1985 USAMO Problems

Problem 1

Determine whether or not there are any positive integral solutions of the simultaneous equations $x_1^2+x_2^2+\cdots+x_{1985}^2=y^3\\\\ x_1^3+x_2^3+\cdots+x_{1985}^3=z^2$ with distinct integers $x_1,x_2,\cdots,x_{1985}$ .

Solution

Problem 2

Determine each real root of

$x^4-(2\cdot10^{10}-1)x^3-x+10^{20}+10^{10}-1=0$

correct to four decimal places.

Solution

Problem 3

Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$ . Determine the maximum value of the sum of the six distances.

Solution

Problem 4

There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\lfloor n/2\rfloor -1$ of them, each of whom knows both or else knows neither of the two. Assume that "know" is a symmetrical relation; $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ .

Solution

Problem 5

Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$ , define $b_m=\min\{n: a_n \ge m\}$ , that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$ . If $a_{19}=85$ , determine the maximum value of $a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}$ .

Solution

1985 USAMO (Problems • Resources)
Preceded by 1984 USAMO	Followed by 1986 USAMO
1 • 2 • 3 • 4 • 5
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1985 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See Also