Advanced book on Mathematics Olympiad

76 2 Algebra

x 3 +x 5 =yx 1 ,x 4 +x 1 =yx 5 ,

whereyis a parameter.

233.Leta, b, c, dbe positive numbers different from 1, andx, y, z, treal numbers
satisfyingax=bcd,by=cda,cz=dab,dt=abc. Prove that
∣
∣∣
∣∣
∣
∣∣

−x 111

1 −y 11

11 −z 1

111 −t

∣

∣∣

∣∣

∣

∣∣

= 0.

234.Given the system of linear equations

a 11 x 1 +a 12 x 2 +a 13 x 3 = 0 ,

a 21 x 1 +a 22 x 2 +a 23 x 3 = 0 ,

a 31 x 1 +a 32 x 2 +a 33 x 3 = 0 ,

whose coefficients satisfy the conditions

(a)a 11 ,a 22 ,a 33 are positive,

(b) all other coefficients are negative,

(c) in each equation, the sum of the coefficients is positive,

prove that the system has the unique solutionx 1 =x 2 =x 3 =0.

235.LetP(x)=xn+xn−^1 +···+x+1. Find the remainder obtained whenP(xn+^1 )
is divided byP(x).

236.Find all functionsf:R{− 1 , 1 }→Rsatisfying

f

(

x− 3

x+ 1

)

+f

(

3 +x

1 −x

)

=x

for allx =±1.

237.Find all positive integer solutions(x,y,z,t)to the Diophantine equation

(x+y)(y+z)(z+x)=txyz

such that gcd(x, y)=gcd(y, z)=gcd(z, x)=1.

238.We havencoins of unknown masses and a balance. We are allowed to place some
of the coins on one side of the balance and an equal number of coins on the other
side. After thus distributing the coins, the balance gives a comparison of the total
mass of each side, either by indicating that the two masses are equal or by indicating
that a particular side is the more massive of the two. Show that at leastn−1 such
comparisons are required to determine whether all of the coins are of equal mass.