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.

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Found problems: 38

PEN S Problems, 29
Tags: algorithm , modular arithmetic , Miscellaneous Problems
What is the rightmost nonzero digit of $1000000!$?

PEN S Problems, 12
Tags: Miscellaneous Problems
Let \[\begin{array}{cccc}a_{1,1}& a_{1,2}& a_{1,3}& \dots \\ a_{2,1}& a_{2,2}& a_{2,3}& \dots \\ a_{3,1}& a_{3,2}& a_{3,3}& \dots \\ \vdots & \vdots & \vdots & \ddots \end{array}\] be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that $a_{m,n}> mn$ for some pair of positive integers $(m,n)$.

PEN S Problems, 16
Tags: modular arithmetic , Miscellaneous Problems
Show that if $a$ and $b$ are positive integers, then \[\left( a+\frac{1}{2}\right)^{n}+\left( b+\frac{1}{2}\right)^{n}\] is an integer for only finitely many positive integer $n$.

PEN S Problems, 22
Tags: Miscellaneous Problems
The decimal expression of the natural number $a$ consists of $n$ digits, while that of $a^3$ consists of $m$ digits. Can $n+m$ be equal to $2001$?

PEN S Problems, 37
Tags: trigonometry , floor function , Miscellaneous Problems
Let $n$ and $k$ are integers with $n>0$. Prove that \[-\frac{1}{2n}\sum^{n-1}_{m=1}\cot \frac{\pi m}{n}\sin \frac{2\pi km}{n}= \begin{cases}\tfrac{k}{n}-\lfloor\tfrac{k}{n}\rfloor-\frac12 & \text{if }k|n \\ 0 & \text{otherwise}\end{cases}.\]

PEN S Problems, 2
Tags: Miscellaneous Problems
It is given that $2^{333}$ is a $101$-digit number whose first digit is $1$. How many of the numbers $2^k$, $1 \le k \le 332$, have first digit $4$?

PEN S Problems, 8
Tags: arithmetic sequence , Miscellaneous Problems
The set $S=\{ \frac{1}{n} \; \vert \; n \in \mathbb{N} \}$ contains arithmetic progressions of various lengths. For instance, $\frac{1}{20}$, $\frac{1}{8}$, $\frac{1}{5}$ is such a progression of length $3$ and common difference $\frac{3}{40}$. Moreover, this is a maximal progression in $S$ since it cannot be extended to the left or the right within $S$ ($\frac{11}{40}$ and $\frac{-1}{40}$ not being members of $S$). Prove that for all $n \in \mathbb{N}$, there exists a maximal arithmetic progression of length $n$ in $S$.

PEN S Problems, 35
Tags: modular arithmetic , Miscellaneous Problems
Counting from the right end, what is the $2500$th digit of $10000!$?

PEN S Problems, 33
Tags: modular arithmetic , Miscellaneous Problems
Four consecutive even numbers are removed from the set \[A=\{ 1, 2, 3, \cdots, n \}.\] If the arithmetic mean of the remaining numbers is $51.5625$, which four numbers were removed?

2004 Vietnam National Olympiad, 3
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, 18
Tags: Miscellaneous Problems
Denote by $S$ the set of all primes $p$ such that the decimal representation of $\frac{1}{p}$ has the fundamental period of divisible by $3$. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write \[\frac{1}{p}= 0.a_{1}a_{2}\cdots a_{3r}a_{1}a_{2}\cdots a_{3r}\cdots,\] where $r=r(p)$. For every $p \in S$ and every integer $k \ge 1$ define \[f(k, p)=a_{k}+a_{k+r(p)}+a_{k+2r(p)}.\] [list=a] [*] Prove that $S$ is finite. [*] Find the highest value of $f(k, p)$ for $k \ge 1$ and $p \in S$.[/list]

PEN S Problems, 21
Tags: Miscellaneous Problems
Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left.

PEN S Problems, 30
Tags: Miscellaneous Problems
For how many positive integers $n$ is \[\left( 1999+\frac{1}{2}\right)^{n}+\left(2000+\frac{1}{2}\right)^{n}\] an integer?

