If $a_1 \le a_2 \le .. \le a_n$ are natural numbers ($n \ge 2$), show that the inequality $$\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0$$ holds for all $n$-tuples $(x_1,...,x_n) \ne (0,..., 0)$ of real numbers if and only if $a_2 \ge 2$.

Let n be a positive integer and $x_1, x_2, ..., x_n > 0$ be real numbers so that $x_1 + x_2 +... + x_n =\frac{1}{x_1^2}+\frac{1}{x_2^2}+...+\frac{1}{x_n^2}$ Show that for each positive integer $k \le n$, there are $k$ numbers among $x_1, x_2, ..., x_n $ whose sum is at least $k$.

Let $\sigma, \tau$ be two permutations of the quantity $\{1, 2,. . . , n\}$. Prove that there is a function $f: \{1, 2,. . . , n\} \to \{-1, 1\}$ such that for any $1 \le i \le j \le n$, we have $\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2$ and $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$

Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

Consider all the tetrahedrons $AXBY$, circumscribed around the sphere. Let $A$ and $B$ points be fixed. Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on points $X$ and $Y$ .

Consider ten concentric circles and ten rays as in the following figure. At the points where the inner circle is intersected by the rays write successively, in direction clockwise, the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$. In the next circle we write the numbers $11, 12, 13, 14, 15, 16, 17, 18, 19,20$ successively, and so on successively until the last round were we write the numbers $91, 92, 93, 94, 95, 96, 97, 98, 99, 100$ successively. In this orde, the numbers $1, 11, 21, 31, 41, 51, 61, 71, 81, 91$ are in the same ray, and similarly for the other rays. In front of $50$ of those $100$ numbers, we use the sign ''$-$'' such as: a) in each of the ten rays, exist exactly $5$ signs ''$-$'' , and also b) in each of the ten concentric circles, to be exactly $5$ signs ''$-$''. Prove that the sum of the $100$ signed numbers that occur, equals zero. [img]https://cdn.artofproblemsolving.com/attachments/9/d/ffee6518fcd1b996c31cf06d0ce484a821b4ae.gif[/img]

Let $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ be real numbers between $1001$ and $2002$ inclusive. Suppose $ \sum_{i=1}^n a_i^2= \sum_{i=1}^n b_i^2$. Prove that $$\sum_{i=1}^n\frac{a_i^3}{b_i} \le \frac{17}{10} \sum_{i=1}^n a_i^2$$ Determine when equality holds.

Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be a) $10$? b) $1$?

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c} \ge 2\sqrt2 \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)$$

The number $665$ is represented as a sum of $18$ natural numbers nenule $a_1, a_2, ..., a_{18}$. Determine the smallest possible value of the smallest common multiple of the numbers $a_1, a_2, ..., a_{18}$.

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

Twenty numbers $a_1,a_2,..,a_{20}$ satisfy: $$a_k \ge 7k \,\,\,\,\, for \,\,\,\,\, k = 1,2,..., 20$$ $$a_1+a_2+...+a_{20}=1518$$ Prove that among the numbers $k = 1,2,... ,20$ there are no more than seventeen, for which $a_k \ge 20k -2k^2$.

Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.

The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$. ( V Senderov)

Consider subsets of the set $1 , 2,..., N$. For each such subset we can compute the product of the reciprocals of each member. Find the sum of all such products.

Given a geometric progression whose denominator $q$ is an integer not equal to $0$ or $-1$, prove that the sum of two or more terms in this progression cannot equal any other term in it.

Let $A$ be a non-empty set of integers with the following property: For each $a \in A$, there exist not necessarily distinct integers $b,c \in A$ so that $a=b+c$. (a) Proof that there are examples of sets $A$ fulfilling above property that do not contain $0$ as element. (b) Proof that there exist $a_1,\ldots,a_r \in A$ with $r \ge 1$ and $a_1+\cdots+a_r=0$. (c) Proof that there exist pairwise distinct $a_1,\ldots,a_r$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.

Does there exist a row of Pascal’s Triangle containing four distinct values $a,b,c$ and $d$ such that $b = 2a$ and $d = 2c$? Recall that Pascal’s triangle is the pattern of numbers that begins as follows [img]https://cdn.artofproblemsolving.com/attachments/2/1/050e56f0f1f1b2a9c78481f03acd65de50c45b.png[/img] where the elements of each row are the sums of pairs of adjacent elements of the prior row. For example, $10 =4+6$. Also note that the last row displayed above contains the four elements $a = 5,b = 10,d = 10,c = 5$, satisfying $b = 2a$ and $d = 2c$, but these four values are NOT distinct.

Is it possible to partition the set $A = \{1, 2, 3, ... , 32, 33\}$ into eleven subsets that contain three integers each, such that for every one of these eleven subsets, one of the integers is equal to the sum of the other two? If so, give such a partition, if not, prove that such a partition cannot exist.

Consider the sequence $a_1, a_2, a_3, \ldots$ with $$ a_n = \frac{1}{n(n+1)}.$$ In how many ways can the number $\frac{1}{1980}$ be represented as the sum of finitely many consecutive terms of this sequence?

For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result? (G Galperin)

An integer is called [i]formidable[/i] if it can be written as a sum of distinct powers of $4$, and [i]successful [/i] if it can be written as a sum of distinct powers of $6$. Can $2005$ be written as a sum of a [i]formidable [/i] number and a [i]successful [/i] number? Prove your answer.

Let $n$ be an integer, $n \ge 2$. For each number $k = 1, 2, ....., n,$ denote by $a _ k$ the number of multiples of $k$ in the set $\{1, 2,. .., n \}$ and let $x _ k = \frac {1} {1} + \frac {1} {2} + \frac {1} {3} _... + \frac {1} {a _ k}$ . Show that: $$\frac {x _ 1 + x _ 2 + ... + x _ n} {n} \le \frac {1} {1 ^ 2} + \frac {1} {2 ^ 2} + ... + \frac {1} {n ^ 2} $$.

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ?