Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that $ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$ Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.

Find the smallest constant $ k$ such that $ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$ for all positive $ x$, $ y$, $ z$.

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Consider a polinomial $p \in \mathbb{R}[x]$ of degree $n$ and with no real roots. Prove that $$\int_{-\infty}^{\infty}\frac{(p'(x))^2}{(p(x))^2+(p'(x))^2}dx$$ converges, and is less or equal than $n^{3/2}\pi.$

Let $p_1,p_2,...,p_n$ be $n\ge 2$ fixed positive real numbers. Let $x_1,x_2,...,x_n$ be nonnegative real numbers such that $$x_1p_1+x_2p_2+...+x_np_n=1.$$ Determine the [i](a)[/i] maximal; [i](b)[/i] minimal possible value of $x_1^2+x_2^2+...+x_n^2$.

Let $a,b,c$ be real numbers distinct from $0$ and $1$, with $a+b+c=1$. Prove that \[8\left(\frac{1}{2}-ab-bc-ca\right)\left(\frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} \right)\ge 9 \]

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

Let $n\ge 2$ be a natural number. Let $a_1\le a_2\le a_3\le \cdots \le a_n$ be real numbers such that $a_1+a_2+\cdots +a_n>0$ and $n(a_1^2+a_2^2+\cdots +a_n^2)=2(a_1+a_2+\cdots +a_n)^2.$ If $m=\lfloor n/2\rfloor+1$, the smallest integer larger than $n/2$, then show that $a_m>0.$

Let $x,y{}$ and $z{}$ be positive real numbers satisfying $x+y+z=1$. Prove that [list=a] [*]\[1-\frac{x^2-yz}{x^2+x}=\frac{(1-y)(1-z)}{x^2+x};\] [*]\[\frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leqslant 0.\] [/list]

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

Given $a$, $b$, and $c$ be positive real numbers with $a+b+c=1$, prove that \[\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le2\]

If $a_1$, $a_2$ and $a_3$ are nonnegative real numbers for which $a_1+a_2+a_3=1$, then prove the inequality $a_1\sqrt{a_2}+a_2\sqrt{a_3}+a_3\sqrt{a_1}\leq \frac{1}{\sqrt{3}}$

Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\] for all $x,y\in\mathbb{R}$.

[b]p1.[/b] Let $f$ be a function such that $f(x + y) = f(x) + f(y)$ for all $x$ and $y$. Assume $f(5) = 9$. Compute $f(2015)$. [b]p2.[/b] There are six cards, with the numbers $2, 2, 4, 4, 6, 6$ on them. If you pick three cards at random, what is the probability that you can make a triangles whose side lengths are the chosen numbers? [b]p3. [/b]A train travels from Berkeley to San Francisco under a tunnel of length $10$ kilometers, and then returns to Berkeley using a bridge of length $7$ kilometers. If the train travels at $30$ km/hr underwater and 60 km/hr above water, what is the train’s average speed in km/hr on the round trip? [b]p4.[/b] Given a string consisting of the characters A, C, G, U, its reverse complement is the string obtained by first reversing the string and then replacing A’s with U’s, C’s with G’s, G’s with C’s, and U’s with A’s. For example, the reverse complement of UAGCAC is GUGCUA. A string is a palindrome if it’s the same as its reverse. A string is called self-conjugate if it’s the same as its reverse complement. For example, UAGGAU is a palindrome and UAGCUA is self-conjugate. How many six letter strings with just the characters A, C, G (no U’s) are either palindromes or self-conjugate? [b]p5.[/b] A scooter has $2$ wheels, a chair has $6$ wheels, and a spaceship has $11$ wheels. If there are $10$ of these objects, with a total of $50$ wheels, how many chairs are there? [b]p6.[/b] How many proper subsets of $\{1, 2, 3, 4, 5, 6\}$ are there such that the sum of the elements in the subset equal twice a number in the subset? [b]p7.[/b] A circle and square share the same center and area. The circle has radius $1$ and intersects the square on one side at points $A$ and $B$. What is the length of $\overline{AB}$ ? [b]p8. [/b]Inside a circle, chords $AB$ and $CD$ intersect at $P$ in right angles. Given that $AP = 6$, $BP = 12$ and $CD = 15$, find the radius of the circle. [b]p9.[/b] Steven makes nonstandard checkerboards that have $29$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard? [b]p10.[/b] John is organizing a race around a circular track and wants to put $3$ water stations at $9$ possible spots around the track. He doesn’t want any $2$ water stations to be next to each other because that would be inefficient. How many ways are possible? [b]p11.[/b] In square $ABCD$, point $E$ is chosen such that $CDE$ is an equilateral triangle. Extend $CE$ and $DE$ to $F$ and $G$ on $AB$. Find the ratio of the area of $\vartriangle EFG$ to the area of $\vartriangle CDE$. [b]p12.[/b] Let $S$ be the number of integers from $2$ to $8462$ (inclusive) which does not contain the digit $1,3,5,7,9$. What is $S$? [b]p13.[/b] Let x, y be non zero solutions to $x^2 + xy + y^2 = 0$. Find $\frac{x^{2016} + (xy)^{1008} + y^{2016}}{(x + y)^{2016}}$ . [b]p14.[/b] A chess contest is held among $10$ players in a single round (each of two players will have a match). The winner of each game earns $2$ points while loser earns none, and each of the two players will get $1$ point for a draw. After the contest, none of the $10$ players gets the same score, and the player of the second place gets a score that equals to $4/5$ of the sum of the last $5$ players. What is the score of the second-place player? [b]p15.[/b] Consider the sequence of positive integers generated by the following formula $a_1 = 3$, $a_{n+1} = a_n + a^2_n$ for $n = 2, 3, ...$ What is the tens digit of $a_{1007}$? [b]p16.[/b] Let $(x, y, z)$ be integer solutions to the following system of equations $x^2z + y^2z + 4xy = 48$ $x^2 + y^2 + xyz = 24$ Find $\sum x + y + z$ where the sum runs over all possible $(x, y, z)$. [b]p17.[/b] Given that $x + y = a$ and $xy = b$ and $1 \le a, b \le 50$, what is the sum of all a such that $x^4 + y^4 - 2x^2y^2$ is a prime squared? [b]p18.[/b] In $\vartriangle ABC$, $M$ is the midpoint of $\overline{AB}$, point $N$ is on side $\overline{BC}$. Line segments $\overline{AN}$ and $\overline{CM}$ intersect at $O$. If $AO = 12$, $CO = 6$, and $ON = 4$, what is the length of $OM$? [b]p19.[/b] Consider the following linear system of equations. $1 + a + b + c + d = 1$ $16 + 8a + 4b + 2c + d = 2$ $81 + 27a + 9b + 3c + d = 3$ $256 + 64a + 16b + 4c + d = 4$ Find $a - b + c - d$. [b]p20.[/b] Consider flipping a fair coin $ 8$ times. How many sequences of coin flips are there such that the string HHH never occurs? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].