Let $N_{10}$ be the answer to problem 10. Compute the number of ordered pairs of integers $(m,n)$ that satisfy the equation \[ m^2+n^2=mn+N_{10}. \]

5. In acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$ and $\tan C = 4\sqrt{21}/17$, the value of $EF$ may be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by winnertakeover and Monkey_king1[/i]

Let $n$ be a positive integer. Prove that there exist an infinity of multiples of $n$ which do not contain the digit “$9$” in their decimal representation

Prove that there exists a sequence $a(1),a(2),\dots,a(n),\dots$ of real numbers such that \[ a(n+m)\le a(n)+a(m)+\frac{n+m}{\log (n+m)} \] for all integers $m,n\ge 1$, and such that the set $\{a(n)/n:n\ge 1\}$ is everywhere dense on the real line. [i]Remark.[/i] A theorem of de Bruijn and Erdős states that if the inequality above holds with $f(n + m)$ in place of the last term on the right-hand side, where $f(n)\ge 0$ is nondecreasing and $\sum_{n=2}^\infty f(n)/n^2<\infty$, then $a(n)/n$ converges or tends to $(-\infty)$.

Circles $k_1,k_2$ intersect at $B,C$ such that $BC$ is diameter of $k_1$.Tangent of $k_1$ at $C$ touches $k_2$ for the second time at $A$.Line $AB$ intersects $k_1$ at $E$ different from $B$, and line $CE$ intersects $k_2$ at F different from $C$. An arbitrary line through $E$ intersects segment $AF$ at $H$ and $k_1$ for the second time at $G$.If $BG$ and $AC$ intersect at $D$, prove $CH//DF$ .

Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \neq 0 = f(1)$. Assume also that $f$ satisfies equations [b](A)[/b] and [b](B)[/b] below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\ f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)} \end{eqnarray*} for all integers $x,y$. [b](i)[/b] Determine explicitly the set $\big\{f(a)~:~a\in\mathbb{Z}\big\}$. [b](ii)[/b] Assuming that there is a non-zero integer $a$ such that $f(a) \neq 0$, prove that the set $\big\{b~:~f(b) \neq 0\big\}$ is infinite.

An airplane wants to go from a point on the equator, and at each moment it will go to the northeast with speed $v$. Suppose the radius of earth is $R$. a) Will the airplane reach to the north pole? If yes how long it will take to reach the north pole? b) Will the airplne rotate finitely many times around the north pole? If yes how many times?

Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$. Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.

For each natural number $n\geq 2$, solve the following system of equations in the integers $x_1, x_2, ..., x_n$: $$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n$$ where $$S=x_1^2+x_2^2+\dots+x_n^2.$$

If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible? $\textbf{(A) }n$ and $m$ are even $\qquad\textbf{(B) }n$ and $m$ are odd $\qquad\textbf{(C) }n+m$ is even $\qquad\textbf{(D) }n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible

Given that $9\times10\times11\times\cdots\times15=32432400$, what is $1\times3\times5\times\cdots\times15$? [i]2017 CCA Math Bonanza Team Round #1[/i]

suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$.

Given the square table $n\times n$ with $(n-1)$ marked fields. Prove that it is possible to move all the marked fields below the diagonal by moving rows and columns.

If $ (0.2)^x \equal{} 2$ and $ \log 2 \equal{} 0.3010$, then the value of $ x$ to the nearest tenth is: $ \textbf{(A)}\ \minus{} 10.0 \qquad\textbf{(B)}\ \minus{} 0.5 \qquad\textbf{(C)}\ \minus{} 0.4 \qquad\textbf{(D)}\ \minus{} 0.2 \qquad\textbf{(E)}\ 10.0$

Given three identical $n$- faced dice whose corresponding faces are identically numbered with arbitrary integers. Prove that if they are tossed at random, the probability that the sum of the bottom three face numbers is divisible by three is greater than or equal to $\frac{1}{4}$.

A set $X\subset \mathbb{R}$ has dimension zero if, for any $\epsilon > 0$ there exists a positive integer $k$ and intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup \cdots \cup I_k$ with $\sum_{j=1}^k |I_j|^{\epsilon} < \epsilon$. Prove that there exist sets $X,Y \subset [0,1]$ both of dimension zero, such that $X+Y = [0,2].$

Five people of heights $65$, $66$, $67$, $68$, and $69$ inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly $1$ inch taller or exactly $1$ inch shorter than himself?

Find all pairs $(x, y)$ of natural numbers $x$ and $y$ such that $\frac{xy^2}{x+y}$ is a prime

A triangle is given on a sphere of radius $1$, the sides of which are arcs of three different circles of radius $1$ centered in the center of a sphere having less than $\pi$ in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere. A. Zaslavsky

We are given a finite, simple, non-directed graph. Ann writes positive real numbers on each edge of the graph such that for all vertices the following is true: the sum of the numbers written on the edges incident to a given vertex is less than one. Bob wants to write non-negative real numbers on the vertices in the following way: if the number written at vertex $v$ is $v_0$, and Ann's numbers on the edges incident to $v$ are $e_1,e_2,\ldots,e_k$, and the numbers on the other endpoints of these edges are $v_1,v_2,\ldots,v_k$, then $v_0=\sum_{i=1}^k e_iv_i+2022$. Prove that Bob can always number the vertices in this way regardless of the graph and the numbers chosen by Ann. Proposed by [i]Boldizsár Varga[/i], Verőce

Hermione and Ron play a game that starts with $129$ hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven. A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses?

Let $f:\mathbb{R} \to\mathbb{R}$ be a function. For every $a\in\mathbb{Z}$ consider the function $f_a : \mathbb{R} \to\mathbb{R}$, $f_a(x) = (x - a)f(x)$. Prove that if there exist infinitely many values $a\in\mathbb{Z}$ for which the functions $f_a$ are increasing, then the function $f$ is monotonic.

Prove that $2^n + 3^n$ is not a perfect cube for any positive integer $n$.