MathDB

6

Part of 2016 India Regional Mathematical Olympiad

Problems(7)

Magic Trick with the deck

Source: RMO Delhi 2016, P6

10/11/2016
A deck of 5252 cards is given. There are four suites each having cards numbered 1,2,…,131,2,\dots, 13. The audience chooses some five cards with distinct numbers written on them. The assistant of the magician comes by, looks at the five cards and turns exactly one of them face down and arranges all five cards in some order. Then the magician enters and with an agreement made beforehand with the assistant, he has to determine the face down card (both suite and number). Explain how the trick can be completed.
combinatoricsGame Theory
A geometric sub-sequence of an arithmetic sequence

Source: RMO Mumbai 2016, P6

10/11/2016
Let (a1,a2,… )(a_1,a_2,\dots) be a strictly increasing sequence of positive integers in arithmetic progression. Prove that there is an infinite sub-sequence of the given sequence whose terms are in a geometric progression.
arithmetic sequencenumber theory
Counting triangles with same centroids

Source: RMO Maharashtra and Goa 2016, P6

10/11/2016
ABCABC is an equilateral triangle with side length 1111 units. Consider the points P1,P2,…,P10P_1,P_2, \dots, P_10 dividing segment BCBC into 1111 parts of unit length. Similarly, define Q1,Q2,…,Q10Q_1, Q_2, \dots, Q_10 for the side CACA and R1,R2,…,R10R_1,R_2,\dots, R_10 for the side ABAB. Find the number of triples (i,j,k)(i,j,k) with i,j,k∈{1,2,…,10}i,j,k \in \{1,2,\dots,10\} such that the centroids of triangles ABCABC and PiQjRkP_iQ_jR_k coincide.
countingMass Pointsgeometry
Number theory

Source: RMO 2016 Karnataka Region P6

10/16/2016
(a). Given any natural number NN, prove that there exists a strictly increasing sequence of NN positive integers in harmonic progression.
(b). Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.
number theory
Finding harmonic progression in (1,4,7,10,...)

Source: RMO 2016 Hyderabad , P6 .

10/12/2016
Show that the infinite arithmetic progression {1,4,7,10…}\{1,4,7,10 \ldots\} has infinitely many 3 -term sub sequences in harmonic progression such that for any two such triples {a1,a2,a3}\{a_1, a_2 , a_3 \} and {b1,b2,b3}\{b_1, b_2 ,b_3\} in harmonic progression , one has a1b1≠a2b2\frac{a_1} {b_1} \ne \frac {a_2}{b_2}.
number theory
Problem 6

Source: India RMO 2016

10/16/2016
(a)Given any natural number N, prove that there exists a strictly increasing sequence of N positive integers in harmonic progression. (b)Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.
RMO 2016algebra
2016 Chandigarh RMO 1/a + 1/b +1/c<1 =>1/a + 1/b +1/c<= 41/42

Source:

8/9/2019
Positive integers a,b,ca, b, c satisfy 1a+1b+1c<1\frac1a +\frac1b +\frac1c<1. Prove that 1a+1b+1c≤4142\frac1a +\frac1b +\frac1c\le \frac{41}{42}. Also prove that equality in fact holds in the second inequality.
inequalitiesalgebra