Difference between revisions of "1976 IMO Problems"
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===Problem 1=== | ===Problem 1=== | ||
In a convex quadrilateral (in the plane) with the area of <math>32 \text{ cm}^{2}</math> the sum of two opposite sides and a diagonal is <math>16 \text{ cm}</math>. Determine all the possible values that the other diagonal can have. | In a convex quadrilateral (in the plane) with the area of <math>32 \text{ cm}^{2}</math> the sum of two opposite sides and a diagonal is <math>16 \text{ cm}</math>. Determine all the possible values that the other diagonal can have. | ||
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+ | [[1976 IMO Problems/Problem 1|Solution]] | ||
===Problem 2=== | ===Problem 2=== | ||
Let <math>P_{1}(x) = x^{2} - 2</math> and <math>P_{j}(x) = P_{1}(P_{j - 1}(x))</math> for <math>j= 2,\ldots</math> Prove that for any positive integer n the roots of the equation <math>P_{n}(x) = x</math> are all real and distinct. | Let <math>P_{1}(x) = x^{2} - 2</math> and <math>P_{j}(x) = P_{1}(P_{j - 1}(x))</math> for <math>j= 2,\ldots</math> Prove that for any positive integer n the roots of the equation <math>P_{n}(x) = x</math> are all real and distinct. | ||
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+ | [[1976 IMO Problems/Problem 2|Solution]] | ||
===Problem 3=== | ===Problem 3=== | ||
A box whose shape is a parallelepiped can be completely filled with cubes of side <math>1.</math> If we put in it the maximum possible number of cubes, each of volume <math>2</math>, with the sides parallel to those of the box, then exactly <math>40</math> percent from the volume of the box is occupied. Determine the possible dimensions of the box. | A box whose shape is a parallelepiped can be completely filled with cubes of side <math>1.</math> If we put in it the maximum possible number of cubes, each of volume <math>2</math>, with the sides parallel to those of the box, then exactly <math>40</math> percent from the volume of the box is occupied. Determine the possible dimensions of the box. | ||
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+ | [[1976 IMO Problems/Problem 3|Solution]] | ||
==Day 2== | ==Day 2== | ||
===Problem 4=== | ===Problem 4=== | ||
Find the largest number obtainable as the product of positive integers whose sum is <math>1976</math>. | Find the largest number obtainable as the product of positive integers whose sum is <math>1976</math>. | ||
+ | [[1976 IMO Problems/Problem 4|Solution]] | ||
===Problem 5=== | ===Problem 5=== | ||
Let a set of <math>p</math> equations be given, | Let a set of <math>p</math> equations be given, | ||
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</cmath> | </cmath> | ||
with coefficients <math>a_{ij}</math> satisfying <math>a_{ij}=-1</math>, <math>0</math>, or <math>+1</math> for all <math>i=1,\dots, p</math>, and <math>j=1,\dots, q</math>. Prove that if <math>q=2p</math>, there exists a solution <math>x_1, \dots, x_q</math> of this system such that all <math>x_j</math> (<math>j=1,\dots, q</math>) are integers satisfying <math>|x_j|\le q</math> and <math>x_j\ne 0</math> for at least one value of <math>j</math>. | with coefficients <math>a_{ij}</math> satisfying <math>a_{ij}=-1</math>, <math>0</math>, or <math>+1</math> for all <math>i=1,\dots, p</math>, and <math>j=1,\dots, q</math>. Prove that if <math>q=2p</math>, there exists a solution <math>x_1, \dots, x_q</math> of this system such that all <math>x_j</math> (<math>j=1,\dots, q</math>) are integers satisfying <math>|x_j|\le q</math> and <math>x_j\ne 0</math> for at least one value of <math>j</math>. | ||
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+ | [[1976 IMO Problems/Problem 5|Solution]] | ||
===Problem 6=== | ===Problem 6=== | ||
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<cmath>3\log_2[u_n]=2^n-(-1)^n,</cmath> | <cmath>3\log_2[u_n]=2^n-(-1)^n,</cmath> | ||
where <math>[x]</math> is the integral part of <math>x</math>. | where <math>[x]</math> is the integral part of <math>x</math>. | ||
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+ | [[1976 IMO Problems/Problem 6|Solution]] | ||
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+ | * [[1976 IMO]] | ||
+ | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1976 IMO 1976 Problems on the Resources page] | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] {{IMO box|year=1976|before=[[1975 IMO]]|after=[[1977 IMO]]}} |
Latest revision as of 15:25, 29 January 2021
Problems of the 18th IMO 1976 in Austria.
Contents
Day 1
Problem 1
In a convex quadrilateral (in the plane) with the area of the sum of two opposite sides and a diagonal is . Determine all the possible values that the other diagonal can have.
Problem 2
Let and for Prove that for any positive integer n the roots of the equation are all real and distinct.
Problem 3
A box whose shape is a parallelepiped can be completely filled with cubes of side If we put in it the maximum possible number of cubes, each of volume , with the sides parallel to those of the box, then exactly percent from the volume of the box is occupied. Determine the possible dimensions of the box.
Day 2
Problem 4
Find the largest number obtainable as the product of positive integers whose sum is .
Problem 5
Let a set of equations be given, with coefficients satisfying , , or for all , and . Prove that if , there exists a solution of this system such that all () are integers satisfying and for at least one value of .
Problem 6
For all positive integral , , , and . Prove that where is the integral part of .
- 1976 IMO
- IMO 1976 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1976 IMO (Problems) • Resources | ||
Preceded by 1975 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1977 IMO |
All IMO Problems and Solutions |