2022 Euler Student Problems Problem 6 Let Th denote the nth triangular number. (a) Use a configuration of dots to show that T7 = 313 + T4. (b) Using algebra, prove that Tan-1 = 3Tn-1 + In. (c) A pattern of solid block towers is shown below. To build the first three of these towers, we require 1, 6 and 15 blocks, respectively. How many blocks are required to make the 10th block tower? Problem 7 Each of the positive integers from 1 to 100 are written on a sheet of paper: 1, 2, 3, . .., 98, 99, 100. Some of these integers are erased. The product of those integers still on the paper leaves a remainder of 4 when divided by 5. Find the least number of integers that could have been erased. Problem 8 A stream of water enters a dam at a constant rate. A series of identical pumps draw water from the dam at a constant rate. If the dam was full, it would be emptied by 27 pumps in 6 days, or by 23 pumps in 9 days. How many pumps should be connected to the dam so that its volume of water remains constant