MathDB

2

Part of 2012 India IMO Training Camp

Problems(6)

Prove that quadratic equation has real roots.

Source:

7/17/2012
Let a≥ba\ge b and c≥dc\ge d be real numbers. Prove that the equation (x+a)(x+d)+(x+b)(x+c)=0(x+a)(x+d)+(x+b)(x+c)=0 has real roots.
quadratics
Prove that two conguence statements are equivalent

Source:

7/17/2012
Let 0<x<y<z<p0<x<y<z<p be integers where pp is a prime. Prove that the following statements are equivalent: (a)x3≡y3(modp) and x3≡z3(modp)(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p (b)y2≡zx(modp) and z2≡xy(modp)(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p
modular arithmeticinequalitiesnumber theory proposednumber theory
Inequality on maximum value of polynomial in interval

Source:

7/12/2012
Let P(z)=anzn+an−1zn−1+…+amzmP(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m be a polynomial with complex coefficients such that am≠0,an≠0a_m\neq 0, a_n\neq 0 and n>mn>m. Prove that max∣z∣=1{∣P(z)∣}≥2∣aman∣+∑k=mn∣ak∣2\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}
inequalitiesalgebrapolynomialrotation
Infinitely many pairs exist satisfying property

Source:

7/22/2012
Show that there exist infinitely many pairs (a,b)(a, b) of positive integers with the property that a+ba+b divides ab+1ab+1, a−ba-b divides ab−1ab-1, b>1b>1 and a>b3−1a>b\sqrt{3}-1
number theorygreatest common divisornumber theory proposed
Set of primes satisfying property for every proper subset

Source:

7/23/2012
Let SS be a nonempty set of primes satisfying the property that for each proper subset PP of SS, all the prime factors of the number (∏p∈Pp)−1\left(\prod_{p\in P}p\right)-1 are also in SS. Determine all possible such sets SS.
pigeonhole principlenumber theory proposednumber theory
Find least positive integer not representable in a form.

Source:

7/23/2012
Find the least positive integer that cannot be represented as 2a−2b2c−2d\frac{2^a-2^b}{2^c-2^d} for some positive integers a,b,c,da, b, c, d.
number theory proposednumber theory