Found problems: 594
Let $S$ be the set of all n-tuples $(X_1,...,X_n)$ of subsets of the set $\{1,2,..,1000\}$, not necessarily different and not necessarily nonempty. For $a = (X_1,...,X_n)$ denote by $E(a)$ the number of elements of $X_1\cup ... \cup X_n$. Find an explicit formula for the sum $\sum_{a\in S} E(a)$
Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ($r > 1, k_i > 0$). Given $m > 1$, we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge 0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$, determine $g(m)$.
Several positive integers $a_0 , a_1 , a_2 , ... , a_n$ are written on a board. On a second board, we write the amount $b_0$ of numbers written on the first board, the amount $b_1$ of numbers on the first board exceeding $1$, the amount $b_2$ of numbers greater than $2$, and so on as long as the $b$s are still positive. Then we stop, so that we do not write any zeros. On a third board we write the numbers $c_0 , c_1 , c_2 , ...$. using the same rules as before, but applied to the numbers $b_0 , b_1 , b_2 , ...$ of the second board. Prove that the same numbers are written on the first and the third boards.
(H. Lebesgue - A Kanel)
Does there exist a subset $E$ of the set $N$ of all positive integers such that none of the elements in $E$ can be presented as a sum of at least two other (not necessarily distinct) elements from $E$ ?
(E. Barabanov)
Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.
Let $m, n, k$ be positive integers with $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$.
Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.
Determine the highest possible value of:
$$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$
where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$.
Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.
We call a positive integer $n$ [i]happy [/i] if there exist integers $a,b$ such that $a^2+b^2 = n$. If $t$ is happy, show that
(a) $2t$ is [i]happy[/i],
(b) $3t$ is not [i]happy[/i]
a) The product of $n$ integers equals $n$, and their sum is zero. Prove that $n$ is divisible by $4$.
b) Let $n$ is divisible by $4$. Prove that there exist $n$ integers such, that their product equals $n$, and their sum is zero.
Let $P$ be an interior point of an equilateral triangle $ABC$, and let $Q,R,S$ be the feet of perpendiculars from $P$ to $AB,BC,CA$, respectively. Show that the sum $PQ+PR+PS$ is independent of the choice of $P$.
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}$ .
Find the least positive integer $k$ making the number $k!\cdot S_{2016}$ an integer.
Let $d_1, d_2, ... , d_k$ be the positive divisors of $n = 1990!$. Show that $\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}$.
There are fi ve distinct real positive numbers. It is known that the total sum of their squares and the total sum of their pairwise products are equal.
(a) Prove that we can choose three numbers such that it would not be possible to make a triangle with sides' lengths equal to these numbers.
(b) Prove that the number of such triples is at least six (triples which consist of the same numbers in different order are considered the same).
Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$. Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.
Fix $a_1, . . . , a_n \in (0, 1)$ and define $$f(I) = \prod_{i \in I} a_i \cdot \prod_{j \notin I} (1 - a_j)$$
for each $I \subseteq \{1, . . . , n\}$. Assuming that $$\sum_{I\subseteq \{1,...,n\}, |I| odd} {f(I)} = \frac12,$$
show that at least one $a_i$ has to be equal to $\frac12$.
(Paolo Leonetti)
Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?
Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.
Answer the following two questions and justify your answers:
(a) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+5^{2012}$?
(b) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+...+2011^{2012}+2012^{2012}$?
In a row there are $51$ written positive integers. Their sum is $100$ . An integer is [i]representable [/i] if it can be expressed as the sum of several consecutive numbers in a row of $51$ integers. Show that for every $k$ , with $1\le k \le 100$ , one of the numbers $k$ and $100-k$ is representable.
Let $a,b,c$ and $d$ be non-negative real numbers such that $a+b+c+d = 4$.
Show that $\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\ge 6$.
Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table.
(R Zhenodarov)
We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.
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 $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$
Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.