Difference between revisions of "1974 USAMO Problems"
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[[1974 USAMO Problems/Problem 5 | Solution]] | [[1974 USAMO Problems/Problem 5 | Solution]] | ||
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*[[USAMO Problems and Solutions]] | *[[USAMO Problems and Solutions]] | ||
{{USAMO box|year=1974|before=[[1973 USAMO]]|after=[[1975 USAMO]]}} | {{USAMO box|year=1974|before=[[1973 USAMO]]|after=[[1975 USAMO]]}} | ||
Revision as of 10:28, 17 September 2012
Problems from the 1974 USAMO.
Problem 1
Let 
, 
, and 
denote three distinct integers, and let 
denote a polynomial having all integral coefficients. Show that it is impossible that 
, 
, and 
.
Problem 2
Prove that if , , and are positive real numbers, then
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.
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.
Problem 5
Consider the two triangles 
and 
shown in Figure 1. In , 
. Prove that 
.
![[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]](http://latex.artofproblemsolving.com/0/2/5/02517a956189e65f4d31ed8ab2e8e8362542cafa.png)
See also
| 1974 USAMO (Problems • Resources) | ||
| Preceded by 1973 USAMO |
Followed by 1975 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
