Practice With Solutions
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Sets $A$ and $B$, shown in the Venn diagram, have the same number of elements. Their union has $2007$ elements and their intersection has $1001$ elements. Find the number of elements in $A$.
How many $3$-digit positive integers have digits whose product equals $24$?
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
How many whole numbers between $1$ and $1000$ do not contain the digit $1$?
If the value of $(9x^2 + k + y^2)$ is a perfect square for any $x$ and $y$, what value can $k$ take?
Evaluate the value of $$\Big(1-\frac{1}{2^2}\Big)\Big(1-\frac{1}{3^2}\Big)\cdots\Big(1-\frac{1}{9^2}\Big)\Big(1-\frac{1}{10^2}\Big)$$
What are the last two digits in the sum of the factorials of the first $100$ positive integers?
Prove: if $a$, $b$, $c$ are all odd integers, then there exists no rational number $x$ which can satisfy the equation $ax^2 + bx + c = 0$.
Prove: randomly select $51$ numbers from $1$, $2$, $3$, $\dots$, $100$, there must exist two numbers for which one is a multiple of the other.
Let four positive integers $a$, $b$, $c$, and $d$ satisfy $a+b+c+d=2019$. Prove $\left(a^3+b^3+c^3+d^3\right)$ cannot be an even number.
Joe wants to measure $6$ liter water using just two containers whose capacities are $27$ liters and $15$ liters, respectively. Can you help him?
Everyday at school, Jo climbs a flight of $6$ stairs. Joe can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs?
What time was it $2011$ minutes after midnight on January $1$, $2011$?
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?
A fair $6$-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?
What is the tens digit of $7^{2019}$?
How many $4$-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of $5$, and $5$ is the largest digit?
In how many ways can $10001$ be written as the sum of two primes?
How many $4$-digit numbers greater than $1000$ are there that use the four digits of $2012$?
What is the units digit of $13^{2019}$?
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of $21$ conference games were played during the $2012$ season, how many teams were members of the BIG N conference?
The smallest number greater than $2$ that leaves a remainder of $2$ when divided by $3$, $4$, $5$, or $6$ lies between what numbers?
A fair coin is tossed $3$ times. What is the probability of at least two consecutive heads?
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?
PolynomialAndEquation
AlgebraBasic
Combinatorics
BasicCountingPrinciple
AdditionPrinciple
MultiplicationPrinciple
InclusionExclusion
IntegerSolution
Recursive (Counting)
Bijection
Probability
NumberTheoryBasic
OddEvenAnalysis
LinearIndeterminateEquation
EndingDigits
MODBasic
ProofByContradiction
PigeonholePrinciple
Symmetry
ManualCountingAndCasework
CircularArrangement