Let $a_1, a_2, \dots$ be a sequence of real numbers satisfying $a_{i+j} \leq a_i+a_j$ for all $i,j=1,2,\dots$. Prove that \[ a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \cdots + \frac{a_n}{n} \geq a_n \] for each positive integer $n$.

Given a finite sequence of real numbers $a_1,a_2,\dots ,a_n$ ($\ast$), we call a segment $a_k,\dots ,a_{k+l-1}$ of the sequence ($\ast$) a “[i]long[/i]”(Chinese dragon) and $a_k$ “[i]head[/i]” of the “[i]long[/i]” if the arithmetic mean of $a_k,\dots ,a_{k+l-1}$ is greater than $1988$. (especially if a single item $a_m>1988$, we still regard $a_m$ as a “[i]long[/i]”). Suppose that there is at least one “[i]long[/i]” among the sequence ($\ast$), show that the arithmetic mean of all those items of sequence ($\ast$) that could be “[i]head[/i]” of a certain “[i]long[/i]” individually is greater than $1988$.

A [u]positive sequence[/u] is a finite sequence of positive integers. [u]Sum of a sequence[/u] is the sum of all the elements in the sequence. We say that a sequence $A$ can be [u]embedded[/u] into another sequence $B$, if there exists a strictly increasing function $$\phi : \{1,2, \ldots, |A|\} \rightarrow \{1,2, \ldots, |B|\},$$ such that $\forall i \in \{1, 2, \ldots ,|A|\}$, $$A[i] \leq B[\phi(i)],$$ where $|S|$ denotes the length of a sequence $S$. For example, $(1,1,2)$ can be embedded in $(1,2,3)$, but $(3,2,1)$ can not be in $(1,2,3)$\\ Given a positive integer $n$, construct a positive sequence $U$ with sum $O(n \, \log \, n)$, such that all the positive sequences with sum $n$, can be embedded into $U$.\\

Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

Two sequences $(a_k)_{k\ge 0}$ and $(b_k)_{k\ge 0}$ of integers are given by $b_k = a_k + 9$ and $a_{k+1} = 8b_k + 8$ for $k\ge 0$. Suppose that the number $1988$ occurs in one of these sequences. Show that the sequence $(a_k)$ does not contain any nonzero perfect square.

Let $n$ be a positive integer. Let us call a sequence $a_1,a_2,\dots,a_n$ interesting if for any $i=1,2,\dots,n$ either $a_i=i$ or $a_i=i+1$. Let us call an interesting sequence even if the sum of its members is even, and odd otherwise. Alice has multiplied all numbers in each odd interesting sequence and has written the result in her notebook. Bob, in his notebook, has done the same for each even interesting sequence. In which notebook is the sum of the numbers greater than by how much? (The answer may depend on $n$.)

A sequence of real numbers $a_0, a_1, \dots$ satisfies the condition\[\sum\limits_{n=0}^{m}a_n\cdot(-1)^n\cdot\dbinom{m}{n}=0\]for all large enough positive integers $m$. Prove that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\ge0$.

Prove that the Fibonacci sequence $\{F_n\}^\infty_{n=1}$ defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for all $n \geq 1$ is a divisibility sequence, that is, if $m\mid n$ then $F_m \mid F_n$ for all positive integers $m$ and $n$.

The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$. Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.

Let $ \left( a_n \right)_{n\ge 1} $ be a sequence of real numbers such that $ a_1>2 $ and $ a_{n+1} =a_1+\frac{2}{a_n} , $ for all natural numbers $ n. $ [b]a)[/b] Show that $ a_{2n-1} +a_{2n} >4 , $ for all natural numbers $ n, $ and $ \lim_{n\to\infty} a_n =2. $ [b]b)[/b] Find the biggest real number $ a $ for which the following inequality is true: $$ \sqrt{x^2+a_1^2} +\sqrt{x^2+a_2^2} +\sqrt{x^2+a_3^2} +\cdots +\sqrt{x^2+a_n^2} > n\sqrt{x^2+a^2}, \quad\forall x\in\mathbb{R} ,\quad\forall n\in\mathbb{N} . $$

Prove that if the numbers $ a_1, a_2,\ldots, a_n $ ($ n $ - natural number $ \geq 2 $) form an arithmetic progression, and none of them is zero, then $$\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \ldots + \frac{1}{a_{n-1}a_n} = \frac{n-1}{a_1a_n}.$$

Let $n \ge 3$ be a fixed integer. The numbers $1,2,3, \cdots , n$ are written on a board. In every move one chooses two numbers and replaces them by their arithmetic mean. This is done until only a single number remains on the board. Determine the least integer that can be reached at the end by an appropriate sequence of moves. (Theresia Eisenkölbl)

There are $mn$ line segments in a plane that connect $n$ given points. Prove that a sequence $V_0$, $V_1$, $...$, $V_m$ of different points can be selected from them such that $V_{i-1}$ and $V_i$ are connected by a line ($1 \le i \le m$).

For a sequence $ a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\}$ we have $ a_0 \equal{} 1$ and \[ a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n} \quad \forall n \in \mathbb{N}.\] Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.

[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].