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Difference between revisions of "1985 USAMO Problems"
(Corrected statements for problems 4 and 5.)
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==Problem 4==
 
==Problem 4==
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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 you <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==
−
UNDETERMINED
+
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]]

Revision as of 09:23, 27 June 2014

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 you $n$ 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

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