1) A regional mathematical olympiad was held on October 23rd, 2016 for 3 hours. Calculators, protractors, and instructions were provided. Six problems were included with equal marks for each.
2) The first problem asks to prove that if ABC is an isosceles triangle satisfying certain conditions related to its circumcircle, then it must be equilateral.
3) The second problem asks to prove an inequality relating the sums of fractions involving three positive real numbers a, b, c.
1) A regional mathematical olympiad was held on October 23rd, 2016 for 3 hours. Calculators, protractors, and instructions were provided. Six problems were included with equal marks for each.
2) The first problem asks to prove that if ABC is an isosceles triangle satisfying certain conditions related to its circumcircle, then it must be equilateral.
3) The second problem asks to prove an inequality relating the sums of fractions involving three positive real numbers a, b, c.
1) A regional mathematical olympiad was held on October 23rd, 2016 for 3 hours. Calculators, protractors, and instructions were provided. Six problems were included with equal marks for each.
2) The first problem asks to prove that if ABC is an isosceles triangle satisfying certain conditions related to its circumcircle, then it must be equilateral.
3) The second problem asks to prove an inequality relating the sums of fractions involving three positive real numbers a, b, c.
1) A regional mathematical olympiad was held on October 23rd, 2016 for 3 hours. Calculators, protractors, and instructions were provided. Six problems were included with equal marks for each.
2) The first problem asks to prove that if ABC is an isosceles triangle satisfying certain conditions related to its circumcircle, then it must be equilateral.
3) The second problem asks to prove an inequality relating the sums of fractions involving three positive real numbers a, b, c.
Instructions: • Calculators (in any form) and protractors are not allowed. • Rulers and compasses are allowed. • Answer all the questions. • All questions carry equal marks. Maximum marks: 102. • Answer to each question should start on a new page. Clearly indicate the question number. 1. Let ABC be an isosceles triangle with AB = AC. Let Γ be its circumcircle and let O be the centre of Γ. Let CO meet Γ in D. Draw a line parallel to AC through D. Let it intersect AB at E. Suppose AE : EB = 2 : 1. Prove that ABC is an equilateral triangle. 2. Let a, b, c be positive real numbers such that ab bc ca + + = 1. 1 + bc 1 + ca 1 + ab 1 1 1 √ Prove that 3 + 3 + 3 ≥ 6 2. a b c 3. The present ages in years of two brothers A and B, and their father C are three distinct b−1 b+1 positive integers a, b, and c respectively. Suppose and are two consecutive a−1 a+1 c−1 c+1 integers, and and are two consecutive integers. If a+b+c ≤ 150 determine b−1 b+1 a, b and c. 4. A box contains 4032 answer scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of the scripts and keep the other outside. If there is one script with even score and the other with odd score, we put back the script with the odd score and keep the other script outside. After following this procedure a number of times, there are 3 scripts left among which there is at least one script each with odd and even scores. Find, with proof, the number of scripts with odd scores among the three left. 5. Let ABC be a triangle, AD an altitude and AE a median. Assume B, D, E, C lie in that order on the line BC. Suppose the incentre of triangle ABE lies on AD and the incentre of ADC lies on AE. Find, with proof, the angles of triangle ABC. 6. (i) Prove that if an infinite sequence of strictly increasing positive integers in arith- metic progression has one cube then it has infinitely many cubes. (ii) Find, with justification, an infinite sequence of strictly increasing positive integers in arithmetic progression which does not have any cube. ———-0000———-