Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

1967 IMO Problems: Difference between revisions

5849206328x (talk | contribs)
editor
1,190 edits
Created page with 'Problems of the 9th IMO 1967 in Yugoslavia. ==Problem 1== Let <math>ABCD</math> be a parallelogram with side lengths <math>AB = a</math>, <math>AD = 1</math>, and with <ma…'
 
Hamstpan38825 (talk | contribs)
editor
270 edits
No edit summary
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
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
Line 8: Line 9:
[[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>.
Line 14: Line 15:
[[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
Line 22: Line 23:
[[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.
Line 28: Line 30:
[[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
Line 40: Line 42:
[[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.

Day I

Problem 1

Let $ABCD$ be a parallelogram with side lengths $AB = a$, $AD = 1$, and with $\angle BAD = \alpha$. If $\triangle ABD$ is acute, prove that the four circles of radius $1$ with centers $A$, $B$, $C$, $D$ cover the parallelogram if and only if \[a\leq \cos \alpha + \sqrt{3} \sin \alpha .\]

Solution

Problem 2

Prove that if one and only one edge of a tetrahedron is greater than $1$, then its volume is $\leq 1/8$.

Solution

Problem 3

Let $k$, $m$, $n$ be natural numbers such that $m + k + 1$ is a prime greater than $n + 1$. Let $c_s = s(s + 1)$. Prove that the product \[(c_{m+1} - c_k)(c_{m+2}- c_k)\cdots (c_{m+n}- c_k)\] is divisible by the product $c_1 c_2\cdots c_n$.

Solution

Day II

Problem 4

Let $A_0 B_0 C_0$ and $A_1 B_1 C_1$ be any two acute-angled triangles. Consider all triangles $ABC$ that are similar to $\triangle A_1 B_1 C_1$ (so that vertices $A_1$, $B_1$, $C_1$ correspond to vertices $A$, $B$, $C$, respectively) and circumscribed about triangle $A_0 B_0 C_0$ (where $A_0$ lies on $BC$, $B_0$ on $CA$, and $C_0$ on $AB$). Of all such possible triangles, determine the one with maximum area, and construct it.

Solution

Problem 5

Consider the sequence $\{ c_n \}$, where \[c_1 = a_1 + a_2 + \cdots + a_8\] \[c_2 = a_1^2 + a_2^2 + \cdots + a_8^2\] \[\cdots\] \[c_n = a_1^n + a_2^n + \cdots + a_8^n\] \[\cdots\] in which $a_1$, $a_2$, $\cdots$, $a_8$ are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence $\{ c_n \}$ are equal to zero. Find all natural numbers $n$ for which $c_n = 0$.

Solution

Problem 6

In a sports contest, there were $m$ medals awarded on $n$ successive days ($n>1$). On the first day, one medal and $1/7$ of the remaining $m - 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n$-th and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?

Solution

1967 IMO (Problems) • Resources
Preceded by
1966 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1968 IMO
All IMO Problems and Solutions
Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=1967_IMO_Problems&oldid=143749"