MathDB

1

Part of 2001 India IMO Training Camp

Problems(5)

Inequality in positive integers!

Source: India TST 2001 Day 1 Problem 1

1/31/2015
Let xx , yy , z>0z>0. Prove that if xyz≥xy+yz+zxxyz\geq xy+yz+zx, then xyz≥3(x+y+z)xyz \geq 3(x+ y+z).
inequalitiesinequalities proposed
Polynomial interpolates powers of 2 !

Source: India TST 2001 Day 2 Problem 1

1/31/2015
For any positive integer nn, show that there exists a polynomial P(x)P(x) of degree nn with integer coefficients such that P(0),P(1),…,P(n)P(0),P(1), \ldots, P(n) are all distinct powers of 22.
algebrapolynomialnumber theory unsolvednumber theory
A nice and easy one {India TST 2001}

Source:

5/11/2009
If on △ABC \triangle ABC, trinagles AEB AEB and AFC AFC are constructed externally such that \angle AEB\equal{}2 \alpha, \angle AFB\equal{} 2 \beta. AE\equal{}EB, AF\equal{}FC. COnstructed externally on BC BC is triangle BDC BDC with \angle DBC\equal{} \beta , \angle BCD\equal{} \alpha. Prove that 1. DA DA is perpendicular to EF EF. 2. If T T is the projection of D D on BC BC, then prove that \frac{DA}{EF}\equal{} 2 \frac{DT}{BC}.
geometrycircumcircleradical axispower of a pointgeometry proposed
Polynomial with integer coefficients!

Source: India TST 2001 Day 4 Problem 1

1/31/2015
Complex numbers α\alpha , β\beta , γ\gamma have the property that αk+βk+γk\alpha^k +\beta^k +\gamma^k is an integer for every natural number kk. Prove that the polynomial (x−α)(x−β)(x−γ)(x-\alpha)(x-\beta )(x-\gamma ) has integer coefficients.
algebrapolynomialcomplex numbersnumber theory proposednumber theory
Not easy

Source: India MO 2001 problem 2

6/30/2004
Let ABCDABCD be a rectangle, and let ω\omega be a circular arc passing through the points AA and CC. Let ω1\omega_{1} be the circle tangent to the lines CDCD and DADA and to the circle ω\omega, and lying completely inside the rectangle ABCDABCD. Similiarly let ω2\omega_{2} be the circle tangent to the lines ABAB and BCBC and to the circle ω\omega, and lying completely inside the rectangle ABCDABCD. Denote by r1r_{1} and r2r_{2} the radii of the circles ω1\omega_{1} and ω2\omega_{2}, respectively, and by rr the inradius of triangle ABCABC. (a) Prove that r1+r2=2rr_{1}+r_{2}=2r. (b) Prove that one of the two common internal tangents of the two circles ω1\omega_{1} and ω2\omega_{2} is parallel to the line ACAC and has the length ∣AB−AC∣\left|AB-AC\right|.
geometryrectangleinradiusgeometry unsolved