1976 IMO Problems: Difference between revisions

Revision as of 22:02, 1 June 2015

Problems of the 18th IMO 1976 in Austria.

Day 1

Problem 1

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$ . Determine all the possible values that the other diagonal can have.

Problem 2

Let $P_{1}(x) = x^{2} - 2$ and $P_{j}(x) = P_{1}(P_{j - 1}(x))$ for $j= 2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x) = x$ are all real and distinct.

Problem 3

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$ , with the sides parallel to those of the box, then exactly $40$ 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 $1976$ .

Problem 5

Let a set of $p$ equations be given, $\begin{array}{ccccccc} a_{11}x_1&+&\cdots&+&a_{1q}x_q&=&0,\\ a_{21}x_1&+&\cdots&+&a_{2q}x_q&=&0,\\ &&&\vdots&&&\\ a_{p1}x_1&+&\cdots&+&a_{pq}x_q&=&0,\\ \end{array}$ with coefficients $a_{ij}$ satisfying $a_{ij}=-1$ , $0$ , or $+1$ for all $i=1,\dots, p$ , and $j=1,\dots, q$ . Prove that if $q=2p$ , there exists a solution $x_1, \dots, x_q$ of this system such that all $x_j$ ( $j=1,\dots, q$ ) are integers satisfying $|x_j|\le q$ and $x_j\ne 0$ for at least one value of $j$ .

Problem 6

For all positive integral $n$ , $u_{n+1}=u_n(u_{n-1}^2-2)-u_1$ , $u_0=2$ , and $u_1=2\frac12$ . Prove that $3\log_2[u_n]=2^n-(-1)^n,$ where $[x]$ is the integral part of $x$ .

@@ Line 13: / Line 13: @@
 ===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===
+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>.

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