Found problems: 594
What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?
Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.
Prove that if the numbers $a, b, c$ are related by the relation $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}= \frac{1}{a+b+c}$ then the sum of some two of them is equal to zero.
A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. The domino may be turned around. There is one domino of each type, so $28$ in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino $(3,4)$, total $3 + 4 = 7$. Then $(1,3)$, total $1 + 3 + 3 + 4 = 11$, then $(4,4)$, total $11 + 4 + 4 = 19$. What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?
Let $P$ be an interior point of a circle and $A_1,A_2...,A_{10}$ be points on the circle such that $\angle A_1PA_2 = \angle A_2PA_3 = ... = \angle A_{10}PA_1 = 36^o$. Prove that $PA_1 + PA_3 + PA_5 + PA_7 +PA_9 = PA_2 + PA_4 + PA_6 + PA_8 + PA_{10}$.
a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $
Determine the maximum value of the sum
$ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $
b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ x_1 + x_2 + ... + x_n = m. $
Determine the maximum value of the sum
$ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $
A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?
Calculate the sum: $nx+(n-1)x^2+...+2x^{n-1}+x^n$
Let $a$ and $b$ be two real numbers and let $n$ be a nonnegative integer. Then prove that
$$\sum_{k=0}^{n} {n \choose k} (a + k)^k (b - k)^{n-k} = \sum_{k=0}^{n} \frac{n!}{t!} (a + b)^t $$
Ten distinct numbers are chosen at random from the set $\{1, 2, 3, ... , 37\}$. Show that one can select four distinct numbers out of those ten so that the sum of two of them is equal to the sum of the other two.
Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.
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.
Find all pairs of naturals $a,b$ with $a <b$, such that the sum of the naturals greater than $a$ and less than $ b$ equals $1998$.
Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 6x^2 + 5x + 12$
The sum $|x_1| + |x_2| + |x_3|$ is
(A): $4$ (B): $6$ (C): $8$ (D): $14$ (E): None of the above.
We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$
Calculate $S_n=\sum_{k=1}^{2^n}T(K)$.
Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$.
(a) May it happen that $d = 2$?
(b) May it happen that $d$ is prime?
(V Senderov)
Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$
Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.
Show that $$5^{2019} \mid \Sigma^{5^{2019}}_{k=1}3^{gcd (5^{2019},k)}$$
A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?
Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$.
(E Turkevich)
*. Positive numbers $x_1, x_2, ..., x_{100}$ satisfy the system $$\begin{cases} x^2_1+ x^2_2+ ... + x^2_{100} > 10 000 \\
x_1 + x_2 + ...+ x_{100} < 300 \end{cases}$$
Prove that among these numbers there are three whose sum is greater than $100$.
For an integer $m\ge 3$, let $S(m)=1+\frac{1}{3}+…+\frac{1}{m}$ (the fraction $\frac12$ does not participate in addition and does participate in fractions $\frac{1}{k}$ for integers from $3$ until $m$). Let $n\ge 3$ and $ k\ge 3$ . Compare the numbers $S(nk)$ and $S(n)+S(k)$
.
Give are the integers $a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111$( with $2n$ digits) with $n > 8$ .
Let $q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n$ the remainder of the division of $a_{i}$ by$ 11$ .
Prove that the sum of nine consecutive quotients: $s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}$ is a multiple of $9$ for any $i= 1,2,3, ... , (n-8)$
Let $a, b, c$ be positive real numbers with $abc = 1$. Find all possible values of the expression $$\frac{1 + a}{1 + a + ab}+\frac{1 + b}{1 + b + bc}+\frac{1 + c}{1 + c + ca}$$ can take.