Simplify $\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$.

Given are $n$ points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with leading coefficient equal to $1$. A parabola is called $good$ if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it). What is the greatest number of $good$ parabolas?

The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$.

$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola. I.Voronovich

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Determine all real numbers $x$ which satisfy \[ x = \sqrt{a - \sqrt{a+x}} \] where $a > 0$ is a parameter.

Let $p > 2$ be a prime number. Prove that there is a permutation $k_1, k_2, ..., k_{p-1}$ of numbers $1,2,...,p-1$ such that the number $1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}$ is divisible by $p$. Note: The numbers $k_1, k_2, ..., k_{p-1}$ are a permutation of the numbers $1,2,...,p-1$ if each of of numbers $1,2,...,p-1$ appears exactly once among the numbers $k_1, k_2, ..., k_{p-1}$.

Let $S$ be a set of positive integers, each of them having exactly $100$ digits in base $10$ representation. An element of $S$ is called [i]atom[/i] if it is not divisible by the sum of any two (not necessarily distinct) elements of $S$. If $S$ contains at most $10$ atoms, at most how many elements can $S$ have?

Let $ABC$ be a triangle. Let $T$ be the point of tangency of the circumcircle of triangle $ABC$ and the $A$-mixtilinear incircle. The incircle of triangle $ABC$ has center $I$ and touches sides $BC,CA$ and $AB$ at points $D,E$ and $F,$ respectively. Let $N$ be the midpoint of line segment $DF.$ Prove that the circumcircle of triangle $BTN,$ line $TI$ and the perpendicular from $D$ to $EF$ are concurrent. [i]Proposed by Diaconescu Tashi, Romania[/i]

A $2$-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number? $\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }9$

Find the number of integer solutions of $||x| - 2023| < 2020$.

The sequence $(x_n)$ is defined by formulas $$ x_1=c,\; x_{n+1} = cx_n + \sqrt{(c^2-1)(x_n^2-1)} \quad\text{ for }\quad n=1,2,\ldots$$ Prove that if $ c $ is a natural number, then all numbers $ x_n $ are natural.

Suppose $a_1,a_2,a_3,a_4$ are distinct integers and $P(x)$ is a polynomial with integer coefficients satisfying $P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1$. (a) Prove that there is no integer $n$ such that $P(n) = 12$. (b) Do there exist such a polynomial and $a_n$ integer $n$ such that $P(n) = 1998$?

[b]8.[/b] Let $(a_n)_{n=1}^{\infty}$ be a sequence of positive numbers and suppose that $\sum_{n=1}^{\infty} a_n^2$ is divergent. Let further $0<\epsilon<\frac{1}{2}$. Show that there exists a sequence $(b_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}b_n^2$ is convergent and $\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}$ for every positive integer $N$. [b](S. 8)[/b]

Prove that if \[ \sum_{n=1}^m a_n \leq Na_m \;(m=1,2,...)\] holds for a sequence $ \{a_n \}$ of nonnegative real numbers with some positive integer $ N$, then $ \alpha_{i+p} \geq p \alpha_i$ for $ i,p=1,2,...,$ where \[ \alpha_i= \sum_{n=(i-1)N+1}^{iN} a_n \;(i=1,2,...)\ .\] [i]L. Leindler[/i]

Denote by $Q$ and $N^*$ the set of all rational and positive integer numbers, respectively. Suppose that $\frac{ax + b}{cx + d} \in Q$ for every $x \in N^*$: Prove that there exist integers $A,B,C,D$ such that $\frac{ax + b}{cx + d}= \frac{Ax + B}{Cx+D}$ for all $x \in N^* $

Acute triangle $\triangle ABC$ with $AB<AC$, circumcircle $\Gamma$ and circumcenter $O$ is given. Midpoint of side $AB$ is $D$. Point $E$ is chosen on side $AC$ so that $BE=CE$. Circumcircle of triangle $BDE$ intersects $\Gamma$ at point $F$ (different from point $B$). Point $K$ is chosen on line $AO$ satisfying $BK \perp AO$ (points $A$ and $K$ lie in different half-planes with respect to line $BE$). Prove that the intersection of lines $DF$ and $CK$ lies on $\Gamma$.

Quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle. Altitude $DE$ in triangle $ABD$ intersects diagonal $AC$ in $F$. Prove that $FB = BC$

Let $ABC$ be a triangle with $\angle A=60^{\circ}$. Point $D, E$ in lines $\overrightarrow{AB}, \overrightarrow{AC}$ respectively satisfies $DB=BC=CE$. ($A,B,D$ lies on this order, and $A,C,E$ likewise) Circle with diameter $BC$ and circle with diameter $DE$ meets at two points $X, Y$. Prove that $\angle XAY\ge 60^{\circ}$

Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$

Let $\mathcal B$ be a family of open balls in $\mathbb R^n$ and $c<\lambda\left(\bigcup\mathcal B\right)$ where $\lambda$ is the $n$-dimensional Lebesgue measure. Show that there exists a finite family of pairwise disjoint balls $\{U_i\}^k_{i=1}\subseteq\mathcal B$ such that $$\sum_{j=1}^k\lambda(U_j)>\frac c{3^n}.$$

Determine all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ so that $\forall x: x\cdot f(\frac x2) - f(\frac2x) = 1$

Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$

Let $\overline{AB}$ be a line segment of length 2, $C_1$ be the circle with diameter $\overline{AB}$, $C_0$ be the circle with radius 2 externally tangent to $C_1$ at $A$, and $C_2$ be the circle with radius 3 externally tangent to $C_1$ at $B$. Duck $D_1$ is located at point $B$, Duck $D_2$ is located on $C_2$, 270 degrees counterclockwise from $B$, and Duck $D_0$ is located on $C_0$, 270 degrees counterclockwise from $A$. At the same time, the ducks all start running counterclockwise around their corresponding circles, with each duck taking the same amount of time to complete a full lap around its circle. When the 3 ducks are first collinear, the distance between $D_0$ and $D_2$ can be expressed as $p\sqrt{q}$. Find $p+q$. [i]2022 CCA Math Bonanza Individual Round #10[/i]

Which of the following cannot be equal to $x^2 + \dfrac 1{4x}$ where $x$ is a positive real number? $ \textbf{a)}\ \sqrt 3 -1 \qquad\textbf{b)}\ 2\sqrt 2 - 2 \qquad\textbf{c)}\ \sqrt 5 - 1 \qquad\textbf{d)}\ 1 \qquad\textbf{e)}\ \text{None of above} $