Let $ x$, $ y$ and $ z$ be non negative numbers. Prove that \[ \frac{x^2\plus{}y^2\plus{}z^2\plus{}xy\plus{}yz\plus{}zx}{6}\le \frac{x\plus{}y\plus{}z}{3}\cdot\sqrt{\frac{x^2\plus{}y^2\plus{}z^2}{3}}\]

If \[ \sum_{k=1}^{1000}\left( \frac{k+1}{k}+\frac{k}{k+1}\right)=\frac{m}{n} \] for relatively prime positive integers $m,n$, compute $m+n$. [i]2020 CCA Math Bonanza Lightning Round #2.4[/i]

Determine all natural numbers $n$ and $k > 1$ such that $k$ divides each of the numbers$\binom{n}{1}$,$\binom{n}{2}$,..........,$\binom{n}{n-1}$

[b]p1.[/b] Solve the equation: $\sqrt{x} +\sqrt{x + 1} - \sqrt{x + 2} = 0$. [b]p2.[/b] Solve the inequality: $\ln (x^2 + 3x + 2) \le 0$. [b]p3.[/b] In the trapezoid $ABCD$ ($AD \parallel BC$) $|AD|+|AB| = |BC|+|CD|$. Find the ratio of the length of the sides $AB$ and $CD$ ($|AB|/|CD|$). [b]p4.[/b] Gollum gave Bilbo a new riddle. He put $64$ stones that are either white or black on an $8 \times 8$ chess board (one piece per each of $64$ squares). At every move Bilbo can replace all stones of any horizontal or vertical row by stones of the opposite color (white by black and black by white). Bilbo can make as many moves as he needs. Bilbo needs to get a position when in every horizontal and in every vertical row the number of white stones is greater than or equal to the number of black stones. Can Bilbo solve the riddle and what should be his solution? [b]p5.[/b] Two trolls Tom and Bert caught Bilbo and offered him a game. Each player got a bag with white, yellow, and black stones. The game started with Tom putting some number of stones from his bag on the table, then Bert added some number of stones from his bag, and then Bilbo added some stones from his bag. After that three players started making moves. At each move a player chooses two stones of different colors, takes them away from the table, and puts on the table a stone of the color different from the colors of chosen stones. Game ends when stones of one color only remain on the table. If the remaining stones are white Tom wins and eats Bilbo, if they are yellow, Bert wins and eats Bilbo, if they are black, Bilbo wins and is set free. Can you help Bilbo to save his life by offering him a winning strategy? [b]p6.[/b] There are four roads in Mirkwood that are straight lines. Bilbo, Gandalf, Legolas, and Thorin were travelling along these roads, each along a different road, at a different constant speed. During their trips Bilbo met Gandalf, and both Bilbo and Gandalf met Legolas and Thorin, but neither three of them met at the same time. When meeting they did not stop and did not change the road, the speed, and the direction. Did Legolas meet Thorin? Justify your answer. PS. You should use hide for answers.

Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that: [LIST] [*] in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and [/*] [*]in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.[/*] [/LIST] [b]Note.[/b] The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$. For $n>1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$ for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.

Let \[f = X^{n}+a_{n-1}X^{n-1}+\ldots+a_{1}X+a_{0}\] be an integer polynomial of degree $n \geq 3$ such that $a_{k}+a_{n-k}$ is even for all $k \in \overline{1,n-1}$ and $a_{0}$ is even. Suppose that $f = gh$, where $g,h$ are integer polynomials and $\deg g \leq \deg h$ and all the coefficients of $h$ are odd. Prove that $f$ has an integer root.

Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was 300/500 = 3/5. The largest possible two-day success ratio that Beta could achieve is $m/n$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Given $m\in\mathbb{Z}^+$. Find all natural numbers $n$ that does not exceed $10^m$ satisfying the following conditions: i) $3|n.$ ii) The digits of $n$ in decimal representation are in the set $\{2,0,1,5\}$.

A square is called [i]proper[/i] if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$. [i]Proposed by Lewis Chen[/i]

Let $S$ be the set of all non-degenerate triangles with integer sidelengths, such that two of the sides are $20$ and $16$. Suppose we pick a triangle, at random, from this set. What is the probability that it is acute?

Find all $f\colon\mathbf{R}\rightarrow\mathbf{R}$ such that \[f(f(x)f(y)) = f(x + y) + f(xy)\] for all $x,y\in\mathbf{R}$.

