Difference between revisions of "1960 IMO Problems"
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=== Problem 1 === | === Problem 1 === | ||
− | + | Determine all three-digit numbers <math>N</math> having the property that <math>N</math> is divisible by 11, and <math>\dfrac{N}{11}</math> is equal to the sum of the squares of the digits of <math>N</math>. | |
[[1960 IMO Problems/Problem 1 | Solution]] | [[1960 IMO Problems/Problem 1 | Solution]] | ||
=== Problem 2 === | === Problem 2 === | ||
+ | For what values of the variable <math>x</math> does the following inequality hold: | ||
+ | <cmath>\dfrac{4x^2}{(1 - \sqrt {2x + 1})^2} < 2x + 9 \ ?</cmath> | ||
Line 15: | Line 17: | ||
=== Problem 3 === | === Problem 3 === | ||
+ | In a given right triangle <math>ABC</math>, the hypotenuse <math>BC</math>, of length <math>a</math>, is divided into <math>n</math> equal parts (<math>n</math> and odd integer). Let <math>\alpha</math> be the acute angle subtending, from <math>A</math>, that segment which contains the midpoint of the hypotenuse. Let <math>h</math> be the length of the altitude to the hypotenuse of the triangle. Prove that: | ||
+ | <center><math> | ||
+ | \tan{\alpha}=\frac{4nh}{(n^2-1)a}. | ||
+ | </math> | ||
+ | </center> | ||
− | |||
− | |||
− | |||
− | |||
[[1960 IMO Problems/Problem 3 | Solution]] | [[1960 IMO Problems/Problem 3 | Solution]] | ||
+ | |||
== Day II == | == Day II == | ||
=== Problem 4 === | === Problem 4 === | ||
− | + | Construct triangle <math>ABC</math>, given <math>h_a</math>, <math>h_b</math> (the altitudes from <math>A</math> and <math>B</math>), and <math>m_a</math>, the median from vertex <math>A</math>. | |
[[1960 IMO Problems/Problem 4 | Solution]] | [[1960 IMO Problems/Problem 4 | Solution]] | ||
=== Problem 5 === | === Problem 5 === | ||
+ | Consider the cube <math>ABCDA'B'C'D'</math> (with face <math>ABCD</math> directly above face <math>A'B'C'D'</math>). | ||
+ | a) Find the locus of the midpoints of the segments <math>XY</math>, where <math>X</math> is any point of <math>AC</math> and <math>Y</math> is any point of <math>B'D'</math>; | ||
+ | |||
+ | b) Find the locus of points <math>Z</math> which lie on the segment <math>XY</math> of part a) with <math>ZY = 2XZ</math>. | ||
[[1960 IMO Problems/Problem 5 | Solution]] | [[1960 IMO Problems/Problem 5 | Solution]] | ||
=== Problem 6 === | === Problem 6 === | ||
+ | Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let <math>V_1</math> be the volume of the cone and <math>V_2</math> be the volume of the cylinder. | ||
+ | |||
+ | a) Prove that <math>V_1 \neq V_2</math>; | ||
+ | |||
+ | b) Find the smallest number <math>k</math> for which <math>V_1 = kV_2</math>; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone. | ||
[[1960 IMO Problems/Problem 6 | Solution]] | [[1960 IMO Problems/Problem 6 | Solution]] | ||
+ | |||
+ | |||
+ | === Problem 7 === | ||
+ | An isosceles trapezoid with bases <math>a</math> and <math>c</math> and altitude <math>h</math> is given. | ||
+ | |||
+ | a) On the axis of symmetry of this trapezoid, find all points <math>P</math> such that both legs of the trapezoid subtend right angles at <math>P</math>; | ||
+ | |||
+ | b) Calculate the distance of <math>p</math> from either base; | ||
+ | |||
+ | c) Determine under what conditions such points <math>P</math> actually exist. Discuss various cases that might arise. | ||
+ | |||
+ | [[1960 IMO Problems/Problem 7 | Solution]] | ||
+ | |||
== Resources == | == Resources == | ||
* [[1960 IMO]] | * [[1960 IMO]] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1960 IMO 1960 Problems on the Resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1960 IMO 1960 Problems on the Resources page] | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{IMO box|year=1960|before=[[1959 IMO]]|after=[[1961 IMO]]}} |
Latest revision as of 21:21, 20 August 2020
Problems of the 2nd IMO 1960 Romania.
Contents
Day I
Problem 1
Determine all three-digit numbers having the property that
is divisible by 11, and
is equal to the sum of the squares of the digits of
.
Problem 2
For what values of the variable does the following inequality hold:
Problem 3
In a given right triangle , the hypotenuse
, of length
, is divided into
equal parts (
and odd integer). Let
be the acute angle subtending, from
, that segment which contains the midpoint of the hypotenuse. Let
be the length of the altitude to the hypotenuse of the triangle. Prove that:
![$\tan{\alpha}=\frac{4nh}{(n^2-1)a}.$](http://latex.artofproblemsolving.com/d/7/6/d7667ee6a4058553b8290f72ffe77f099ab473bf.png)
Day II
Problem 4
Construct triangle , given
,
(the altitudes from
and
), and
, the median from vertex
.
Problem 5
Consider the cube (with face
directly above face
).
a) Find the locus of the midpoints of the segments , where
is any point of
and
is any point of
;
b) Find the locus of points which lie on the segment
of part a) with
.
Problem 6
Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let be the volume of the cone and
be the volume of the cylinder.
a) Prove that ;
b) Find the smallest number for which
; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.
Problem 7
An isosceles trapezoid with bases and
and altitude
is given.
a) On the axis of symmetry of this trapezoid, find all points such that both legs of the trapezoid subtend right angles at
;
b) Calculate the distance of from either base;
c) Determine under what conditions such points actually exist. Discuss various cases that might arise.
Resources
- 1960 IMO
- IMO 1960 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1960 IMO (Problems) • Resources | ||
Preceded by 1959 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1961 IMO |
All IMO Problems and Solutions |