Found problems: 594
A polynomial $x^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n$ has $n$ distinct real roots $x_1, x_2,...,x_n$, where $n > 1$. The polynomial $nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1}$ has roots $y_1, y_2,..., y_{n_1}$.
Prove that $\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}$
Compute the sum $$\sum_{k=0}^{1273}\frac{1}{1 + tan^{2548}\left(\frac{k\pi}{2548}\right)}$$
Let $ a(k) $ be the largest odd number by which $ k $ is divisible. Prove that
$$
\sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$
Calculate the sum $\sum_{n=1}^{1.000.000}[ \sqrt{n} ]$ .
You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.
Each vertex of a convex polygon has integer coordinates, and no side of this polygon is horizontal or vertical. Prove that the sum of the lengths of the segments of lines of the form $x = m$, $m$ an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form $y = n$, $n$ an integer, that lie within the polygon.
(G Galperin)
Prove that for all positive integers $n$,
$$\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+ ...+ \frac{1}{(2n - 1)2n}=\frac{1}{n + 1}\frac{1}{n + 2}+ ... +\frac{1}{2n}$$
Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that
$$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c} \ge 2\sqrt2 \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)$$
Find the value of the expression
$$f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right)$$
assuming $f(x) =\frac{x^2}{1 + x^2}$ .
For any integer $n\ge2$ evaluate the sum \[\sum_{k=1}^{n^2-1}\bigl\lfloor\sqrt k\bigr\rfloor.\]
Call a positive integer $N \ge 2$ [i]special [/i] if for every k such that $2 \le k \le N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.
$\bullet$ Determine a natural $n$ such that the constant sum $S$ of a magic square of $ n \times n$ (that is, the sum of its elements in any column, or the diagonal) differs as little as possible from $1992$.
$\bullet$ Construct or describe the construction of this magic square.
In the beginning, number $1$ has been written to point $(0,0)$ and $0$ has been written to any other point of integral coordinates. After every second, all numbers are replaced with the sum of the numbers in four neighbouring points at the previous second. Find the sum of numbers in all points of integral coordinates after $n$ seconds.
Initially, the number $10$ is written on the board. In each subsequent moves, you can either
(i) erase the number $1$ and replace it with a $10$, or
(ii) erase the number $10$ and replace it with a $1$ and a $25$ or
(iii) erase a $25$ and replace it with two $10$.
After sometime, you notice that there are exactly one hundred copies of $1$ on the board. What is the least possible sum of all the numbers on the board at that moment?
Let $a_i$, $i=1,2,...,n$ be non-negative real numbers and $\sum_{i=1}^na_i =1$. Find
$\max S=\sum_{i\mid j}a_i a_j $.
Let $T$ be a real number satisfying the property:
For any nonnegative real numbers $a, b, c,d, e$ with their sum equal to $1$, it is possible to arrange them around a circle such that the products of any two neighboring numbers are no greater than $T$.
Determine the minimum value of $T$.
We have thrown $k$ white dice and $m$ black dice. Find the probability that the remainder modulo $7$ of the sum of the numbers on the white dice is equal to the remainder modulo $7$ of the sum of the numbers on the black dice.
In the diagram below $a, b, c, d, e, f, g, h, i, j$ are distinct positive integers and each (except $a, e, h$ and $j$) is the sum of the two numbers to the left and above. For example, $b = a + e, f = e + h, i = h + j$. What is the smallest possible value of $d$?
j
h i
e f g
a b c d
Let $p$ be an odd positive integer. Find all values of the natural numbers $n\ge 2$ for which holds $$\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0$$ where $x_1,x_2,..,x_n$ are any real numbers.
A [i]trio [/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio [/i] contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all [i]trios [/i] with the maximum sum.
Fill in the corners of the square, so that the sum of the numbers in each one of the $5$ lines of the square is the same and the sum of the four corners is $123$.
Are there four positive integers whose sum is $2^{1002}$ and product is $5^{1002}$? Justify your answer.
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.
Find the number of ways to write the number $2013$ as the sum of two integers greater than or equal to zero so that when adding there is no carry over.
Clarification: In the sum $2008+5=2013$ there is carry over from the units to the tens
Let $n$ be a nonnegative integer, and let $p$ be a prime number that is congruent to $7$ modulo $8$. Prove that
$$\sum_{k=1}^{p} \left\{ \frac{k^{2n}}{p} - \frac{1}{2} \right\} = \frac{p-1}{2}$$
Let $S$ be a subset of $\{1,2,3,...,24\}$ with $n(S)=10$. Show that $S$ has two $2$-element subsets $\{x,y\}$ and $\{u,v\}$ such that $x+y=u+v$