A man in a rowing boat is at point A at a distance of $2$ kilometers from a straight coastline. By first rowing in to a point P and then strolling along the coast he reaches point B, which is at a distance of $5$ kilometers from C, which is the point on the coast closest to A. The man's speed at rest is $3$ kilometers per hour and his strolling speed is $5$ kilometers per hour. Decide where P should go be placed between C and B so that the man gets from A to B in the shortest possible time.

Let $ a,b,c$ be positive reals with $ ab \plus{} bc \plus{} ca \equal{} 3$. Prove that: \[ \frac {1}{1 \plus{} a^2(b \plus{} c)} \plus{} \frac {1}{1 \plus{} b^2(a \plus{} c)} \plus{} \frac {1}{1 \plus{} c^2(b \plus{} a)}\le \frac {1}{abc}. \]

Place numbers from $1$ to $9$ in the circles of the figure (see Fig. ) so that the sum of four numbers, finding located in the circles at the tops of all squares (there are six of them), was constant , [img]https://cdn.artofproblemsolving.com/attachments/8/8/5fe1e8c5949903dd9500b992c8139277cebe7f.png[/img]

Find $k$ such that $k\pi$ is the area of the region of points in the plane satisfying $$\frac{x^2+y^2+1}{11} \le x \le \frac{x^2+y^2+1}{7}.$$

Find the last two digits of $\tbinom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g. if the last two digits are $05$, just write $5$.)

Suppose a class contains $100$ students. Let, for $1\le i\le 100$, the $i^{\text{th}}$ student have $a_i$ many friends. For $0\le j\le 99$ let us define $c_j$ to be the number of students who have strictly more than $j$ friends. Show that \begin{align*} & \sum_{i=1}^{100}a_i=\sum_{j=0}^{99}c_j \end{align*}

A function $f(x)$ satisfies $f(x)=f\left(\frac{c}{x}\right)$ for some real number $c(>1)$ and all positive number $x$. If $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$, evaluate $\int_1^c \frac{f(x)}{x} dx$

Find all real values of the parameter $a$ for which the system of equations \[x^4 = yz - x^2 + a,\] \[y^4 = zx - y^2 + a,\] \[z^4 = xy - z^2 + a,\] has at most one real solution.

Zan starts with a rational number $\tfrac{a}{b}$ written on the board in lowest terms. Then, every second, Zan adds $1$ to both the numerator and denominator of the latest fraction and writes the result in lowest terms. Zan stops as soon as he writes a fraction of the form $\tfrac{n}{n+1}$, for some positive integer $n$. If $\tfrac{a}{b}$ started in that form, Zan does nothing. As an example, if Zan starts with $\tfrac{13}{19}$, then after one second he writes $\tfrac{14}{20} = \tfrac{7}{10}$, then after two seconds $\tfrac{8}{11}$, then $\tfrac{9}{12} = \tfrac{3}{4}$, at which point he stops. (a) Prove that Zan will stop in less than $b-a$ seconds. (b) Show that if $\tfrac{n}{n+1}$ is the final number, then \[\frac{n-1}{n} < \frac{a}{b} \le \frac{n}{n+1}.\] [i](Proposed by Michael Tang.)[/i]

Natural numbers $m$ and $n$ satisfy $$gcd(m,n)+lcm(m,n) = m+n. $$Prove that one of numbers $m,n$ divides the other.

Compute $\frac{55}{21}\times \frac{28} 5\times \frac 3 2$.

2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.

Let $\dots, a_{-1}, a_0, a_1, a_2, \dots$ be a sequence of positive integers satisfying the folloring relations: $a_n = 0$ for $n < 0$, $a_0 = 1$, and for $n \ge 1$, \[a_n = a_{n - 1} + 2(n - 1)a_{n - 2} + 9(n - 1)(n - 2)a_{n - 3} + 8(n - 1)(n - 2)(n - 3)a_{n - 4}.\] Compute \[\sum_{n \ge 0} \frac{10^n a_n}{n!}.\]

In $\triangle ABC$, $E\in AC$, $D\in AB$, $P=BE\cap CD$. Given that $S\triangle BPC=12$, while the areas of $\triangle BPD$, $\triangle CPE$ and quadrilateral $AEPD$ are all the same, which is $x$. Find the value of $x$.