Found problems: 85335
The six sides of a convex hexagon $A_1 A_2 A_3 A_4 A_5 A_6$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. If $N$ is the number of colorings such that every triangle $A_i A_j A_k$, where $1 \leq i<j<k \leq 6$, has at least one red side, find the sum of the squares of the digits of $N$.
Prove that, for any triangle with side lengths $a,b,c$, the following inequality holds $$\frac{a}{(b+c)^2}+\frac{b}{(c+a)^2}+\frac{c}{(a+b)^2}\geq\frac9{8p}$$ ($p$ denotes the semiperimeter of a triangle)
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that
$$3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),$$
depending only on $r$ and $s.$
Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.
Prove or disprove that $\forall a,b,c,d \in \mathbb{R}^+$ we have the following inequality:
\[3 \leq \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} < \frac{33}{4}\]
$ABCD$ is a parallelogram with $\angle BAD \neq 90$. Circle centered at $A$ radius $BA$ denoted as $\omega _1$ intersects the extended side of $AB,CB$ at points $E,F$ respectively. Suppose the circle centered at $D$ with radius $DA$, denoted as $\omega _2$, intersects $AD,CD$ at points $M,N$ respectively. Suppose $EN,FM$ intersects at $G$, $AG$ intersects $ME$ at point $T$. $MF$ intersects $\omega _1$ at $Q \neq F$, and $EN$ intersects $\omega _2$ at $P \neq N$. Prove that $G,P,T,Q$ concyclic.
The lines joining the incenter of a triangle to the vertices divide the triangle into three triangles. If one of these triangles is similar to the initial one,determine the angles of the triangle.
Are there positive integers $a, b, c$ greater than $2011$ such that:
$(a+ \sqrt{b})^c=...2010,2011...$?
If $a,b\neq 0$ are real numbers such that $a^2b^2(a^2b^2+4)=2(a^6+b^6)$ , then show that $a,b$ can’t be both of them rational .
Initially, there are 2018 distinct boxes on a table. In the first stage, Yazan and Bozan, starting with Yazan, take turns make $2016$ moves each, such that, in each move, the person whose turn selects a pair of boxes that is not written on the board, and writes the pair on the board.
In the second stage, Bozan enumerates the $4032$ pairs with numbers from $1,2,\dots,4032$, in whichever order he wants, and puts $k$ balls in each boxes written contained in the $k^{th}$ pair. Is there a strategy for Bozan that guarantees that the number of balls in each box are distinct?
Let $p$ be a prime such that $3|p+1$. Show that $p|a-b$ if and only if $p|a^3-b^3$
Let $ABC$ be convex quadrilateral and $X$ lying inside it such that $XA \cdot XC^2 = XB \cdot XD^2$ and $\angle AXD + \angle BXC = \angle CXD$. Prove that $\angle XAD + \angle XCD = \angle XBC + \angle XDC$.
Given that positive integers $a,b$ satisfy \[\frac{1}{a+\sqrt{b}}=\sum_{i=0}^\infty \frac{\sin^2\left(\frac{10^\circ}{3^i}\right)}{\cos\left(\frac{30^\circ}{3^i}\right)},\] where all angles are in degrees, compute $a+b$.
[i]2021 CCA Math Bonanza Team Round #10[/i]
Let $P(x)$ a grade 3 polynomial such that:
$$P(1)=1, P(2)=4, P(3)=9$$
Find the value of $P(10)+P(-6)$
We look at an octahedron, a regular octahedron, having painted one of the side surfaces red and the other seven surfaces blue. We throw the octahedron like a die. The surface that comes up is painted: if it is red it is painted blue and if it is blue it is painted red. Then we throw the octahedron again and paint it again according to the above rule. In total we throw the octahedron $10$ times. How many different octahedra can we get after finishing the $10$th time?
[i]Two octahedra are different if they cannot be converted into each other by rotation.[/i]
Find all functions $f\colon \mathbb{R}^+ \to \mathbb{R}^+$ such that
\[
f(xf(x)+y^2) = x^2+yf(y)
\]
for any positive reals $x,y$.
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real $x,y$:
$$ f(x^2) + xf(y) = f(x) f(x + f(y)) \, . $$
Let $ABC$ be an equilateral triangle with circumcircle of center $O$ and radius $R$. Point $M$ is exterior to the triangle such that $S_bS_c = S_aS_b+S_aS_c$, where $S_a, S_b, S_c$ are the areas of triangles $MBC,MCA,MAB$ respectively. Prove that $OM \ge R$.
Nguyễn Tiến Lâm
Find all pairs of prime natural numbers $(p, q)$ for which the value of the expression $\frac{p}{q}+\frac{p+1}{q+1}$ is an integer.
Let $a$, $b$, $c$ be nonnegative reals such that $ab+bc+ca = \frac{1}{3}$. Prove that
\[\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3 \]
If $a,b,c$ are the sides of a triangle and $m_a , m_b, m_c$ are the medians prove that
\[4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)\]
Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying:
(1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$
(2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$
(3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$
Find all positive integers $n$ such that $7^n+147$ is a perfect square.
Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Lewis Chen
[/i]