IME ITA PRACTICE Imo 1959 2009, Notas de estudo de Matemática

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International Mathematical Olympiad

Problems and Solutions

IMO

The most important and prestigious mathematical competition for high-school students

Contents

• 1. Introduction to IMO…………………………………………………………………………………
• 2. IMO Problems 1959 – 2009……………………………………………………………………..
• 3. IMO Solutions 1970 – 2003 & 2006………………………………………………………….
• 4. IMO Training Materials……………………………………………………………………………

IMO Problems

1959 - 2009

First International Olympiad, 1959

Prove that the fraction 2114 nn+4+3 is irreducible for every natural number n.

For what real values of x is √ (x +

2 x − 1) +

√ (x −

2 x − 1) = A,

given (a) A =

2 , (b) A = 1, (c) A = 2, where only non-negative real numbers are admitted for square roots?

Let a, b, c be real numbers. Consider the quadratic equation in cos x :

a cos^2 x + b cos x + c = 0.

Using the numbers a, b, c, form a quadratic equation in cos 2x, whose roots are the same as those of the original equation. Compare the equations in cos x and cos 2x for a = 4, b = 2, c = − 1.

Construct a right triangle with given hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

An arbitrary point M is selected in the interior of the segment AB. The squares AM CD and M BEF are constructed on the same side of AB, with the segments AM and M B as their respective bases. The circles circum- scribed about these squares, with centers P and Q, intersect at M and also at another point N. Let N ′^ denote the point of intersection of the straight lines AF and BC. (a) Prove that the points N and N ′^ coincide. (b) Prove that the straight lines M N pass through a fixed point S indepen- dent of the choice of M. (c) Find the locus of the midpoints of the segments P Q as M varies between A and B.

Two planes, P and Q, intersect along the line p. The point A is given in the plane P, and the point C in the plane Q; neither of these points lies on the straight line p. Construct an isosceles trapezoid ABCD (with AB parallel to CD) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively.

An isosceles trapezoid with bases a and c and altitude h is given. (a) On the axis of symmetry of this trapezoid, find all points P such that both legs of the trapezoid subtend right angles at P. (b) Calculate the distance of P from either base. (c) Determine under what conditions such points P actually exist. (Discuss various cases that might arise.)

Third International Olympiad, 1961

Solve the system of equations:

x + y + z = a x^2 + y^2 + z^2 = b^2 xy = z^2

where a and b are constants. Give the conditions that a and b must satisfy so that x, y, z (the solutions of the system) are distinct positive numbers.

Let a, b, c be the sides of a triangle, and T its area. Prove: a^2 +b^2 +c^2 ≥ 4

3 T.

In what case does equality hold?

Solve the equation cosn^ x − sinn^ x = 1, where n is a natural number.

Consider triangle P 1 P 2 P 3 and a point P within the triangle. Lines P 1 P, P 2 P, P 3 P intersect the opposite sides in points Q 1 , Q 2 , Q 3 respectively. Prove that, of the numbers P 1 P P Q 1

P 2 P

P Q 2

P 3 P

P Q 3

at least one is ≤ 2 and at least one is ≥ 2.

Construct triangle ABC if AC = b, AB = c and 6 AM B = ω, where M is the midpoint of segment BC and ω < 90 ◦. Prove that a solution exists if and only if

b tan

ω 2

≤ c < b.

In what case does the equality hold?

Fourth International Olympiad, 1962

Find the smallest natural number n which has the following properties: (a) Its decimal representation has 6 as the last digit. (b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number n.

Determine all real numbers x which satisfy the inequality:

√ 3 − x −

x + 1 >

Consider the cube ABCDA′B′C′D′^ (ABCD and A′B′C′D′^ are the upper and lower bases, respectively, and edges AA′, BB′, CC′, DD′^ are parallel). The point X moves at constant speed along the perimeter of the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B′C′CB in the direction B′C′CBB′. Points X and Y begin their motion at the same instant from the starting positions A and B′, respectively. Determine and draw the locus of the midpoints of the segments XY.

Solve the equation cos^2 x + cos^2 2 x + cos^2 3 x = 1.

On the circle K there are given three distinct points A, B, C. Construct (using only straightedge and compasses) a fourth point D on K such that a circle can be inscribed in the quadrilateral thus obtained.

Consider an isosceles triangle. Let r be the radius of its circumscribed circle and ρ the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is

d =

√ r(r − 2 ρ).

The tetrahedron SABC has the following property: there exist five spheres, each tangent to the edges SA, SB, SC, BCCA, AB, or to their extensions. (a) Prove that the tetrahedron SABC is regular. (b) Prove conversely that for every regular tetrahedron five such spheres exist.

Five students, A, B, C, D, E, took part in a contest. One prediction was that the contestants would finish in the order ABCDE. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second pre- diction had the contestants finishing in the order DAECB. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

Sixth International Olympiad, 1964

(a) Find all positive integers n for which 2n^ − 1 is divisible by 7. (b) Prove that there is no positive integer n for which 2n^ + 1 is divisible by

Suppose a, b, c are the sides of a triangle. Prove that

a^2 (b + c − a) + b^2 (c + a − b) + c^2 (a + b − c) ≤ 3 abc.

A circle is inscribed in triangle ABC with sides a, b, c. Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts off a triangle from ∆ABC. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of a, b, c).

Seventeen people correspond by mail with one another - each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

Suppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point per- pendiculars are drawn to all the lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have.

In tetrahedron ABCD, vertex D is connected with D 0 the centroid of ∆ABC. Lines parallel to DD 0 are drawn through A, B and C. These lines intersect the planes BCD, CAD and ABD in points A 1 , B 1 and C 1 , respectively. Prove that the volume of ABCD is one third the volume of A 1 B 1 C 1 D 0. Is the result true if point D 0 is selected anywhere within ∆ABC?

In a plane a set of n points (n ≥ 3) is given. Each pair of points is connected by a segment. Let d be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length d. Prove that the number of diameters of the given set is at most n.

Eighth International Olympiad, 1966

In a mathematical contest, three problems, A, B, C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem A, the number who solved B was twice the number who solved C. The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. Of all students who solved just one problem, half did not solve problem A. How many students solved only problem B?

Let a, b, c be the lengths of the sides of a triangle, and α, β, γ, respectively, the angles opposite these sides. Prove that if

a + b = tan

γ 2

(a tan α + b tan β),

the triangle is isosceles.

Prove: The sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

Prove that for every natural number n, and for every real number x 6 = kπ/ 2 t(t = 0, 1 , ..., n; k any integer)

1 sin 2x

sin 4x

sin 2nx

= cot x − cot 2nx.

Solve the system of equations

|a 1 − a 2 | x 2 + |a 1 − a 3 | x 3 + |a 1 − a 4 | x 4 = 1 |a 2 − a 1 | x 1 + |a 2 − a 3 | x 3 + |a 2 − a 3 | x 3 = 1 |a 3 − a 1 | x 1 + |a 3 − a 2 | x 2 = 1 |a 4 − a 1 | x 1 + |a 4 − a 2 | x 2 + |a 4 − a 3 | x 3 = 1

where a 1 , a 2 , a 3 , a 4 are four different real numbers.