$ \lim_{n\to\infty } \frac{1}{2^n}\left( \left( \frac{a}{a+b}+\frac{b}{b+c} \right)^n +\left( \frac{b}{b+c}+\frac{c}{c+a} \right)^n +\left( \frac{c}{c+a}+\frac{a}{a+b} \right)^n \right) ,\quad a,b,c>0 $

Find the real numbers $x$ and $y$ such that $$(x^2 -x +1)(3y^2-2y + 3) -2=0.$$

Let $\Delta ABC$ be a triangle with orthocenter $H$ and $\Gamma$ be the circumcircle of $\Delta ABC$ with center $O$. Consider $N$ the center of the circle that passes through the feet of the heights of $\Delta ABC$ and $P$ the intersection of the line $AN$ with the circle $\Gamma$. Suppose that the line $AP$ is perpendicular to the line $OH$. Prove that $P$ belongs to the reflection of the line $OH$ by the line $BC$.

Let the nonnegative numbers $a_1, a_2,... a_9$, where $a_1 = a_9 = 0$ and let at least one of the numbers is nonzero. Denote the sentence $(P)$: '' For $2 \le i \le 8$ there is a number $a_i$, such that $a_{i - 1} + a_{i + 1} <ka_i $”. a) Show that the sentence $(P)$ is true for $k = 2$. b) Determine whether is the sentence $(P)$ true for $k = \frac{19}{10}$

Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths \[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\] (A triangle is non-degenerate if its vertices are not collinear.) [i]Proposed by Bruno Le Floch, France[/i]

Find all values of the parameter $a$ for which there exist exactly two integer values of $x$ that satisfy the inequality $$x^2+5\sqrt2 x+a<0.$$

Given positive integer $n (n \geq 2)$, find the largest positive integer $\lambda$ satisfying : For $n$ bags, if every bag contains some balls whose weights are all integer powers of $2$ (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least $\lambda$.

Find all pairs of positive integers $x,y$ such that $\frac{4}{x}+\frac{2}{y}=1$

$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.

Students in a class of $n$ students had to solve $2^{n-1}$ problems on an exam. It turned out that for each pair of distinct problems: • there is at least one student who has solved both • there is at least one student who has solved one of them, but not the other. Show that there is a problem solved by all the students in the class.

How many values of $x$ satisfy the equation $$(x^2 - 9x + 19)^{x^2 + 16x + 60 }= 1?$$

$2003$ dollars are placed into $N$ purses, and the purses are placed into $M$ pockets. It is known that $N$ is greater than the number of dollars in any pocket. Is it true that there is a purse with less than $M$ dollars in it?

Let $n \geq 3$ be an integer. Find the minimum degree of one algebraic (polynomial) equation that defines the set of vertices of the correct $n$-gon on plane $R^2$.

A positive integer $n$ is called "foursic'' if there exists a placement of $0$ in the digits of $n$ such that the resulting number a multiple of $4.$ For example, $14$ is foursic because $104$ is a multiple of $4.$ Find the number of two-digit foursic numbers.

The circle inscribed in the triangle $ABC$ touches the side $[AC]$ in the point $K$. Prove that the line connecting the midpoint of the side $[AC]$ with the centre of the circle halves the segment $[BK]$ .

Compute the smallest positive integer that gives a remainder of $1$ when divided by $11$, a remainder of $2$ when divided by $21$, and a remainder of $5$ when divided by $51$. [i]2021 CCA Math Bonanza Lightning Round #3.3[/i]

Find all non-negative integers $n$ so that $\sqrt{n + 3}+ \sqrt{n +\sqrt{n + 3}} $ is an integer.

What mass of the compound $\ce{CrO3}$ $\text{(M = 100.0)}$ contains $4.5\times10^{23}$ oxygen atoms? $ \textbf{(A) }\text{2.25 g}\qquad\textbf{(B) }\text{12.0 g}\qquad\textbf{(C) }\text{25.0 g}\qquad\textbf{(D) }\text{75.0 g}\qquad$

Let the points $A,B,C$ lie on a line in this order. $AB$ is the diameter of semicircle $\omega_1$, $AC$ is the diameter of semicircle $\omega_2$. Assume both $\omega_1$ and $\omega_2$ lie on the same side of $AC$. $D$ is a point on $\omega_2$ such that $BD\perp AC$. A circle centered at $B$ with radius $BD$ intersects $\omega_1$ at $E$. $F$ is on $AC$ such that $EF\perp AC$. Prove that $BC=BF$.

On the fields of a chesstable of dimensions $ n\times n$, where $ n\geq 4$ is a natural number, are being put coins. We shall consider a [i]diagonal[/i] of table each diagonal formed by at least $ 2$ fields. What is the minimum number of coins put on the table, s.t. on each column, row and diagonal there is at least one coin? Explain your answer.

Show that, if the real numbers $a, b, c, A, B, C$ satisfy $$aC -2bB + cA = 0 \ \ and \ \ ac - b^2 > 0,$$ then $$AC - B^2 \le 0.$$

Each edge of a cube is increased by $50 \%$. The percent of increase of the surface area of the cube is: $ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 750 $

Georg has a circular game board with 100 squares labelled $1, 2, . . . , 100$. Georg chooses three numbers $a, b, c$ among the numbers $1, 2, . . . , 99$. The numbers need not be distinct. Initially there is a piece on the square labelled $100$. First, Georg moves the piece $a$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Then he moves the piece $b$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Finally, he moves the piece $c$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Thus he puts a total of $99$ caramels on the board. Georg wins all the caramels on square number $1$. How many caramels can Georg win, at most? [img]https://cdn.artofproblemsolving.com/attachments/d/c/af438e5feadca5b1bfc98ae427f6fc24655e29.png[/img]

Compute $\sin^4\left(7.5^\circ\right)+\sin^4\left(82.5^\circ\right)$. [i]2019 CCA Math Bonanza Lightning Round #2.3[/i]

Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.