This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 38

PEN S Problems, 14
Tags: inequalities , Miscellaneous Problems
Let $p$ be an odd prime. Determine positive integers $x$ and $y$ for which $x \le y$ and $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is nonnegative and as small as possible.

PEN S Problems, 17
Tags: Miscellaneous Problems
Determine the maximum value of $m^{2}+n^{2}$, where $m$ and $n$ are integers satisfying $m,n\in \{1,2,...,1981\}$ and $(n^{2}-mn-m^{2})^{2}=1.$

PEN S Problems, 28
Tags: Miscellaneous Problems
Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.

PEN S Problems, 34
Tags: modular arithmetic , Miscellaneous Problems
Let $S_{n}$ be the sum of the digits of $2^n$. Prove or disprove that $S_{n+1}=S_{n}$ for some positive integer $n$.

PEN S Problems, 20
Tags: floor function , Miscellaneous Problems
Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a$, $b$, $c$ by \[a=\sqrt[3]{n}, \; b=\frac{1}{a-\lfloor a\rfloor}, \; c=\frac{1}{b-\lfloor b\rfloor}.\] Prove that there are infinitely many such integers $n$ with the property that there exist integers $r$, $s$, $t$, not all zero, such that $ra+sb+tc=0$.

PEN S Problems, 26
Tags: Miscellaneous Problems
Prove that there does not exist a natural number which, upon transfer of its initial digit to the end, is increased five, six or eight times.

PEN S Problems, 6
Tags: algebra , polynomial , induction , complex numbers , Miscellaneous Problems , pen
Suppose that $x$ and $y$ are complex numbers such that \[\frac{x^{n}-y^{n}}{x-y}\] are integers for some four consecutive positive integers $n$. Prove that it is an integer for all positive integers $n$.

PEN S Problems, 15
Tags: inequalities , limit , induction , Miscellaneous Problems
Let $\alpha(n)$ be the number of digits equal to one in the dyadic representation of a positive integer $n$. Prove that [list=a] [*] the inequality $\alpha(n^2 ) \le \frac{1}{2} \alpha(n) (1+\alpha(n))$ holds, [*] equality is attained for infinitely $n\in\mathbb{N}$, [*] there exists a sequence $\{n_i\}$ such that $\lim_{i \to \infty} \frac{ \alpha({n_{i}}^2 )}{ \alpha(n_{i}) } = 0$.[/list]

PEN S Problems, 36
Tags: modular arithmetic , induction , limit , Miscellaneous Problems
For every natural number $n$, denote $Q(n)$ the sum of the digits in the decimal representation of $n$. Prove that there are infinitely many natural numbers $k$ with $Q(3^{k})>Q(3^{k+1})$.

PEN S Problems, 24
Tags: Miscellaneous Problems
A number $n$ is called a Niven number, named for Ivan Niven, if it is divisible by the sum of its digits. For example, $24$ is a Niven number. Show that it is not possible to have more than $20$ consecutive Niven numbers.

PEN S Problems, 4
Tags: quadratics , induction , binomial coefficients , algebra , Miscellaneous Problems
If $x$ is a real number such that $x^2 -x$ is an integer, and for some $n \ge 3$, $x^n -x$ is also an integer, prove that $x$ is an integer.

PEN S Problems, 27
Tags: Miscellaneous Problems
Which integers have the following property? If the final digit is deleted, the integer is divisible by the new number.

PEN S Problems, 11
Tags: Miscellaneous Problems
For each positive integer $n$, prove that there are two consecutive positive integers each of which is the product of $n$ positive integers greater than $1$.