Let $a$, $b$, $c$ be real numbers with $|a-b|=3$, $|b-c|=4$, and $|c-d|=5$. What is the sum of all possible values of $|a-d|$?
What is the greatest integer less than or equal to $$\dfrac{5^{200}+3^{200}}{5^{197}+3^{197}}$$
How many negative integers can be written in the form of $$a_4\cdot 5^4+a_3\cdot 5^3 +a_2\cdot 5^2+a_2\cdot 5^1+a_0\cdot 5^0$$ where $a_i \in \{-2,-1,0,1,2\}$ for $0\le i \le 4$.
What is the sum of all real numbers $x$ for which the median of the numbers $4$, $6$, $8$, $17$, and $x$ is equal to the mean of those five numbers?
What is the remainder when $2024^{4050}+2025$ is divided by $2024^{2025}+2024^{1013}+1$?
The number $2023$ is expressed in the form $$2023=\dfrac{a_1!a_2!…a_m!}{b_1!b_2!…b_n!}$$ where $a_1\ge a_2\ge … \ge a_m$ and $b_1\ge b_2\ge …\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
Two increasing sequences of positive integers have different first terms. Each sequence has the property that each term beginning with the third term is the sum of the previous two terms, and the 7th term of each sequence is $N$. What is the smallest possible value of $N$?
In a certain card game, a player is dealt a hand of 13 cards from a deck of 52 distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $A35013559A00$. What is the digit of $A$?
In the following equation, each letter represents uniquely a different digit in base 10: $$XY\times YX=X20Y$$ Find the value of $XY$.
A digital clock is malfunctioning as it always skips digit 3. For example, if the clock is displaying $04:29$, the next minute its display would show $04:40$. The clock was reset as $00:00$ at midnight. Later that day, its display shows the current time as $12:56$. What should be the correct time?
