Difference between revisions of "1967 IMO Problems"
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Problems of the 9th [[IMO]] 1967 in Yugoslavia. | Problems of the 9th [[IMO]] 1967 in Yugoslavia. | ||
− | ==Problem 1== | + | ==Day I== |
+ | ===Problem 1=== | ||
Let <math>ABCD</math> be a parallelogram with side lengths <math>AB = a</math>, <math>AD = 1</math>, and with <math>\angle BAD = \alpha </math>. If <math>\triangle ABD</math> is acute, prove that the four circles of radius <math>1</math> with centers <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math> cover the parallelogram if and only if | Let <math>ABCD</math> be a parallelogram with side lengths <math>AB = a</math>, <math>AD = 1</math>, and with <math>\angle BAD = \alpha </math>. If <math>\triangle ABD</math> is acute, prove that the four circles of radius <math>1</math> with centers <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math> cover the parallelogram if and only if | ||
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[[1967 IMO Problems/Problem 1|Solution]] | [[1967 IMO Problems/Problem 1|Solution]] | ||
− | ==Problem 2== | + | ===Problem 2=== |
Prove that if one and only one edge of a tetrahedron is greater than <math>1</math>, then its volume is <math>\leq 1/8</math>. | Prove that if one and only one edge of a tetrahedron is greater than <math>1</math>, then its volume is <math>\leq 1/8</math>. | ||
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[[1967 IMO Problems/Problem 2|Solution]] | [[1967 IMO Problems/Problem 2|Solution]] | ||
− | ==Problem 3== | + | ===Problem 3=== |
Let <math>k</math>, <math>m</math>, <math>n</math> be natural numbers such that <math>m + k + 1</math> is a prime greater than <math>n + 1</math>. Let <math>c_s = s(s + 1)</math>. Prove that the product | Let <math>k</math>, <math>m</math>, <math>n</math> be natural numbers such that <math>m + k + 1</math> is a prime greater than <math>n + 1</math>. Let <math>c_s = s(s + 1)</math>. Prove that the product | ||
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[[1967 IMO Problems/Problem 3|Solution]] | [[1967 IMO Problems/Problem 3|Solution]] | ||
− | ==Problem 4== | + | ==Day II== |
+ | ===Problem 4=== | ||
Let <math>A_0 B_0 C_0</math> and <math>A_1 B_1 C_1</math> be any two acute-angled triangles. Consider all triangles <math>ABC</math> that are similar to <math>\triangle A_1 B_1 C_1</math> (so that vertices <math>A_1</math>, <math>B_1</math>, <math>C_1</math> correspond to vertices <math>A</math>, <math>B</math>, <math>C</math>, respectively) and circumscribed about triangle <math>A_0 B_0 C_0</math> (where <math>A_0</math> lies on <math>BC</math>, <math>B_0</math> on <math>CA</math>, and <math>C_0</math> on <math>AB</math>). Of all such possible triangles, determine the one with maximum area, and construct it. | Let <math>A_0 B_0 C_0</math> and <math>A_1 B_1 C_1</math> be any two acute-angled triangles. Consider all triangles <math>ABC</math> that are similar to <math>\triangle A_1 B_1 C_1</math> (so that vertices <math>A_1</math>, <math>B_1</math>, <math>C_1</math> correspond to vertices <math>A</math>, <math>B</math>, <math>C</math>, respectively) and circumscribed about triangle <math>A_0 B_0 C_0</math> (where <math>A_0</math> lies on <math>BC</math>, <math>B_0</math> on <math>CA</math>, and <math>C_0</math> on <math>AB</math>). Of all such possible triangles, determine the one with maximum area, and construct it. | ||
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[[1967 IMO Problems/Problem 4|Solution]] | [[1967 IMO Problems/Problem 4|Solution]] | ||
− | ==Problem 5== | + | ===Problem 5=== |
Consider the sequence <math>\{ c_n \}</math>, where | Consider the sequence <math>\{ c_n \}</math>, where | ||
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[[1967 IMO Problems/Problem 5|Solution]] | [[1967 IMO Problems/Problem 5|Solution]] | ||
− | ==Problem 6== | + | ===Problem 6=== |
In a sports contest, there were <math>m</math> medals awarded on <math>n</math> successive days (<math>n>1</math>). On the first day, one medal and <math>1/7</math> of the remaining <math>m - 1</math> medals were awarded. On the second day, two medals and <math>1/7</math> of the now remaining medals were awarded; and so on. On the <math>n</math>-th and last day, the remaining <math>n</math> medals were awarded. How many days did the contest last, and how many medals were awarded altogether? | In a sports contest, there were <math>m</math> medals awarded on <math>n</math> successive days (<math>n>1</math>). On the first day, one medal and <math>1/7</math> of the remaining <math>m - 1</math> medals were awarded. On the second day, two medals and <math>1/7</math> of the now remaining medals were awarded; and so on. On the <math>n</math>-th and last day, the remaining <math>n</math> medals were awarded. How many days did the contest last, and how many medals were awarded altogether? | ||
[[1967 IMO Problems/Problem 6|Solution]] | [[1967 IMO Problems/Problem 6|Solution]] | ||
+ | |||
+ | * [[1967 IMO]] | ||
+ | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1967 IMO 1967 Problems on the Resources page] | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{IMO box|year=1967|before=[[1966 IMO]]|after=[[1968 IMO]]}} |
Latest revision as of 11:40, 29 January 2021
Problems of the 9th IMO 1967 in Yugoslavia.
Contents
[hide]Day I
Problem 1
Let be a parallelogram with side lengths
,
, and with
. If
is acute, prove that the four circles of radius
with centers
,
,
,
cover the parallelogram if and only if
Problem 2
Prove that if one and only one edge of a tetrahedron is greater than , then its volume is
.
Problem 3
Let ,
,
be natural numbers such that
is a prime greater than
. Let
. Prove that the product
is divisible by the product
.
Day II
Problem 4
Let and
be any two acute-angled triangles. Consider all triangles
that are similar to
(so that vertices
,
,
correspond to vertices
,
,
, respectively) and circumscribed about triangle
(where
lies on
,
on
, and
on
). Of all such possible triangles, determine the one with maximum area, and construct it.
Problem 5
Consider the sequence , where
in which
,
,
,
are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence
are equal to zero. Find all natural numbers
for which
.
Problem 6
In a sports contest, there were medals awarded on
successive days (
). On the first day, one medal and
of the remaining
medals were awarded. On the second day, two medals and
of the now remaining medals were awarded; and so on. On the
-th and last day, the remaining
medals were awarded. How many days did the contest last, and how many medals were awarded altogether?
- 1967 IMO
- IMO 1967 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1967 IMO (Problems) • Resources | ||
Preceded by 1966 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1968 IMO |
All IMO Problems and Solutions |