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Difference between revisions of "1974 USAMO Problems"

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Revision as of 13:34, 17 September 2012

Problems from the 1974 USAMO.

Problem 1

Let $a$, $b$, and $c$ denote three distinct integers, and let $P$ denote a polynomial having all integral coefficients. Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$.

Solution

Problem 2

Prove that if $a$, $b$, and $c$ are positive real numbers, then

$a^ab^bc^c\ge (abc)^{(a+b+c)/3}$

Solution

Problem 3

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.

Solution

Problem 4

A father, 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.

Solution

Problem 5

Consider the two triangles $\triangle ABC$ and $\triangle PQR$ shown in Figure 1. In $\triangle ABC$, $\angle ADB = \angle BDC = \angle CDA = 120^\circ$. Prove that $x=u+v+w$. [asy] size(400); defaultpen(1); pair C=(0,1), A=(1,6), B=(4,1), D=(1.5,2); draw(D--A--B--C--D--B); draw(A--C); label("\(A\)",A,N); label("$B$",B,ESE); label("$C$",C,SW); label("$a$",(B+C)/2,S); label("$b$",(C+A)/2,WNW); label("$c$",(A+B)/2,NE); label("$u$",(A+D)/2,E); label("$v$",(B+D)/2,N); label("$w$",(C+D)/2,N); pair P=(7,0), Q=(14,0), R=P+7expi(pi/3), M=(10,1.2); draw(M--P--Q--R--M--Q); draw(P--R); label("$P$",P,W); label("$Q$",Q,E); label("$R$",R,W); label("$M$",M,NE); label("$x$",(P+Q)/2,S); label("$x$",(Q+R)/2,E); label("$x$",(R+P)/2,W); label("$a$",(P+M)/2,SE); label("$b$",(Q+M)/2,N); label("$c$",(R+M)/2,E); label("Figure 1",P-(0,1),S); [/asy]

Solution

See Also

1974 USAMO (Problems • Resources)
Preceded by
1973 USAMO 		Followed by
1975 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions
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