Found problems: 85335
If $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $CDX$ is equilateral, compute $\angle{AXE}$ in degrees.
[i]2020 CCA Math Bonanza Lightning Round #1.3[/i]
There is a unique degree-$10$ monic polynomial with integer coefficients $f(x)$ such that
$$f \left( \sum^9_{j=0}\sqrt[10]{2021^j}\right)= 0.$$
Find the remainder when $f(1)$ is divided by $1000$.
Find $x$ if $\frac{1+\frac{3}{x}}{2-\frac{2}{x}}=7$.
Prove that a graph containing a copy of each possible tree on $n$ vertices as a subgraph has at least $n(\ln n - 2)$ edges.
There are $n$ students $A_1,A_2,...,A_n$ and some of them shaked hands with each other. ($A_i$ and $A-j$ can shake hands more than one time.) Let the student $A_i$ shaked hands $d_i$ times. Suppose $d_1 + d_2 +... + d_n > 0$. Prove that there exist $1 \le i < j \le n$ satisfying the following conditions:
(a) Two students $A_i$ and $A_j$ shaked hands each other.
(b) $\frac{(d_1 + d_2 +... + d_n)^2}{n^2}\le d_id_j$
Mom brought Andriy and Olesya $4$ balls with the numbers $1, 2, 3$ and $4$ written on them (one on each ball). She held $2$ balls in each hand and did not know which numbers were written on the balls in each hand. The mother asked Andriy to take a ball with a higher number from each hand, and then to keep the ball with the lower number from the two balls he took. After that, she asked Olesya to take two other balls, and out of these two, keep the ball with the higher number.
Does the mother know with certainty, which child has the ball with the higher number?
[i]Proposed by Bogdan Rublov[/i]
Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that:
[i](i)[/i] $i$ and $n - i$ always receive the same color, and
[i](ii)[/i] for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$
Prove that all numbers in $N$ must receive the same color.
The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$
Positive real sequences $\{ a_n \}$ and $\{ b_n \}$ satisfy the following conditions for all positive integers $n$.
[list]
[*] $a_{n+1}b_{n+1}= a_n^2 + b_n^2$
[*] $a_{n+1}+b_{n+1}=a_nb_n$
[*] $a_n \geq b_n$
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Prove that there exists positive integer $n$ such that $\frac{a_n}{b_n}>2023^{2023}.$
Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$, find the remainder when $m+n$ is divided by 1000 .
Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$.
(Here we always assume that an angle bisector is a ray.)
[i]Proposed by Sergey Berlov, Russia[/i]
Given positive integers $a,b,$ find the least positive integer $m$ such that among any $m$ distinct integers in the interval $[-a,b]$ there are three pair-wise distinct numbers that their sum is zero.
[i]Proposed by Marian Tetiva, Romania[/i]
Find all non-negative integers $a, b, c$ such that the roots of equations: $\begin{cases}x^2 - 2ax + b = 0 \\
x^2- 2bx + c = 0 \\
x^2 - 2cx + a = 0 \end{cases}$ are non-negative integers.
a) Determine if there exists a convex hexagon $ABCDEF$ with
$$\angle ABD + \angle AED > 180^{\circ},$$
$$\angle BCE + \angle BFE > 180^{\circ},$$
$$\angle CDF + \angle CAF > 180^{\circ}.$$
b) The same question, with additional condition, that diagonals $AD, BE,$ and $CF$ are concurrent.
Consider polynomial functions $ax^2 -bx +c$ with integer coefficients which have two distinct zeros in the open interval $(0,1).$ Exhibit with proof the least positive integer value of $a$ for which such a polynomial exists.
For real numbers $x \ge 0$ and $y \ge 0$, prove the inequality $$x^4+y^3+x^2+y+1 >\frac92 xy.$$
Let \[T=\{(tq,1-t) \in\mathbb R^{2}| t \in [0,1],q\in\mathbb Q\}\]Prove that each continuous function $f: T\longrightarrow T$ has a fixed point.
Prove that any quadratic expression $$Q(x) = Ax^2 + Bx + C$$
(a) can be put into the form $$Q(x) = k \frac{x(x- 1)}{1 \cdot 2} + \ell x + m$$ where $k, \ell, m$ depend on the coefficients $A,B,C$ and
(b) $Q(x)$ takes on integral values for every integer $x$ if and only if $k, \ell, m$ are integers.
Kylie is trying to count to $202250$. However, this would take way too long, so she decides to only write down positive integers from $1$ to $202250$, inclusive, that are divisible by $125$. How many times does she write down the digit $2$?
Thirty two pairs of identical twins are lined up in an $8\times 8$ formation. Prove that it is possible to choose $32 $ persons, one from each pair of twins, so that there is at least one chosen person in each row and in each column
Solve the equation
\[(x+1)\log^2_{3}x+4x\log_{3}x-16=0\]
For real numbers $x$ and $y$ we define $M(x, y)$ to be the maximum of the three numbers $xy$, $(x- 1)(y - 1)$, and $x + y - 2xy$. Determine the smallest possible value of $M(x, y)$ where $x$ and $y$ range over all real numbers satisfying $0 \le x, y \le 1$.
A coin is tossed 9 times. Hence $2^{9}$ different outcomes are possible. In how many cases 2 consecutive heads does not appear?
[list=1]
[*] 34
[*] 55
[*] 89
[*] None of these
[/list]