Difference between revisions of "1986 INMO"
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| − | This was the first INMO conducted, it was conducted by the Indian Institute of Science, Department of Applied Mathematics | + | This was the first INMO conducted, it was conducted by the Indian Institute of Science, Department of Applied Mathematics. It had 9 problems to be completed in 3 hours. |
==Problem 1== | ==Problem 1== | ||
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<cmath>\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}</cmath> | <cmath>\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}</cmath> | ||
| + | |||
| + | ==Problem 4== | ||
| + | Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number | ||
| + | ==Problem 5== | ||
| + | If P(x) is a polynomial with integer coefficients and a, b, c three distinct integers, then show that it is impossible to have P(a)=b, P(b)=c, P(c)=a. | ||
| + | ==Problem 6== | ||
| + | Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal | ||
| + | ==Problem 7== | ||
| + | If a,b,x,y are integers greater than 1, such that a and b have no common factor except 1 and <math>x^a = y^b</math> , show that <math>x = n^b, \; y = n^a</math> for some integer n greater than 1 | ||
| + | ==Problem 8== | ||
| + | Suppose <math>A_1, A_2, \dots, A_6</math> are six sets, each with four elements, and <math>B_1, B_2, \dots, B_n</math> are <math>n</math> sets, each with two elements. | ||
| + | |||
| + | Let | ||
| + | <cmath>A_1 \cup A_2 \cup \cdots \cup A_6 \;=\; B_1 \cup B_2 \cup \cdots \cup B_n.</cmath> | ||
| + | |||
| + | It is given that each element of <math>S</math> belongs to exactly four of the <math>A_i</math>’s and to exactly three of the <math>B_j</math>’s. Find <math>n</math>. | ||
| + | ==Problem 9== | ||
| + | Show that among all quadrilateral of a given Perimeter the square has the largest area | ||
Revision as of 07:35, 2 October 2025
This was the first INMO conducted, it was conducted by the Indian Institute of Science, Department of Applied Mathematics. It had 9 problems to be completed in 3 hours.
Contents
Problem 1
A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exchanged places. When did he go out ?
Problem 2
Solve:
\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z = 2 \\[6pt] \log_{3} y + \log_{9} z + \log_{9} x = 2 \\[6pt] \log_{4} z + \log_{16} x + \log_{16} y = 2 \end{cases}
Problem 3
Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles Prove that:
![\[\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}\]](http://latex.artofproblemsolving.com/6/f/a/6fa429a041db3675cc939c4d84aa176e54f9b696.png)
Problem 4
Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number
Problem 5
If P(x) is a polynomial with integer coefficients and a, b, c three distinct integers, then show that it is impossible to have P(a)=b, P(b)=c, P(c)=a.
Problem 6
Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal
Problem 7
If a,b,x,y are integers greater than 1, such that a and b have no common factor except 1 and 
, show that 
for some integer n greater than 1
Problem 8
Suppose 
are six sets, each with four elements, and 
are 
sets, each with two elements.
Let
![\[A_1 \cup A_2 \cup \cdots \cup A_6 \;=\; B_1 \cup B_2 \cup \cdots \cup B_n.\]](http://latex.artofproblemsolving.com/7/8/c/78c1bb2ea5b6fccf381087d7cc9c86db252bfc6a.png)
It is given that each element of 
belongs to exactly four of the 
’s and to exactly three of the 
’s. Find .
Problem 9
Show that among all quadrilateral of a given Perimeter the square has the largest area
