# Difference between revisions of "1976 IMO Problems"

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===Problem 5=== | ===Problem 5=== | ||

+ | Let a set of <math>p</math> equations be given, | ||

+ | <cmath> | ||

+ | \begin{array}{ccccccc} | ||

+ | a_{11}x_1&+&\cdots&+&a_{1q}x_q&=&0,\\ | ||

+ | a_{21}x_1&+&\cdots&+&a_{2q}x_q&=&0,\\ | ||

+ | &&&\vdots&&&\\ | ||

+ | a_{p1}x_1&+&\cdots&+&a_{pq}x_q&=&0,\\ | ||

+ | \end{array} | ||

+ | </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>. | ||

===Problem 6=== | ===Problem 6=== | ||

+ | |||

+ | For all positive integral <math>n</math>, <math>u_{n+1}=u_n(u_{n-1}^2-2)-u_1</math>, <math>u_0=2</math>, and <math>u_1=2\frac12</math>. Prove that | ||

+ | <cmath>3\log_2[u_n]=2^n-(-1)^n,</cmath> | ||

+ | where <math>[x]</math> is the integral part of <math>x</math>. |

## Revision as of 23:02, 1 June 2015

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 .