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Given a 1*1 square.
Find the area of the subset of points that are closer to its center than to any of the square’s sides.
Find the sum of all of the three digit positive integers with three distinct digits.
In the problem Cables provided by K Sengupta, the solution required the knowledge that a hanging flexible cable takes the form of a catenary, the general equation of which is y=a*cosh(x/a), combined with calculating the overall length of the catenary between the two fixed points of the cable.
This time we have a cable strung between two towers on a flat level plane. The low point of the cable is 10m above the plane. Later, the cable heats up and expands to a length of 80m. The towers also expand and gain 0.2m in height – they are now exactly 50m high. The low point of the cable is still 10m above the plane. How far apart are the towers?
Let x+1/x = √(47)
Find the value of:
x2023 - 2023*x2019 + x2015
What set of positive integers with sum 2024 has highest possible product?
Same question for 2025.
What is the highest product in this century?
Inspired by: Putnam 1979
Find all possible non-negative integer solution (x,y) of the following equation
x! + 2y = (x+1)!
There are infinitely many triplets (n, n+1, n+2) such that each member of the triplet is a square or a sum of two squares.
Prove it.
Source: Putnam 2000
Best-Of-Five
(in General)
Rating: 5.00
In a best-of-five tennis match Pete Shamprowess led André Allgassy the whole way during the first two sets. André won more games in the first set than in the second. Pete won the first game in the third set. Pete won the match, even though André won 25% more games.
What were the scores in each of the 5 sets?
Find the smallest distinct whole numbers, M and N such that you can rearrange the digits of M to get N, you can rearrange the digits of M2 to get N2, you can rearrange the digits of M3 to get N3, and where M does not contain the digit zero.
Find positive integers X, Y, and Z, where:
· X and Y each contain exactly three 3s, and
· X * Y = Z, and
· Z is the smallest positive integer containing exactly nine 9s while also meeting the other conditions.
For each positive integer n, S(n) is defined to be the greatest integer such that, for every positive integer k ≤ S(n), n^2 can be written as the sum of k positive square integers.
(a) Prove that S(n) ≤ n^2 − 14
for each n ≥ 4.
(b) Find an integer n such that
S(n) = n^2 − 14.
(c) Prove that there are infinitely many integers n such that S(n) = n^2 − 14.
Find the smallest integer N such that when you move the last 3 decimal digits (in the same order) to the front the result is 2N.
Cables
(in General)
Rating: 5.00
A flexible cable was hung across the Black Canyon between two points that were exactly 1 km apart and at the same elevation.
During the cool night the cable length was calculated to contract by .2 meters. The cable dip was actually measured to decrease by .2 meters!
What is the length of the cable after cooling?
Find the smallest distinct positive integers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M 3 to get N 3. [Leading zeroes are not allowed.]
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