MathDB

3

Part of 2004 India IMO Training Camp

Problems(5)

Least value(very easy)

Source: my notes

1/4/2005
For a,b,ca,b,c positive reals find the minimum value of a2+b2c2+ab+b2+c2a2+bc+c2+a2b2+ca. \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}.
inequalitiesinequalities proposed
A polynomial inequality

Source: Indian IMOTC 2004 Practice Test 2 Problem 3

9/23/2005
Suppose the polynomial P(x)≡x3+ax2+bx+cP(x) \equiv x^3 + ax^2 + bx +c has only real zeroes and let Q(x)≡5x2−16x+2004Q(x) \equiv 5x^2 - 16x + 2004. Assume that P(Q(x))=0P(Q(x)) = 0 has no real roots. Prove that P(2004)>2004P(2004) > 2004
algebrapolynomialinequalitiesalgebra unsolved
A game of pebbles

Source: Indian IMOTC 2004 Day 1 Problem 3

9/23/2005
The game of pebblespebbles is played on an infinite board of lattice points (i,j)(i,j). Initially there is a pebblepebble at (0,0)(0,0). A move consists of removing a pebblepebble from point (i,j)(i,j)and placing a pebblepebble at each of the points (i+1,j)(i+1,j) and (i,j+1)(i,j+1) provided both are vacant. Show taht at any stage of the game there is a pebblepebble at some lattice point (a,b)(a,b) with 0≤a+b≤30 \leq a+b \leq 3
functiongeometrycombinatorics unsolvedcombinatorics
Func eqn

Source: Indian IMOTC 2004 Day 2 Problem 3

9/23/2005
Determine all functionf f:R↦Rf : \mathbb{R} \mapsto \mathbb{R} such that f(x+y)=f(x)f(y)−csin⁡xsin⁡y f(x+y) = f(x)f(y) - c \sin{x} \sin{y} for all reals x,yx,y where c>1c> 1 is a given constant.
trigonometryfunctionalgebrafunctional equationsystem of equationsalgebra unsolved
2 runners

Source: Indian IMOTC 2004 Day 3 Problem 3

9/23/2005
Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds v1v_1 and v2v_2 respectively, where v1v_1 and v2v_2 are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that (a) at any given time tt, at least one of the runners is at a distance not more than [v1+v22]v1+v2\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2} units from the starting point. (b) there is a time tt such that both the runners are at least [v1+v22]v1+v2\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2} units away from the starting point. (All disstances are measured along the track). [x][x] is the greatest integer function.
number theoryrelatively primecombinatorics unsolvedcombinatorics