MathDB

1

Part of 2016 India IMO Training Camp

Problems(5)

Geometry

Source: All Russian Grade 9 Day 2 P 3

12/12/2015
An acute-angled ABC (AB<AC)ABC \ (AB<AC) is inscribed into a circle ω\omega. Let MM be the centroid of ABCABC, and let AHAH be an altitude of this triangle. A ray MHMH meets ω\omega at A′A'. Prove that the circumcircle of the triangle A′HBA'HB is tangent to ABAB. (A.I. Golovanov , A.Yakubov)
geometrycircumcircle
Odd perfect number has 3 prime factors

Source: India IMO Training Camp 2016, Practice Test 2, Problem 1

7/22/2016
We say a natural number nn is perfect if the sum of all the positive divisors of nn is equal to 2n2n. For example, 66 is perfect since its positive divisors 1,2,3,61,2,3,6 add up to 12=2×612=2\times 6. Show that an odd perfect number has at least 33 distinct prime divisors.
Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result.
number theory
n+3 is a power of 2

Source: India TST 2016 Day 3 Problem 1

7/22/2016
Let nn be a natural number. We define sequences ⟨ai⟩\langle a_i\rangle and ⟨bi⟩\langle b_i\rangle of integers as follows. We let a0=1a_0=1 and b0=nb_0=n. For i>0i>0, we let (ai,bi)={(2ai−1+1,bi−1−ai−1−1)if ai−1<bi−1,(ai−1−bi−1−1,2bi−1+1)if ai−1>bi−1,(ai−1,bi−1)if ai−1=bi−1.\left( a_i,b_i\right)=\begin{cases} \left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\right) & \text{if } a_{i-1}<b_{i-1},\\ \left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\right) & \text{if } a_{i-1}>b_{i-1},\\ \left(a_{i-1},b_{i-1}\right) & \text{if } a_{i-1}=b_{i-1}.\end{cases} Given that ak=bka_k=b_k for some natural number kk, prove that n+3n+3 is a power of two.
number theoryrecurrence relationpower of 2geometrybir tinga qimmat ekan boru
Sum is rational implies numbers are rational

Source: India TST 2016 Day 2 Problem 1

7/22/2016
Suppose α,β\alpha, \beta are two positive rational numbers. Assume for some positive integers m,nm,n, it is known that α1n+β1m\alpha^{\frac 1n}+\beta^{\frac 1m} is a rational number. Prove that each of α1n\alpha^{\frac 1n} and β1m\beta^{\frac 1m} is a rational number.
algebrairrational number
Comparing inradii of triangles

Source: India TST 2016 Day 4 Problem 1

7/22/2016
Let ABCABC be an acute triangle with circumcircle Γ\Gamma. Let A1,B1A_1,B_1 and C1C_1 be respectively the midpoints of the arcs BAC,CBABAC,CBA and ACBACB of Γ\Gamma. Show that the inradius of triangle A1B1C1A_1B_1C_1 is not less than the inradius of triangle ABCABC.
geometryinradiuscircumcircleinequalitiesgeometric inequality