $n$ is not divisible by any primes larger than $10$. [*] For some positive integer $k$, the positive divisors of $n$ are \[1=d_1<d_2<d_3\cdots<d_{2k}=n.\] [*] The divisors of $n$ have the property that \[d_1+d_2+\cdots+d_k=3k.\] [/list] [i]2021 CCA Math Bonanza Team Round #7[/i]

Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer. Here $\tau (n)$ denotes the number of positive divisor of $n$.

Let $a_1,a_2,\dots$ be real numbers. Suppose there is a constant $A$ such that for all $n,$ \[\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.\] Prove there is a constant $B>0$ such that for all $n,$ \[\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.\]

In an acute traingle $ABC$ with $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.

In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that (1) $IO$ is perpendicular to $BI$; (2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.

Calculate the following indefinite integrals. [1] $\int \sin x\sin 2x dx$ [2] $\int \frac{e^{2x}}{e^x-1}dx$ [3] $\int \frac{\tan ^2 x}{\cos ^2 x}dx$ [4] $\int \frac{e^x+e^{-x}}{e^x-e^{-x}}dx$ [5] $\int \frac{e^x}{e^x+1}dx$

Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$.

The radius of the circumscribed circle of an acute-angled triangle is $23$ and the radius of its Inscribed circle is $9$. Common external tangents to its ex-circles, other than straight lines containing the sides of the original triangle, form a triangle. Find the radius of its inscribed circle.

Given a triangle $ABC$ where $|BC| = a, |CA| = b$ and $|AB| = c$, prove that the equality $\frac{1}{a + b}+\frac{1}{b + c}=\frac{3}{a + b + c}$ holds if and only if $\angle ABC = 60^o$.

If the number $K = \frac{9n^2+31}{n^2+7}$ is integer, find the possible values of $n \in Z$.

Prove that $n!\cdot(n+1)!\cdot(n+2)!$ divides $(3n)!$ for every integer $n \geq 3$.

Prove that $A=10^{n^3-n+2}$ can be written as a sum of four perfect cubes.

Prove that it is impossible to arrange the numbers $1,2,\ldots,1997$ around a circle in such a way that, if $x$ and $y$ are any two neighboring numbers, then $499\le|x-y|\le997$.