Difference between revisions of "1961 IMO Problems"

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==Day I==
 
==Day I==
 
===Problem 1===
 
===Problem 1===
 +
(''Hungary'')
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Solve the system of equations:
 +
 +
<center>
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<math>
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\begin{matrix}
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\quad x + y + z \!\!\! &= a \; \, \\
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x^2 +y^2+z^2 \!\!\! &=b^2 \\
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\qquad \qquad xy \!\!\!  &= z^2
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\end{matrix}
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</math>
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</center>
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 +
where <math>a </math> and <math>b </math> are constants.  Give the conditions that <math>a </math> and <math>b </math> must satisfy so that <math>x, y, z </math> (the solutions of the system) are distinct positive numbers.
  
 
[[1961 IMO Problems/Problem 1 | Solution]]
 
[[1961 IMO Problems/Problem 1 | Solution]]
  
 +
===Problem 2===
 +
Let ''a'',''b'', and ''c'' be the lengths of a triangle whose area is ''S''.  Prove that
  
 +
<math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math>
  
===Problem 2===
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In what case does equality hold?
  
 
[[1961 IMO Problems/Problem 2 | Solution]]
 
[[1961 IMO Problems/Problem 2 | Solution]]
  
 +
===Problem 3===
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Solve the equation
  
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<math>\cos^n{x} - \sin^n{x} = 1</math>
  
===Problem 3===
+
where ''n'' is a given positive integer.
  
 
[[1961 IMO Problems/Problem 3 | Solution]]
 
[[1961 IMO Problems/Problem 3 | Solution]]
 
  
  
 
==Day 2==
 
==Day 2==
 
===Problem 4===
 
===Problem 4===
 +
 +
In the interior of [[triangle]] <math>P_1 P_2 P_3</math> a [[point]] ''P'' is given.  Let <math>Q_1,Q_2,Q_3</math> be the [[intersection]]s of <math>PP_1, PP_2,PP_3</math> with the opposing [[edge]]s of triangle <math>ABC</math>.  Prove that among the [[ratio]]s <math>\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}</math> there exists one not larger than 2 and one not smaller than 2.
  
 
[[1961 IMO Problems/Problem 4 | Solution]]
 
[[1961 IMO Problems/Problem 4 | Solution]]
  
 +
===Problem 5===
 +
 +
Construct a triangle ''ABC'' if the following elements are given: <math>AC = b, AB = c</math>, and <math>\angle AMB = \omega \left(\omega < 90^{\circ}\right)</math> where ''M'' is the midpoint of ''BC''.  Prove that the construction has a solution if and only if
  
 +
<math>b \tan{\frac{\omega}{2}} \le c < b</math>
  
===Problem 5===
+
In what case does equality hold?
  
 
[[1961 IMO Problems/Problem 5 | Solution]]
 
[[1961 IMO Problems/Problem 5 | Solution]]
 
 
  
 
===Problem 6===
 
===Problem 6===
 +
Consider a plane <math>\epsilon</math> and three non-collinear points <math>A,B,C</math> on the same side of <math>\epsilon</math>; suppose the plane determined by these three points is not parallel to <math>\epsilon</math>. In plane <math>\epsilon</math> take three arbitrary points <math>A',B',C'</math>. Let <math>L,M,N</math> be the midpoints of segments <math>AA', BB', CC'</math>; Let <math>G</math> be the centroid of the triangle <math>LMN</math>. (We will not consider positions of the points <math>A', B', C'</math> such that the points <math>L,M,N</math> do not form a triangle.) What is the locus of point <math>G</math> as <math>A', B', C'</math> range independently over the plane <math>\epsilon</math>?
  
 
[[1961 IMO Problems/Problem 6 | Solution]]
 
[[1961 IMO Problems/Problem 6 | Solution]]
  
  
 +
== Resources ==
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* [[IMO Problems and Solutions, with authors]]
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* [[Mathematics competition resources]]
  
 
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{{IMO box|year=1961|before=[[1960 IMO]]|after=[[1962 IMO]]}}
==See Also==
 

Latest revision as of 20:18, 20 August 2020

Day I

Problem 1

(Hungary) Solve the system of equations:

$\begin{matrix} \quad x + y + z \!\!\! &= a \; \, \\ x^2 +y^2+z^2 \!\!\! &=b^2 \\ \qquad \qquad xy \!\!\!  &= z^2 \end{matrix}$

where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x, y, z$ (the solutions of the system) are distinct positive numbers.

Solution

Problem 2

Let a,b, and c be the lengths of a triangle whose area is S. Prove that

$a^2 + b^2 + c^2 \ge 4S\sqrt{3}$

In what case does equality hold?

Solution

Problem 3

Solve the equation

$\cos^n{x} - \sin^n{x} = 1$

where n is a given positive integer.

Solution


Day 2

Problem 4

In the interior of triangle $P_1 P_2 P_3$ a point P is given. Let $Q_1,Q_2,Q_3$ be the intersections of $PP_1, PP_2,PP_3$ with the opposing edges of triangle $ABC$. Prove that among the ratios $\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}$ there exists one not larger than 2 and one not smaller than 2.

Solution

Problem 5

Construct a triangle ABC if the following elements are given: $AC = b, AB = c$, and $\angle AMB = \omega \left(\omega < 90^{\circ}\right)$ where M is the midpoint of BC. Prove that the construction has a solution if and only if

$b \tan{\frac{\omega}{2}} \le c < b$

In what case does equality hold?

Solution

Problem 6

Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?

Solution


Resources

1961 IMO (Problems) • Resources
Preceded by
1960 IMO
1 2 3 4 5 6 Followed by
1962 IMO
All IMO Problems and Solutions
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