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1985 USAMO Problems: Difference between revisions

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Problems from the '''1985 [[USAMO]].'''
==Problem 1==
==Problem 1==
Determine whether or not there are any positive integral solutions of the simultaneous equations  
Determine whether or not there are any positive integral solutions of the simultaneous equations  
<cmath>x_1^2+x_2^2+\cdots+x_{1985}^2=y^3\\\\
<cmath>x_1^2+x_2^2+\cdots+x_{1985}^2=y^3,
\hspace{20pt}
x_1^3+x_2^3+\cdots+x_{1985}^3=z^2</cmath>
x_1^3+x_2^3+\cdots+x_{1985}^3=z^2</cmath>
with distinct integers <math>x_1,x_2,\cdots,x_{1985}</math>.
with distinct integers <math>x_1,x_2,\cdots,x_{1985}</math>.
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Determine each real root of
Determine each real root of


<math>x^4-(2\cdot10^{10}-1)x^3-x+10^{20}+10^{10}-1=0</math>
<math>x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0</math>


correct to four decimal places.
correct to four decimal places.
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==Problem 4==
==Problem 4==
Let <math>a_1,a_2,a_3,\cdots</math> be a non-decreasing sequence of positive integers. For <math>m\ge1</math>, define <math>b_m=\min\{n: a_n \ge m\}</math>, that is, <math>b_m</math> is the minimum value of <math>n</math> such that <math>a_n\ge m</math>. If <math>a_{19}=85</math>, determine the maximum value of
There are <math>n</math> people at a party. Prove that there are two people such that, of the remaining <math>n-2</math> people, there are at least <math>\lfloor n/2\rfloor -1</math> of them, each of whom knows both or else  knows neither of the two. Assume that "know" is a symmetrical relation; <math>\lfloor x\rfloor</math> denotes the greatest integer less than or equal to <math>x</math>.
 
<math>a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}</math>.


[[1985 USAMO Problems/Problem 4 | Solution]]
[[1985 USAMO Problems/Problem 4 | Solution]]


==Problem 5==
==Problem 5==
<math>0\le a_1\le a_2\le a_3\le \cdots</math> is an unbounded sequence of integers. Let <math>b_n \equal{} m</math> if <math>a_m</math> is the first member of the sequence to equal or exceed <math>n</math>. Given that <math>a_{19}=85</math>, what is the maximum possible value of <math>a_1+a_2+\cdots a_{19}+b_1+b_2+\cdots b_{85}</math>?
Let <math>a_1,a_2,a_3,\cdots</math> be a non-decreasing sequence of positive integers. For <math>m\ge1</math>, define <math>b_m=\min\{n: a_n \ge m\}</math>, that is, <math>b_m</math> is the minimum value of <math>n</math> such that <math>a_n\ge m</math>. If <math>a_{19}=85</math>, determine the maximum value of
<math>a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}</math>.


[[1985 USAMO Problems/Problem 5 | Solution]]
[[1985 USAMO Problems/Problem 5 | Solution]]
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== See Also ==
== See Also ==
{{USAMO box|year=1985|before=[[1984 USAMO]]|after=[[1986 USAMO]]}}
{{USAMO box|year=1985|before=[[1984 USAMO]]|after=[[1986 USAMO]]}}
{{MAA Notice}}

Latest revision as of 13:44, 18 July 2016

Problems from the 1985 USAMO.

Problem 1

Determine whether or not there are any positive integral solutions of the simultaneous equations \[x_1^2+x_2^2+\cdots+x_{1985}^2=y^3, \hspace{20pt} x_1^3+x_2^3+\cdots+x_{1985}^3=z^2\] with distinct integers $x_1,x_2,\cdots,x_{1985}$.

Solution

Problem 2

Determine each real root of

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

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

See Also

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

These problems are copyrighted © by the Mathematical Association of America.

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