Practice (90/1000)
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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?
There exist $5$ consecutive positive integers such that their sum is a square, and the sum of the middle three is a cube. What is the smallest one of these five numbers?
There are four wolves standing on the four corners of a square, and a rabbit standing at the center of that square. If a wolf can run at $1.4$ times of the rabbit's speed, but can only move along the sides of this square, can the rabbit escape to outside the square?
Let $a$ and $b$ be non-negative real numbers such that $a + b = 2$. Show that: $$\frac{1}{a^2+1}+\frac{1}{b^2 +1} \le \frac{2}{ab+1}$$
Let integers $a$, $b$ and $c$ satisfy $a + b + c = 0$, show that $\vert{a^3 + b^3 + c^3}\vert$ cannot be a prime number.
A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, respectively. How many rounds will there be in the game?
A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
