Find all pairs of integers $(x, y)$ such that $5\times (x^2 + 3)= y^2$.
If we arrange all the square numbers ascendingly as a queue: $1491625364964\cdots$ What is the $612^{th}$ digit?
Mary typed a six-digit number, but the two 1s she typed didn't show. What appeared was 2002. How many different six-digit numbers could she have typed?
How many ordered triples of positive integers $(x, y, z)$ satisfy $(x^y)^z = 64$?
Let $P(x) = kx^3 + 2k^2x^2 + k^3$. Find the sum of all real numbers $k$ for which $x - 2$ is a factor of $P(x)$.
The vertex $E$ of square $EFGH$ is at the center of square $ABC$D. The length of a side of $ABCD$ is 1 and the length of a side of $EFGH$ is 2. Side $EF$ intersects $CD$ at $I$ and $EH$ intersects $AD$ at $J$. If angle $EID = 60^\circ$, what is the area of quadrilateral $EIDJ$?
What is the smallest integer n for which any subset of {1, 2, 3, . . . , 20} of size $n$ must contain two numbers that differ by 8?
Let $f$ be a real-valued function such that $f(x) + 2f(\frac{2002}{x}) = 3x$ for all $x > 0$. Find $f(2)$.
In how many zeros does the number $\frac{2002!}{(1001!)^2}$ end?
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1$, $k_2$, $\dots$ , $k_n$ for which $$k_1^2 + k_2^2 + \cdots +k_n^2 = 2002$$
Under the new AMC 10, 12 scoring method, 6 points are given for each correct answer, 2.5 points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between 0 and 150
can be obtained in only one way, for example, the only way to obtain a score of 146.5 is to have 24 correct answers and one unanswered question. Some scores can be obtained in exactly two ways; for example, a score of 104.5 can
be obtained with 17 correct answers, 1 unanswered question, and 7 incorrect, and also with 12 correct answers and 13 unanswered questions. There are three scores that can be obtained in exactly three ways. What is their sum?
How many three-digit numbers have at least one $2$ and at least one $3$?
Solve the following question in integers $$x^6 + 3x^3 +1 = y^4$$
Solve in positive integers the equation $x^2y + y^2z +z^2x = 3xyz$
Let $A$ and $B$ be two positive integers and $A=B^2$. If $A$ satisfies the following conditions, find the value of $B$:
- $A$'s thousands digit is $4$
- $A$'s tens digit is $9$
- The sum of all $A$'s digits is $19$
Is it possible to find four positive integers such that $2002$ plus the product of any two of them is always a square? If yes, find such four positive integers. If no, explain.
If the middle term of three consecutive integers is a perfect square, then the product of these three numbers is called a $\textit{beautiful}$ number. What is the greatest common divisor of all the $\textit{beautiful}$ numbers?
Find the smallest square whose last three digits are the same but not equal $0$.
Let $\overline{ABCA}$ be a four-digit number. If $\overline{AB}$ is a prime, $\overline{BC}$ is a square, and $\overline{CA}$ is the product of a prime and a greater-than-one square. Find all such $\overline{ABCA}$.
Let $A$ be a two-digit number, multiplying $A$ by 6 yields a three-digit number $B$. The difference of the two five-digit numbers obtained by appending $A$ to the left and right of $B$, respectively, is a perfect square. Find the sum of all such possible $A$s.
Let both $A$ and $B$ be two-digit numbers, and their difference is $14$. If the last two digits of $A^2$ and $B^2$ are the same, what are all the possible values of $A$ and $B$.
Find such a positive integer $n$ such that both $(n-100)$ and $(n-63)$ are square numbers.
Find such a positive integer $n$ such that both $(n+23)$ and $(n-30)$ are square numbers.
Find the smallest positive integer $n$ such that $\frac{12!}{n}$ is a square.
Consider the following $32$ numbers: $1!, 2!, 3!, \cdots, 32!$. If one of them is removed, then the product of the remaining $31$ numbers is a perfect squre. What is that removed number?