MOPSS, 23rd August 2025
(IOQM 2024 P8, AoPS) Let \(n\) be the smallest integer such that the sum of the digits of \(n\) is divisible by \(5\), as well as the sum of the digits of \(n + 1\) is divisible by \(5\). What are the first two digits of \(n\) in the same order?
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- Show that the last digit of \(n\) is \(9\).
- Note that \(n\) has at least two digits. If \(n\) is a two-digit number, then show that it is not equal to any integer other than \(19\) or \(69\).
- Conclude that \(n\) has at least three digits, and its second last digit is \(9\).
- If \(n\) is a three-digit number, then show that it is not equal to any integer other than \(299\) or \(799\).
- Argue similarly to show that \(n\) has at least four digits, and its third last digit is \(9\).
- Prove that \(n\) has at least five digits, and its fourth last digit is \(9\).
- If \(n\) is a five-digit number, then show that it is equal to \(49999\).
(IOQM 2024 P10, AoPS) Determine the number of positive integral values of \(p\) for which there exists a triangle with sides \(a, b\) and \(c\) which satisfy \[ a^2 + (p^2 + 9)b^2 + 9c^2 - 6ab - 6pbc = 0 \]
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- Note that \[ a^2 + (p^2 + 9)b^2 + 9c^2 - 6ab - 6pbc = (a - 3b)^2 + (pb - 3c)^2. \]
- This gives \(a = 3b\) and \[ c = \frac{p}{3}b. \]
- Use the inequality1 \(a + b > c\), to obtain \[ 3b + b > \frac{p}{3}b, \] which yields \(p \leq 11\).
- Use the inequality2 \(b + c > a\), to get \[ b + \frac{p}{3}b > 3b, \] implying \(p \geq 7\).
- It reduces to finding the number of positive integers \(p\) such that \(7 \leq p \leq 11\).
(IOQM 2024 P15, AoPS) Let \(X\) be the set consisting of twenty positive integers \(n, n + 2, \dots, n + 38\). The smallest value of \(n\) for which any three numbers \(a, b, c \in X\), not necessarily distinct, form the sides of an acute-angled triangle is:
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- Use the fact that1 if \(a, b, c\) are the sides of an acute-angled triangle with \(c\) denoting the largest side, then \(a^2 + b^2 > c^2\) holds.
- Show that for any such \(n\), the inequality \[ (n + 38)^2 < 2n^2 \] holds, which yields \(n \geq 92\).
- Prove that for \(n = 92\), the inequality \[ 2 (n + 2i)^2 > (n + 2i + 2)^2 \] holds for \(i = 0, 1, \dots, 18\).
- Note that if suffices to show that \[ 2m^2 > (m + 2)^2 \] for any \(m \in \{92, 94, \dots, 128\}\).
What is the role of the third part? Does it guarantee that for \(n = 92\), the set \(X\) has the stated property?
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How to prove it? Does the converse of this statement hold? In other words, if \(a, b, c\) are positive real numbers satisfying \(a^2 + b^2 > c^2\), then does it follow that \(a, b, c\) are the sides of an acute-angled triangle? Even if it doesn’t, would it be true under additional hypothesis? ↩
(INMO 2025 P5, AoPS, proposed by Pranjal Srivastava and Rohan Goyal) Greedy goblin Griphook has a regular \(2000\)-gon, whose every vertex has a single coin. In a move, he chooses a vertex, removes one coin each from the two adjacent vertices, and adds one coin to the chosen vertex, keeping the remaining coin for himself. He can only make such a move if both adjacent vertices have at least one coin. Griphook stops only when he cannot make any more moves. What is the maximum and minimum number of coins that he could have collected?
(USAJMO 2013 P1, proposed by Titu Andreescu) Are there integers \(a, b\) such that \(a^5b + 3\) and \(ab^5 + 3\) are both perfect cubes of integers?
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- If \(3\) divides \(ab\), then consider one of the integers \(a^5b + 3, ab^5 + 3\) modulo \(9\).
- If \(3\) does not divide \(ab\), then assuming the integers \(a^5b + 3, ab^5 + 3\) to be perfect cubes, determine the integers \(a^5b, ab^5\) modulo \(9\).
(USAMO 2000 P4) Find the smallest positive integer \(n\) such that if \(n\) squares of a \(1000 \times 1000\) chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
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Does Fig. 1 help?
- Consider a suitable coloring to show that \(n \geq 1999\).
- Does coloring \(1999\) squares suffice?
- What happens when there are two columns, each containing at least two colored squares, and some row contains one colored square from each of these columns?
- Consider the columns which contain at least two colored squares. How many colored squares can they contain together?
- What about the remaining colored squares? How many are they in total?
(USAMO 2018 P4, proposed by Ankan Bhattacharya) Let \(p\) be a prime, and let \(a_1, a_2, \dots, a_p\) be integers. Show that there exists an integer \(k\) such that the numbers \[ a_1 + k, a_2 + 2k, \dots, a_p + pk \] produce at least \(p/2\) distinct remainders upon division by \(p\).
Lecture notes for Math Olympiad (IOQM, RMO, INMO)
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