PEN S Problems, 10
Tags: geometry , perimeter , Miscellaneous Problems
Let $p$ be an odd prime. Show that there is at most one non-degenerate integer triangle with perimeter $4p$ and integer area. Characterize those primes for which such triangle exist.

PEN S Problems, 38
Tags: function , trigonometry , calculus , integration , Miscellaneous Problems
The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.

PEN S Problems, 23
Tags: Miscellaneous Problems
Observe that \[\frac{1}{1}+\frac{1}{3}=\frac{4}{3}, \;\; 4^{2}+3^{2}=5^{2},\] \[\frac{1}{3}+\frac{1}{5}=\frac{8}{15}, \;\; 8^{2}+{15}^{2}={17}^{2},\] \[\frac{1}{5}+\frac{1}{7}=\frac{12}{35}, \;\;{12}^{2}+{35}^{2}={37}^{2}.\] State and prove a generalization suggested by these examples.

PEN S Problems, 9
Tags: Miscellaneous Problems
Suppose that \[\prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1+a_{1}x^{k_{1}}+a_{2}x^{k_{2}}+\cdots+a_{m}x^{k_{m}}\] where $a_{1}$, $a_{2}$,..., $a_{m}$ are nonzero and $k_{1}< k_{2}< \cdots < k_{m}$. Find $a_{1996}$.

PEN S Problems, 3
Tags: Miscellaneous Problems
Is there a power of $2$ such that it is possible to rearrange the digits giving another power of $2$?

PEN S Problems, 5
Tags: algebra , polynomial , number theory , greatest common divisor , Miscellaneous Problems
Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.

PEN S Problems, 1
Tags: number theory , Diophantine equation , Miscellaneous Problems
a) Two positive integers are chosen. The sum is revealed to logician $A$, and the sum of squares is revealed to logician $B$. Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` Now I can tell what they are.' What are the two numbers? b) When $B$ first says that he cannot tell what the two numbers are, $A$ receives a large amount of information. But when $A$ first says that he cannot tell what the two numbers are, $B$ already knows that $A$ cannot tell what the two numbers are. What good does it do $B$ to listen to $A$?

PEN S Problems, 32
Tags: calculus , integration , Miscellaneous Problems
Alice and Bob play the following number-guessing game. Alice writes down a list of positive integers $x_{1}$, $\cdots$, $x_{n}$, but does not reveal them to Bob, who will try to determine the numbers by asking Alice questions. Bob chooses a list of positive integers $a_{1}$, $\cdots$, $a_{n}$ and asks Alice to tell him the value of $a_{1}x_{1}+\cdots+a_{n}x_{n}$. Then Bob chooses another list of positive integers $b_{1}$, $\cdots$, $b_{n}$ and asks Alice for $b_{1}x_{1}+\cdots+b_{n}x_{n}$. Play continues in this way until Bob is able to determine Alice's numbers. How many rounds will Bob need in order to determine Alice's numbers?

PEN S Problems, 13
Tags: function , inequalities , induction , Miscellaneous Problems , pen
The sum of the digits of a natural number $n$ is denoted by $S(n)$. Prove that $S(8n) \ge \frac{1}{8} S(n)$ for each $n$.

PEN S Problems, 19
Tags: floor function , calculus , integration , induction , Miscellaneous Problems
Determine all pairs $(a, b)$ of real numbers such that $a\lfloor bn\rfloor =b\lfloor an\rfloor$ for all positive integer $n$.

PEN S Problems, 7
Tags: trigonometry , Miscellaneous Problems
Let $n$ be a positive integer. Show that \[\sum^{n}_{k=1}\tan^{2}\frac{k \pi}{2n+1}\] is an odd integer.

PEN S Problems, 31
Tags: Miscellaneous Problems
Is there a $3 \times 3$ magic square consisting of distinct Fibonacci numbers (both $f_{1}$ and $f_{2}$ may be used; thus two $1$s are allowed)?