Tiêu đề ALMO 2025 رفع صفحة ALI MATHS ✅

People’s Democratic Republic of Algeria Ministry of National Education Directorate of Specialized Education and Private Education National Committee of Olympiads of Educational Disciplines Algerian Mathematical Olympiad - Second Edition 2025 Category: Benjamin July 3rd, 2025 Problem 1. Find all natural numbers n such that : n 6 + n + 61 n2 + n + 1 is an integer. Problem 2. Find all positive real numbers a1, a2, ..., a45 such that: X 45 k=1 k 2 ak + 1 2025ak = 46 Problem 3. Let ABC be a right triangle in A. Let ω be its circumcircle and D the midpoint of BC. Let E be the intersection of AD with ω and let F be the intersection of the perpendicular bisector of AC with (BDE). The line CF intersects (BDE) in P and the let Q be the intersection of DP with AB. Let S and T be the symmetric points of P and F with respect to D respectively. Prove that the intersection point of the lines AT and QF is the circumcenter of ∆QAS. Language: English Time: 4 hours and 30 minutes Each problem is worth 7 points. The problems are ordered by difficulty.
People’s Democratic Republic of Algeria Ministry of National Education Directorate of Specialized Education and Private Education National Committee of Olympiads of Educational Disciplines Algerian Mathematical Olympiad - Second Edition 2025 Category: Benjamin July 4th, 2025 Problem 4. Let a, b, c be positive integers such that a > b > c, prove that one of the numbers a − b, b − c, c − a is even. Problem 5. Let ABC be a triangle such that AB = 2, AC = 4, let D be a point on [AC] such that AD = 1, suppose that the peremeter of triangle BCD is 9, compute the length of the side [BC]. Problem 6. Prove that for all nonzero numbers x, y, z, we get: x 2 y 2 z 2 + x 2 z 2 y 2 + y 2 z 2 x 2 + 3 ≥ 2(x + y + z) Language: English Time: 4 hours and 30 minutes Each problem is worth 7 points. The problems are ordered by difficulty.
People’s Democratic Republic of Algeria Ministry of National Education Directorate of Specialized Education and Private Education National Committee of Olympiads of Educational Disciplines Algerian Mathematical Olympiad - Second Edition 2025 Category: Junior July 3rd, 2025 Problem 1. Find all functions f : R → R such that: f(f(2x + y)) + f(x) = 2x + f(x + y) For all real numbers x, y ∈ R Problem 2. Let ABC be a triangle such that ∠ABC = 3∠ACB. In the circumcir- cle of this triangle, let D, E and F be points such that: (AD) ∥ (BC), (DE) ∥ (CA) and (EF) ∥ (AB). Let J be the intersection of (DF) and (AC) and let ω be the circle that passes through J and is tangent to (BD) at D. Finally, let L be the intersection point of ω and (ABC). Prove that points E, J and L are collinear. Problem 6. Let a0, a1, . . . , an be positive divisors of the number 20242025 such that • a0 < a1 < a2 · · · < an • a0 | a1, a1 | a2, . . . , an−1 | an Find the largest possible value of the positive integer n. Language: English Time: 4 hours and 30 minutes Each problem is worth 7 points. The problems are ordered by difficulty.
People’s Democratic Republic of Algeria Ministry of National Education Directorate of Specialized Education and Private Education National Committee of Olympiads of Educational Disciplines Algerian Mathematical Olympiad - Second Edition 2025 Category: Junior July 4th, 2025 Problem 4. Mohamed wrote 9 distinct natural numbers around the circumference of a circle. Each time, he chooses two numbers and writes all positive divisors of their differ- ence inside the circle, the process ends when all possible pairs have been chosen. In this case, prove that all numbers less than 9 have been written inside the circle (not necessarily alone). Problem 5. find all nonnegative integers x, y ∈ N and primes p such that: x 2 = p − 2, y2 = 2p 2 − 2 Problem 6. Let ABC be a triangle. Consider the points D, E, F as the feet of the alti- tudes from A, B, C, respectively and H its orthocenter which we suppose is the midpoint of CF. Let M be the midpoint of BC, N be the midpoint of BE, and X = (AN)∩(MF). Prove that ∠HXM = 90◦ . Language: English Time: 4 hours and 30 minutes Each problem is worth 7 points. The problems are ordered by difficulty.