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Difference between revisions of "1963 IMO Problems"

(New page: Problems of the 5th IMO 1963 Poland. == Day I == === Problem 1 === Find all real roots of the equation <center><math>\sqrt{x^2-p}+2\sqrt{x^2-1}=x</math>,</center>where <math>p</math...)
 
m (→‎Resources)
 
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=== Problem 4 ===
 
=== Problem 4 ===
 
Find all solutions <math>x_1,x_2,x_3,x_4,x_5</math> of the system
 
Find all solutions <math>x_1,x_2,x_3,x_4,x_5</math> of the system
−
<center><math>\begin{eqnarray}
+
<cmath>\begin{eqnarray*}
 
x_5+x_2&=&yx_1\\
 
x_5+x_2&=&yx_1\\
 
x_1+x_3&=&yx_2\\
 
x_1+x_3&=&yx_2\\
 
x_2+x_4&=&yx_3\\
 
x_2+x_4&=&yx_3\\
 
x_3+x_5&=&yx_4\\
 
x_3+x_5&=&yx_4\\
−
x_4+x_1&=&yx_5,\end{eqnarray}</math></center>
+
x_4+x_1&=&yx_5,\end{eqnarray*}</cmath>
 
where <math>y</math> is a parameter.
 
where <math>y</math> is a parameter.
  
Line 48: Line 48:
 
* [[1963 IMO]]
 
* [[1963 IMO]]
 
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1963 IMO 1963 Problems on the Resources page]
 
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1963 IMO 1963 Problems on the Resources page]
 +
* [[IMO Problems and Solutions, with authors]]
 +
* [[Mathematics competition resources]]
 +
 +
{{IMO box|year=1963|before=[[1962 IMO]]|after=[[1964 IMO]]}}

Latest revision as of 20:16, 20 August 2020

Problems of the 5th IMO 1963 Poland.

Day I

Problem 1

Find all real roots of the equation

$\sqrt{x^2-p}+2\sqrt{x^2-1}=x$,

where $p$ is a real parameter.

Solution

Problem 2

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are the vertices of right angles with one side passing through $A$, and the other side intersecting the segment $BC$.

Solution

Problem 3

In an $n$-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation

$a_1\ge a_2\ge \cdots \ge a_n$.

Prove that $a_1=a_2=\cdots = a_n$.

Solution

Day II

Problem 4

Find all solutions $x_1,x_2,x_3,x_4,x_5$ of the system \begin{eqnarray*} x_5+x_2&=&yx_1\\ x_1+x_3&=&yx_2\\ x_2+x_4&=&yx_3\\ x_3+x_5&=&yx_4\\ x_4+x_1&=&yx_5,\end{eqnarray*} where $y$ is a parameter.

Solution

Problem 5

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$.

Solution

Problem 6

Five students, $A,B,C,D,E$, took part in a contest. One prediction was that the contestants would finish in the order $ABCDE$. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

Solution

Resources

1963 IMO (Problems) • Resources
Preceded by
1962 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1964 IMO
All IMO Problems and Solutions
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