Pi School

Problem (Preparation for Waterloo Math Contests)[AHSME 1976]: If ​\( \, p\, \)​ and ​\( \, q\, \)​ are primes and ​\( \, x^2 -px +q=0\, \)​ has distinct positive integral roots, find ​\( \, p\, \)​ and ​\( \, q. \)​...

Problem of the Day (March 14, 2025)-(Preparation for Waterloo Math Contests) Problem (Preparation for Waterloo Math Contests)[AHSME 1976]: If ​\( \, p\, \)​ and ​\( \, q\, \)​ are primes and ​\( \, x^2 -px +q=0\, \)​ has distinct positive integral roots, find ​\( \, p\, \)​…

Problem (Preparation for Waterloo Math Contests): Find all positive integers ​\( \, n\, \)​ for which ​\( \, 3n -4, \,\, 4n - 5, \,\, \mbox{and}\, \,\, 5n-3\, \)​ are all prime numbers....

Problem of the Day (March 11, 2025)-(Preparation for Waterloo Math Contests) Problem (Preparation for Waterloo Math Contests): Find all positive integers ​\( \, n\, \)​ for which ​\( \, 3n -4, \,\, 4n – 5, \,\, \mbox{and}\, \,\, 5n-3\, \)​ are all prime numbers.…

Problem (Preparation for Waterloo Math Contests): A bug moves in the coordinate plane, starting at (0,0). On the first turn, the bug moves one unit up, down, left, or right, each with equal probability. On subsequent turns the bug moves one unit up, down, left, or right, choosing with equal probability among the three directions other than that of its previous move. For example, if the first move was one unit up then the second move has to be either one unit down or one unit left or one unit right. After four moves, what is the probability that the bug is at (2,2)?...

Problem of the Day (March 7, 2025)-(Preparation for Waterloo Math Contests) Problem (Preparation for Waterloo Math Contests): A bug moves in the coordinate plane, starting at (0,0). On the first turn, the bug moves one unit up, down, left, or right, each with equal probabi…

Problem (Preparation for Waterloo Math Contests): A point with coordinates (a,2a) lies in the 3rd quadrant and on the curve given by the equation 3x2 + y2 = 28. Find a....

Problem of the Day (March 5, 2025)-(Preparation for Waterloo Math Contests) Problem (Preparation for Waterloo Math Contests): A point with coordinates (a,2a) lies in the 3rd quadrant and on the curve given by the equation 3×2 + y2 = 28. Find a.

Problem (Preparation for Waterloo Math Contests): Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a 2 × 4 grid-structured island. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid [a, b]×[c, d] with 1 ≤ a ≤ b ≤ 2 and 1 ≤ c ≤ d ≤ 4. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up Q detectors, which may only be run simultaneously after all Q detectors are ready. What is the minimum Q required to guarantee your crew can determine the location of Blackboard’s legendary treasure?...

Problem of the Day (January 13, 2025)-(Preparation for Waterloo Math Contests) Problem (Preparation for Waterloo Math Contests): Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a 2 × 4 grid-structured island. You and your crew…

Problem (Preparation for Waterloo Math Contests): A bug moves in the coordinate plane, starting at (0,0). On the first turn, the bug moves one unit up, down, left, or right, each with equal probability. On subsequent turns the bug moves one unit up, down, left, or right, choosing with equal probability among the three directions other than that of its previous move. For example, if the first move was one unit up then the second move has to be either one unit down or one unit left or one unit right....

Problem of the Day (January 8, 2025)-(Preparation for Waterloo Math Contests) Problem (Preparation for Waterloo Math Contests): A bug moves in the coordinate plane, starting at (0,0). On the first turn, the bug moves one unit up, down, left, or right, each with equal probabi…

Problem (Preparation for Waterloo Math Contests): A number n has sum of digits 100, whilst 44n has sum of digits 800. Find the sum of the digits of 3n....

Problem of the Day (November 27, 2024)-(Preparation for Waterloo Math Contests) Problem (Preparation for Waterloo Math Contests): A number n has sum of digits 100, whilst 44n has sum of digits 800. Find the sum of the digits of 3n.

Problem (Preparation for Waterloo Math Contests): Are there more positive integers under a million for which the nearest square is odd or for which it is even?...

Problem of the Day (November 19, 2024)-(Preparation for Waterloo Math Contests) Problem (Preparation for Waterloo Math Contests): Are there more positive integers under a million for which the nearest square is odd or for which it is even?

Problem (Preparation for Waterloo Math Contest): Some pairs of towns are connected by a road. At least 3 roads leave each town. Show that there is a cycle containing a number of towns which is not a multiple of 3....

Problem of the Day (November 14, 2024)-(Preparation for Waterloo Math Contests) Problem (Preparation for Waterloo Math Contest): Some pairs of towns are connected by a road. At least 3 roads leave each town. Show that there is a cycle containing a number of towns which is not …

Problem: A chooses a positive integer X ≤ 100. B has to find it. B is allowed to ask 7 questions of the form "What is the greatest common divisor of X + m and n?" for positive integers m, n < 100. Show that he can find X....

Problem of the Day (November 11, 2024)-(Preparation for Waterloo Math Contest) Problem: A chooses a positive integer X ≤ 100. B has to find it. B is allowed to ask 7 questions of the form “What is the greatest common divisor of X + m and n?” for positive integers …

Problem: If ​\( \, a+b+c=0\, \)​and ​\( \, abc=4,\, \)​ find ​\( \,a^3+b^3+c^3.\, \)​ Originally from the 21st W.J. Blundon Mathematics Contest (2004), problem 7....

Problem of the Day (November 4, 2024) Math Contest Problem: If ​\( \, a+b+c=0\, \)​and ​\( \, abc=4,\, \)​ find ​\( \,a^3+b^3+c^3.\, \)​ Originally from the 21st W.J. Blundon Mathematics Contest (2004), problem 7.

Problem: A palindrome is a whole number whose digits are the same when read from left to right as from right to left. For example, 565 and 7887 are palindromes. Find the smallest six-digit palindrome divisible by 12....

Problem of the Day (November 2, 2024)COM(Canadian Open Mathematics Challenge 2022) Problem: A palindrome is a whole number whose digits are the same when read from left to right as from right to left. For example, 565 and 7887 are palindromes. Find the smallest six-digit palindro…