Found problems: 85335
In each of the squares of an $n \times n$ board, with $n \ge 3$, a positive integer is written in such a way that the absolute value of the difference of the numbers written in any two neighboring cells is less than or equal to $2$ (two neighboring cells are those that have a common side).
a) Show a $5 \times 5$ board on which $15$ integers have been written different following the indicated rule.
b) Find, as a function of $n$, the maximum number of different numbers that can have the board of $n \times n$ squares.
All the angles of the hexagon $ABCDEF$ are equal. Prove that
\[AB-DE=EF-BC=CD-FA \]
Find all triples $(x,y,z)$ of real numbers that satisfy \[ \begin{aligned} & x\left(1 - y^2\right)\left(1 - z^2\right) + y\left(1 - z^2\right)\left(1 - x^2\right) + z\left(1 - x^2\right)\left(1 - y^2\right) \\ & = 4xyz \\ & = 4(x + y + z). \end{aligned} \]
Nine tiles are numbered $1, 2, 3, \ldots, 9,$ respectively. Each of three players randomly selects and keeps three of the tile, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Let $a_0,a_1,\ldots$ be a sequence of positive integers such that $a_0=1$, and for all positive integers $n$, $a_n$ is the smallest composite number relatively prime to all of $a_0,a_1,\ldots,a_{n-1}$. Compute $a_{10}$.
[i]2020 CCA Math Bonanza Lightning Round #4.2[/i]
An acute triangle $ABC$ is given. Let us denote by $D$ and $E$ the orthogonal projections, respectively of points $ B$ and $C$ on the bisector of the external angle $BAC$. Let $F$ be the point of intersection of the lines $BE$ and $CD$. Show that the lines $AF$ and $DE$ are perpendicular.
The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$.
Let $x, y, z$ be positive real numbers. Prove that $\frac{y^2 + z^2}{x}+\frac{z^2 + x^2}{y}+\frac{x^2 + y^2}{z}\ge 2(x + y + z)$.
Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$
It is known that the angles of the triangle $ABC$ are $1: 3: 5$. Find the angle between the bisector of the largest angle of the triangle and the line containing the altitude drawn to the smallest side of the triangle.
What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?
Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.
It is known that the number of real solutions of the following system if finite. Prove that this system has an even number of solutions:
$(y^2+6)(x-1)=y(x^2+1)$
$(x^2+6)(y-1)=x(y^2+1)$
What non-zero real value for $ x$ satisfies $ (7x)^{14} \equal{} (14x)^7$?
$ \textbf{(A) } \frac 17 \qquad \textbf{(B) } \frac 27 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 14$
Arnold has plates weighing $5$, $15$, $25$, $35$, or $45$ pounds. He lifts a barbell, which consists of a $45$-pound bar and any number of plates that he has. Vlad looks at Arnold's bar and is impressed to see him bench-press $600$ pounds. Unfortunately, Vlad mistook each plate on Arnold's bar for the plate one size heavier, and Arnold was actually lifting $470$ pounds. How many plates did Arnold have on the bar?
[i]2018 CCA Math Bonanza Team Round #2[/i]
Let $ABCDEF$ be a hexagon. Sides and diagonals of hexagon are colored in two colors: blue and yellow. Prove that there exist a triangle with vertices from set $\{A,B,C,D,E,F\}$ which sides are all same colour
Given $n$ points $X_1,X_2,\ldots, X_n$ in the interval $0 \leq X_i \leq 1, i = 1, 2,\ldots, n$, show that there is a point $y, 0 \leq y \leq 1$, such that
\[\frac{1}{n} \sum_{i=1}^{n} | y - X_i | = \frac 12.\]
What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
For a positive integer $n$ define a sequence of zeros and ones to be [i]balanced[/i] if it contains $n$ zeros and $n$ ones. Two balanced sequences $a$ and $b$ are [i]neighbors[/i] if you can move one of the $2n$ symbols of $a$ to another position to form $b$. For instance, when $n = 4$, the balanced sequences $01101001$ and $00110101$ are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set $S$ of at most $\frac{1}{n+1} \binom{2n}{n}$ balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in $S$.
In a triangle $ABC$ with $AB<AC$, $D$ is a point on segment $AC$ such that $BD = CD$.
A line parallel to $BD$ meets segment $BC$ at $E$ and line $AB$ at $F$. Point $G$ is the intersection of $AE$ and $BD$.
Show that $\angle BCG = \angle BCF$. [i](Côte d’Ivoire)[/i]
Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?
Let $n \in \mathbb{N} , n \ge 2$ and $ a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0 $. Then:
[b][size=100][i]P.[/i][/size][/b] $|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|$ for any $z \in \mathbb{C}, |z|=1$
[b][size=100][i]Q[/i][/size][/b]. $a_1=a_2=\cdots=a_{n-1}=0$ and $a_0/a_n \in [0,\infty)$
Prove that $ P \Longleftrightarrow Q$
In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$ positions clockwise from its initial position.
Find the smallest real number \(a\) such that for any positive integer number \(n > 2\) and any arrangement of the numbers from 1 to \(n\) on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than \(a\).
[i]Proposed by Mykhailo Shtandenko[/i]
Find
\[\sum_{n=1}^{\infty} \frac{\varphi(n)}{(-4)^n-1},\]where $\varphi(n)$ is the number of positive integers $k \le n$ relatively prime to $n$. (Note $\varphi(1)=1$.)