Sharing your mathematics in MathYouLike

Find $n\in N$ such that: $\left(n2^{2n}+1\right)⋮3$

Prove that: $3^{2n+3}+40n-27⋮64\left(\forall n\in N\right)$

Prove that: $\left(16^n-15n-1\right)⋮225$ with n $\in$ N.

Calculate: $34^2+66^2+68.66$ in a quick way

Prove that: $55^{n+1}-55^n⋮54\left(\forall n\in N\right)$

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Question: Jennie Weiner has p pennies, n nickels, d dimes and q quarters with a total value of $1.08. If the numbers p, n, d and q are distinct and positive, and the greatest common divisor of each pair of these numbers is 1, what is the least possible value of p + n + d + q?

A seesaw is in balance when the weight on one side of the fulcrum times its distance from the fulcrum is equal to the weight on the other side of the fulcrum times its distance from the fulcrum. Shandra weighs 96 pounds. Her little sister weighs 72 pounds. The seesaw at their playground has a beam with seats 14 feet apart. The position of the fulcrum can be adjusted as required. If each girl sits in her seat, how far should the fulcrum be from Shandra’s seat to achieve perfect balance with her sister?

Let S be the set of all integers N such that both N and the number formed by reversing the digits of N are three-digit perfect squares. What is the sum of the integers in S?

What is the greatest possible absolute difference between the median and mean of a list of 10 positive integers that are at most 20? Express your answer as a decimal to the nearest tenth.

Jeffrey Pribble needs to buy 6 pairs of socks. The Sock Shop is running a limited time promotion: buy 3 pairs of socks and get 3 pairs at half off the regular price. What percent savings does Jeffrey get with the promotion compared to the regular price without the promotion?

The number of seats per row in an auditorium increases from the front to the back. The first row has 15 seats, the second row has 2 more seats than the first row, the third row has 3 more seats than the second row, the fourth row has 2 more seats than the third row, the fifth row has 3 more seats than the fourth row. This pattern continues, with successive rows alternating between 2 more seats and then 3 more seats than the previous row. How many seats are in the auditorium if there are 30 rows total?

Let there be a pile of $19$ stones. You take the following actions repeatedly until you can no longer repeat them.

1. If you have a pile of $x>1$ stones, you may divide it into piles of $k<x$ and $x-k$ stones.
2. Add $k\cdot (x-k)$ to your running total.

What is the minimum and maximum total you can achieve? Prove that this is the minimum and maximum, and generalize this to any pile of $n$ stones.

Circles $w_1$ and $w_2$ intersect at $A_1$ and $A_2$ . Circles $w_1$ and $w_3$ intersect at $B_1$ and $B_2$ . Circles $w_2$ and $w_3$ intersect at $C_1$ and $C_3$ . Let $P$ be a point on $w_1$ on arc $A_1B_1$ not including $B_2$ or $A_2$ . Let line $PA_1$ intersect $w_2$ again at $X$ , and $PB_1$ intersect $w_3$ again at $Y$ . Prove that $X, C_1, Y$ are collinear if and only if $A_2, B_2, C_2$ are all the same point.

Granny has 10 grandchildren, all of different ages. Alice is the eldest. The sum of the ages of all grandchildren is 180. At least how old is Alice?

Rabbit Vasya loves cabbage and carrots. In a day, he eats either 9 carrots, or 2 heads of cabbage, or 1 head of cabbage and 4 carrots. But some days he only eats grass. Over the last 10 days, Vasya ate a total of 30 carrots and 9 heads of cabbage. On how many of these 10 days did he eat only grass?

Andy made a playlist with 5 songs (A, B, C, D and E). Song A is 3 minutes long, song B is 2 minutes and 30 seconds, song C is 2 minutes, song D is 1 minute and 30 seconds, and song E is 4 minutes. The songs play in this order in a continuous loop. If Andy left the house when song C was just starting, what song was playing when he got back home 1 hour later?

Describe all constants $\color[rgb]{0.35,0.35,0.35}a$ and $\color[rgb]{0.35,0.35,0.35}b$ such that $\color[rgb]{0.35,0.35,0.35}f(x) = \dfrac{2x + a}{bx - 2}$ and $\color[rgb]{0.35,0.35,0.35}f(x) = f^{-1}(x)$ for all $\color[rgb]{0.35,0.35,0.35}x$ in the domain of $\color[rgb]{0.35,0.35,0.35}f$ .