Found problems: 85335
Given a $101\times 200$ sheet of graph paper, we start moving from a corner square in the direction of the square’s diagonal (not the sheet’s diagonal) to the border of the sheet, then change direction obeying the laws of light’s reflection. Will we ever reach a corner square?
[img]https://cdn.artofproblemsolving.com/attachments/b/8/4ec2f4583f406feda004c7fb4f11a424c9b9ae.png[/img]
Let $ k$ and $ m$ be positive integers such that $ \displaystyle\frac12\left(\sqrt{k\plus{}4\sqrt{m}}\minus{}\sqrt{k}\right)$ is an integer.
(a) Prove that $ \sqrt{k}$ is rational.
(b) Prove that $ \sqrt{k}$ is a positive integer.
An integer is given $N> 1$. Arne and Britt play the following game:
(1) Arne says a positive integer $A$.
(2) Britt says an integer $B> 1$ that is either a divisor of $A$ or a multiple of $A$. ($A$ itself is a possibility.)
(3) Arne says a new number $A$ that is either $B - 1, B$ or $B + 1$.
The game continues by repeating steps 2 and 3. Britt wins if she is okay with being told the number $N$ before the $50$th has been said. Otherwise, Arne wins.
a) Show that Arne has a winning strategy if $N = 10$.
b) Show that Britt has a winning strategy if $N = 24$.
c) For which $N$ does Britt have a winning strategy?
What is the minimum distance between $(2019, 470)$ and $(21a - 19b, 19b + 21a)$ for $a, b \in Z$?
A connected graph is given. Prove that its vertices can be coloured
blue and green and some of its edges marked so that every two vertices
are connected by a path of marked edges, every marked edge connects
two vertices of different colour and no two green vertices are connected
by an edge of the original graph.
Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$. For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $$f(x_1+x_2)\ge f(x_1)+f(x_2).$$
a) Prove that for all $x$ holds $f(x) \le 2x$.
b) Is the inequality $f(x) \le 1.9x$ valid?
A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$.
Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I.$ Let the internal angle bisectors of $\angle A,\angle B,\angle C$ meet $\Gamma$ in $A',B',C'$ respectively. Let $B'C'$ intersect $AA'$ at $P,$ and $AC$ in $Q.$ Let $BB'$ intersect $AC$ in $R.$ Suppose the quadrilateral $PIRQ$ is a kite; that is, $IP=IR$ and $QP=QR.$ Prove that $ABC$ is an equilateral triangle.
What is the smallest number ?
(A) $3$ (B) $2^{\sqrt2}$ (C) $2^{1+\frac{1}{\sqrt2}}$ (D) $2^{\frac12} + 2^{\frac23}$ (E) $2^{\frac53}$
Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality
\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]
holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.
Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k,$ the integer
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016.$
Determine all functions $f : R \to R$ such that $f(xy + f(x)) = xf(y) + f(x)$ holds for any $x, y \in R$.
Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.
Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$.
[i]2021 CCA Math Bonanza Lightning Round #2.4[/i]
Determine all pairs $ (m,n)$ of natural numbers for which $ m^2\equal{}nk\plus{}2$ where $ k\equal{}\overline{n1}$.
EDIT. [color=#FF0000]It has been discovered the correct statement is with $ k\equal{}\overline{1n}$.[/color]
Find all pairs of complex numbers $(z,w) \in \mathbb{C}^2$ such that the relation \[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \] holds for all positive integers $n$.
Find the smallest positive integer $n\neq 2004$ for which there exists a polynomial $f\in\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions.
The least value of the function $ ax^2\plus{}bx\plus{}c$ with $a>0$ is:
$\textbf{(A)}\ -\dfrac{b}{a} \qquad
\textbf{(B)}\ -\dfrac{b}{2a} \qquad
\textbf{(C)}\ b^2-4ac \qquad
\textbf{(D)}\ \dfrac{4ac-b^2}{4a}\qquad
\textbf{(E)}\ \text{None of these}$
Let $n$ be a fixed natural number.
Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have
\[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]
Let $p = 2^{16}+1$ be an odd prime. Define $H_n = 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. Compute the remainder when \[ (p-1)! \sum_{n = 1}^{p-1} H_n \cdot 4^n \cdot \binom{2p-2n}{p-n} \] is divided by $p$.
[i]Proposed by Yang Liu[/i]
Geri plays chess against himself. White has a 5% chance of winning, Black has a 5% chance of winning, and there is a 90% chance of a draw. What is the expected number of games Geri will have to play against himself for one of the colors to win four times?
[i]Proposed by Matthew Weiss
For which $n \in N$ is it possible to arrange a tennis tournament for doubles with $n$ players such that each player has every other player as an opponent exactly once?
$\boxed{A1}$ Find all ordered triplets of $(x,y,z)$ real numbers that satisfy the following system of equation $x^3=\frac{z}{y}-\frac {2y}{z}$ $y^3=\frac{x}{z}-\frac{2z}{x}$ $z^3=\frac{y}{x}-\frac{2x}{y}$
In the table are written the positive integers $1, 2,3,...,2018$. John and Mary have the ability to make together the following move:
[i]They select two of the written numbers in the table, let $a,b$ and they replace them with the numbers $5a-2b$ and $3a-4b$.[/i]
John claims that after a finite number of such moves, it is possible to triple all the numbers in the table, e.g. have the numbers: $3, 6, 9,...,6054$. Mary thinks a while and replies that this is not possible. Who of them is right?
Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]