Found problems: 54
Determine all even positive integers that can be written as the sum of odd composite positive integers.
John takes seven positive integers $a_1,a_2,...,a_7$ and writes the numbers $a_i a_j$, $a_i+a_j$ and $|a_i -a_j |$ for all $i \ne j$ on the blackboard. Find the greatest possible number of distinct odd integers on the blackboard.
Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .
Let's call a function $f : R \to R$ [i]cool[/i] if there are real numbers $a$ and $b$ such that $f(x + a)$ is an even function and $f(x + b)$ is an odd function.
(a) Prove that every cool function is periodic.
(b) Give an example of a periodic function that is not cool.
Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.
A sequence $a_1, a_2, a_3, \ldots $ is defined as follows: $a_1$ is a positive integer and
$$a_{n+1} = \left\lfloor \frac{3}{2} a_n \right\rfloor +1$$
for all $n \in \mathbb{N}$. Can $a_1$ be chosen in such a way that the first $100000$ terms of the sequence are even, but the $100001$-th term is odd?
Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation
$k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.
Let $n \in N$ and $k$ be such that $1 \le k \le n$. Find the number of ordered $k$-tuples $(a_1, a_2,...,a_k)$ of integers such the $1 \le a_j \le n$, for $1 \le j \le k$ and [u]either [/u] there exist $l,m \in \{1, 2,..., k\}$ such that $l < m$ but $a_l > a_m$ [u]or [/u] there exists $l \in \{1, 2,..., k\}$ such that $a_l - l$ is an odd number.
Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the [i]odd part[/i] of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m > 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.
Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.
If you take a subset of $4002$ numbers from the whole numbers $1$ to $6003$, then there is always a subset of $2001$ numbers within that subset with the following property:
If you order the $2001$ numbers from small to large, the numbers are alternately even and odd (or odd and even).
Prove this.
Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$
Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?
a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$
b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd.
(I. Voronovich)
Let $n$ be a positive integer. Let $G (n)$ be the number of $x_1,..., x_n, y_1,...,y_n \in \{0,1\}$, for which the number $x_1y_1 + x_2y_2 +...+ x_ny_n$ is even, and similarly let $U (n)$ be the number for which this sum is odd. Prove that $$\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.$$
To each three-digit number, Matías added the number obtained by inverting its digits. For example, he added $729$ to the number $927$. Calculate in how many cases the result of the sum of Matías is a number with all its digits odd.
Prove that the polynomial $P (x)$ with integer coefficients, taking odd values for $x = 0$ and $x= 1$, has no integer roots.
Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite.
(V Senderov)
Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .
Let $A$ and $B$ be the number of odd positive integers $n<1000$ for which the number formed by the last three digits of $n^{2015}$ is greater and smaller than $n$, respectively. Prove that $A=B$.
Let $x$ be the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $b$, $c$, $d$ are non-negative integers. Prove that $x$ is odd.
For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$.
Compute $T =r_{100}+ r_{101} + r_{102}+...+r_{200}$
For a positive integer $n$ that is not a power of two, we de fine $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.
Find a pair prime numbers $(p, q)$, $p> q$ of , if any, such that $\frac{p^2 - q^2}{4}$ is an odd integer.
Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.