1974 USAMO

Part of USAMO

Subcontests

(5)

1

Triangle Lengths

Consider the two triangles

ABC

PQR

shown below. In triangle ABC, \angle ADB \equal{} \angle BDC \equal{} \angle CDA \equal{} 120^\circ. Prove that x\equal{}u\plus{}v\plus{}w. [asy]unitsize(7mm); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair C=(0,0), B=4*dir(5); pair A=intersectionpoints(Circle(C,5), Circle(B,6))[0];
pair Oc=scale(sqrt(3)/3)*rotate(30)*(B-A)+A; pair Ob=scale(sqrt(3)/3)*rotate(30)*(A-C)+C;
pair D=intersectionpoints(Circle(Ob,length(Ob-C)), Circle(Oc,length(Oc-B)))[1];
real s=length(A-D)+length(B-D)+length(C-D); pair P=(6,0), Q=P+(s,0), R=rotate(60)*(s,0)+P; pair M=intersectionpoints(Circle(P,length(B-C)), Circle(Q,length(A-C)))[0];
draw(A--B--C--A--D--B); draw(D--C);
label("

B

",B,SE); label("

C

",C,SW); label("

A

",A,N); label("

D

",D,NE); label("

a

",midpoint(B--C),S); label("

b

",midpoint(A--C),WNW); label("

c

",midpoint(A--B),NE); label("

u

",midpoint(A--D),E); label("

v

",midpoint(B--D),N); label("

w

",midpoint(C--D),NNW);
draw(P--Q--R--P--M--Q); draw(M--R);
label("

P

",P,SW); label("

Q

",Q,SE); label("

R

",R,N); label("

M

",M,NW); label("

x

",midpoint(P--R),NW); label("

x

",midpoint(P--Q),S); label("

x

",midpoint(Q--R),NE); label("

c

",midpoint(R--M),ESE); label("

a

",midpoint(P--M),NW); label("

b

",midpoint(Q--M),NE);[/asy]

1

Family Tournament

A father, a mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: (i) The weakest player chooses the first two contestants. (ii) The winner of any game plays the next game against the person left out. (iii) The first person to win two games wins the tournament. The father is the weakest player, the son the strongest, and it is assumed that any player's probability of winning an individual game from another player does not change during the tournament. Prove that the father's optimal strategy for winning the tournament is to play the first game with his wife.

1

Curve in a Sphere

Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.

1

Exponent Inequality

a,b,

c

are positive real numbers, then a^ab^bc^c \ge (abc)^{(a\plus{}b\plus{}c)/3}.

1

Integer Polynomial with P(a)=b, P(b)=c, and P(c)=a

a,b,

c

denote three distinct integers, and let

P

denote a polynomial having integer coefficients. Show that it is impossible that P(a) \equal{} b, P(b) \equal{} c, and P(c) \equal{} a.