Let $m$ and $n$ be integers greater than $2$, and let $A$ and $B$ be non-constant polynomials with complex coefficients, at least one of which has a degree greater than $1$. Prove that if the degree of the polynomial $A^m-B^n$ is less than $\min(m,n)$, then $A^m=B^n$. [i]Proposed by Tobi Moektijono, Indonesia[/i]

Prove that for all positive integers $n$, $n>1$ the number $\sqrt{ \overline{ 11\ldots 44 \ldots 4 }}$, where 1 appears $n$ times, and 4 appears $2n$ times, is irrational.

Show that for every integer $k \ge 1$ there is a positive integer $n$ such that the decimal representation of $2^n$ contains a block of exactly $k$ zeros, i.e. $2^n = \dots a00 \dots 0b \cdots$ with $k$ zeros and $a, b \ne 0$.

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is $\textbf{(A)}\ \text{500 thousand} \qquad \textbf{(B)}\ \text{5 million} \qquad \textbf{(C)}\ \text{50 million} \qquad \textbf{(D)}\ \text{500 million} \qquad \textbf{(E)}\ \text{5 billion}$

Without using a calculator, prove that $$2^{1995} > 5^{856}$$

What is the imaginary part of the complex number $\frac{-4+7i}{1+2i}$? $\text{(A) }-\frac{1}{2}\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }\frac{7}{2}\qquad\text{(E) }-\frac{18}{5}$

Find the smallest value that the expression can take $$|a-1|+|b-2|+c-3|+|3a+2b+c|$$ ($a$, $b$ and $c$ are arbitrary numbers).

Find all prime numbers $p,q,r,s$ such that their sum is a prime number and $p^2+qs$ and $p^2 +qr$ are squares of integers.

In the acute triangle $ABC$ point $M$ is the midpoint of $AC$ and $N$ is the foot of the height of $A$ on $BC$. Point $D$ is on the circumcircle of triangle $BMN$ such that $AD$ and $BM$ are parallel and $AC$ is between the points $B$ and $D$. Prove that $BD=BC$.

Prove that for any natural number $ n, $ there exists a number having $ n+1 $ decimal digits, namely, $ k_0,k_1,k_2,\ldots ,k_n $, and a $ \text{(n+1)-tuple}, $ say $\left( \epsilon_0 ,\epsilon_1 ,\epsilon_2\ldots ,\epsilon_n \right)\in\{-1,1\}^{n+1} , $ that satisfies: $$ 1\le \prod_{j=0}^n (2+j)^{k_j\cdot \epsilon_j}\le \sqrt[10^n-1]{2} $$ [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

The $100$th degree polynomial $P(x)$ satisfies $P(2^k) = k$ for $k = 0, 1, . . . 100$. Let $a$ denote the leading coefficient of $P(x)$. Find the unique integer $M$ such that $2^M < |a| < 2^{M+1}$. .

Let $a, b, c$ be integers satisfying $ab + bc + ca = 1.$ Prove that $(1+ a^2 )(1+ b^2 )(1+ c^2 )$ is a perfect square.

Welcome to the [b]USAYNO[/b], where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them [b]all[/b] correct, you will receive $\max(0, (n-1)(n-2))$ points. If any of them are wrong (or you leave them all blank), you will receive $0$ points. Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1, 2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive $12$ points if all five answers are correct, 0 points if any are wrong). (a) Does there exist a finite set of points, not all collinear, such that a line between any two points in the set passes through a third point in the set? (b) Let $ABC$ be a triangle and $P$ be a point. The [i]isogonal conjugate[/i] of $P$ is the intersection of the reflection of line $AP$ over the $A$-angle bisector, the reflection of line $BP$ over the $B$-angle bisector, and the reflection of line $CP$ over the $C$-angle bisector. Clearly the incenter is its own isogonal conjugate. Does there exist another point that is its own isogonal conjugate? (c) Let $F$ be a convex figure in a plane, and let $P$ be the largest pentagon that can be inscribed in $F$. Is it necessarily true that the area of $P$ is at least $\frac{3}{4}$ the area of $F$? (d) Is it possible to cut an equilateral triangle into $2017$ pieces, and rearrange the pieces into a square? (e) Let $ABC$ be an acute triangle and $P$ be a point in its interior. Let $D,E,F$ lie on $BC, CA, AB$ respectively so that $PD$ bisects $\angle{BPC}$, $PE$ bisects $\angle{CPA}$, and $PF$ bisects $\angle{APB}$. Is it necessarily true that $AP+BP+CP\ge 2(PD+PE+PF)$? (f) Let $P_{2018}$ be the surface area of the $2018$-dimensional unit sphere, and let $P_{2017}$ be the surface area of the $2017$-dimensional unit sphere. Is $P_{2018}>P_{2017}$? [color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.[/color]

For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$, $G = \sqrt[3]{abc}$, $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that \[ \left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.\]

The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$. Let $k=a-1$. If the $k$-th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$, find the highest possible value of $n$.

Prove that if the numbers $a, b, c$ are the lengths of the sides of some nondegenerate triangle, then the equation $$b^2x^2 + (b^2 + c^2 - a^2) x + c^2 = 0$$ has imaginary roots.

Calculate the $100$th derivative of the function \[\frac{1}{x^2+3x+2}\] at $x=0$ with $10\%$ relative error.

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$? ${ \textbf{(A)}\ 55\qquad\textbf{(B)}\ 89\qquad\textbf{(C)}\ 104\qquad\textbf{(D}}\ 144\qquad\textbf{(E)}\ 273 $

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

Find the sum of all positive integers $n$ such that $1+2+\cdots+n$ divides \[15\left[(n+1)^2+(n+2)^2+\cdots+(2n)^2\right].\]

A square-table of dimensions $ n\times n$, where $ n\in N^*$, is filled arbitrarly with the numbers $ 1,2,...,n^2$ such that every number appears on the table exactly one time. From each row of the table is chosen the least number and then denote by $ x$ the biggest number from the numbers chosen. From each column of the table is chosen the least number and then denote by $ y$ the biggest number from the numbers chosen. The table is called [i]balanced [/i]iff $ x \equal{} y$. How many balanced tables we can obtain?

The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.

Let $T=\text{TNFTPP}$. How many positive integers are within $T$ of exactly $\lfloor \sqrt T\rfloor$ perfect squares? (Note: $0^2=0$ is considered a perfect square.)

Let $k$ be a natural number. For each natural number $n$ we define $f_k (n)$ to be the least number, greater than $kn$, for which $nf_k (n)$ is a perfect square. Prove that $f_k (n)$ is injective.