1989 APMO

Part of APMO

Subcontests

(5)

1

Prove no solutions other than 0

Prove that the equation

6(6a^2 + 3b^2 + c^2) = 5n^2

has no solutions in integers except

a = b = c = n = 0

1

Inequality from the first apmo

x_1

x_2

\cdots

x_n

be positive real numbers, and let

S = x_1 + x_2 + \cdots + x_n.

(1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!}

1

Area ratios

A_1

A_2

A_3

be three points in the plane, and for convenience, let

A_4= A_1

A_5 = A_2

n = 1

2

3

B_n

is the midpoint of

A_n A_{n+1}

, and suppose that

C_n

is the midpoint of

A_n B_n

A_n C_{n+1}

B_n A_{n+2}

D_n

A_n B_{n+1}

C_n A_{n+2}

E_n

. Calculate the ratio of the area of triangle

D_1 D_2 D_3

to the area of triangle

E_1 E_2 E_3

1

Triples (a,b,c) belong to s

S

be a set consisting of

m

(a,b)

of positive integers with the property that

1 \leq a < b \leq n

. Show that there are at least

4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n}

(a,b,c)

(a,b)

(a,c)

(b,c)

S

1

Functional equation

Determine all functions

f

from the reals to the reals for which (1)

f(x)

is strictly increasing and (2)

f(x) + g(x) = 2x

x

g(x)

is the composition inverse function to

f(x)

f

g

are said to be composition inverses if

f(g(x)) = x

g(f(x)) = x

x