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 $A$?

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 $4$?

Is there a power of $2$ such that it is possible to rearrange the digits giving another power of $2$?

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.

Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.

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 $n$.

Let $n$ be a positive integer. Show that \[\sum^{n}_{k=1}\tan^{2}\frac{k \pi}{2n+1}\] is an odd integer.

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 $S$.

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 $a_{1996}$.

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.

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 $1$.

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 $(m,n)$.

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 $n$.

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.

Let $\alpha(n)$ be the number of digits equal to one in the dyadic representation of a positive integer $n$. Prove that 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 $\lim_{i \to \infty} \frac{ \alpha({n_{i}}^2 )}{ \alpha(n_{i}) } = 0$.

Click for solution I assume "dyadic" here means "binary". It is easy to prove by induction on the binary length of numbers that \[ a(x+y)\leq a(x)+a(y).\] Let $ n=2^{b_1} + 2^{b_2} + \dots + 2^{b_m}$ be binary representation of an integer $ n$, where $ b_1<b_2<\dots<b_m$. Then $ a(n)=m$ and \[ n^2 = \sum_{i=1}^m 2^{2b_i} + \sum_{i=1}^m\sum_{j=i+1}^m 2^{b_i+b_j+1},\] implying that a proof for the problem a.: \[ a(n^2)\leq \sum_{i=1}^m 1 + \sum_{i=1}^m\sum_{j=i+1}^m 1=\frac{m(m+1)}{2}.\] To establish b., it is enough to show that there is infinitely many sets of numbers $ b_1<\dots<b_m$ for which $ |\{ 2b_i, b_i+b_j+1 \}|=\frac{m(m+1)}{2}$. It is clear that this requirement holds for any sequence $ b_1<\dots<b_m$ that grows fast enough. For example, any sequence for which $ b_{i+1}>2b_i$ for every $ i=1,\dots,m-1$ works. To prove c., it is enough to take \[ n_i = 2^{2^i-1} - \sum_{j=1}^i 2^{2^i-2^j}.\]

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 $n$.

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

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 \[f(k, p)=a_{k}+a_{k+r(p)}+a_{k+2r(p)}.\] Prove that $S$ is finite. Find the highest value of $f(k, p)$ for $k \ge 1$ and $p \in S$.

Determine all pairs $(a, b)$ of real numbers such that $a\lfloor bn\rfloor =b\lfloor an\rfloor$ for all positive integer $n$.

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 $ra+sb+tc=0$.

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.

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 $2001$?

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.

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.

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.

Which integers have the following property? If the final digit is deleted, the integer is divisible by the new number.

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.

What is the rightmost nonzero digit of $1000000!$?

For how many positive integers $n$ is \[\left( 1999+\frac{1}{2}\right)^{n}+\left(2000+\frac{1}{2}\right)^{n}\] an integer?

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)?

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?

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?

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 $n$.

Counting from the right end, what is the $2500$th digit of $10000!$?

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 $Q(3^{k})>Q(3^{k+1})$.

Click for solution The problem must be $ Q(3^k)\geq Q(3^{k + 1})$. In that case, the solution is rather well-known; suppose the statement doesn't happen. Then there exists some $ n_0$ such that for all integers $ n\geq n_0$, $ Q(3^n) < Q(3^{n + 1})$. But we all know that $ Q(k)\equiv k \pmod {9}$, so basically $ Q(3^n ) \equiv Q(3^{n + 1}) \pmod 9$, therefore $ Q(3^{n + 1})\geq 9 + Q(3^n)$, and by induction we get \[ Q(3^{n + k}) \geq 9k + Q(3^n), \] for all positive integers $ k$. Problem is that if $ k\to\infty$, then $ \lim_{k\to\infty} \frac { 3^{n + k} }{10^k} = 0$, so at some point $ k_0$ we have $ 3^{n + k_0} < 10^{k_0}$, which means that $ 3^{n + k_0}$ has at most $ k_0$ digits, so $ Q(3^{n + k_0})\leq 9\cdot k_0$, which is a contradiction.

Let $n$ and $k$ are integers with $n>0$. Prove that \[-\frac{1}{2n}\sum^{n-1}_{m=1}\cot \frac{\pi m}{n}\sin \frac{2\pi km}{n}= \begin{cases}\tfrac{k}{n}-\lfloor\tfrac{k}{n}\rfloor-\frac12 & \text{if }k|n \\ 0 & \text{otherwise}\end{cases}.\]

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 $n$.