1984 IMO Longlists

Part of IMO Longlists

Subcontests

(50)

1

Performing operations on a matrix by keeping invariance

(p_{ij})

m\times n

with real entries, set

a_i =\displaystyle\sum_{j=1}^n p_{ij}\text{ for }i = 1,\cdots,m\text{ and }b_j =\displaystyle\sum_{i=1}^m p_{ij}\text{ for }j = 1, . . . , n\longrightarrow(1)

By integering a real number, we mean replacing the number with the integer closest to it. Prove that integering the numbers

a_i, b_j, p_{ij}

can be done in such a way that

(1)

1

Collinearity of points being intersection of circles

P

exterior to a circle

K

, two rays are drawn intersecting

K

in the respective pairs of points

A, A'

B,B'

. For any other pair of points

C, C'

K

D

be the point of intersection of the circumcircles of triangles

PAC

PB'C'

other than point

P

. Similarly, let

D'

be the point of intersection of the circumcircles of triangles

PA'C'

PBC

other than point

P

. Prove that the points

P, D

D'

1

Two cyclists and a pedestrian in a circular runway

Two cyclists leave simultaneously a point

P

in a circular runway with constant velocities

v_1, v_2 (v_1 > v_2)

and in the same sense. A pedestrian leaves

P

at the same time, moving with velocity

v_3 = \frac{v_1+v_2}{12}

. If the pedestrian and the cyclists move in opposite directions, the pedestrian meets the second cyclist

91

seconds after he meets the first. If the pedestrian moves in the same direction as the cyclists, the first cyclist overtakes him

187

seconds before the second does. Find the point where the first cyclist overtakes the second cyclist the first time.

1

Prove CAP=2CPA if BC=AC+0.5AB and BP:PA=1:3

ABC

is given for which

BC = AC + \frac{1}{2}AB

P

AB

BP : PA = 1 : 3

\angle CAP = 2\angle CPA

1

Diagonal of cuboid passes through 1984 unit cubes

Determine positive integers

p, q

r

such that the diagonal of a block consisting of

p\times q\times r

unit cubes passes through exactly

1984

of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.)

1

Finding three nonnegative integers...

a, b, c

be nonnegative integers such that

a \le b \le c, 2b \neq a + c

\frac{a+b+c}{3}

is an integer. Is it possible to find three nonnegative integers

d, e

f

d \le e \le f, f \neq c

, and such that

a^2+b^2+c^2 = d^2 + e^2 + f^2

1

Prove that k is composite and 1984 divides k

a, b, c

be natural numbers such that

a+b+c = 2pq(p^{30}+q^{30}), p > q

being two given positive integers.

(a)

k = a^3 + b^3 + c^3

is not a prime number.

(b)

a\cdot b\cdot c

is maximum, then

1984

k

1

Find necessary and sufficient condition for regular polygon.

P

be a convex planar polygon with equal angles. Let

l_1,\cdots, l_n

be its sides. Show that a necessary and sufficient condition for

P

to be regular is that the sum of the ratios

\frac{l_i}{l_{i+1}} (i = 1,\cdots, n; l_{n+1}= l_1)

equals the number of sides.

1

Union of A_i s is a subset of union of B_i s

X

be an arbitrary nonempty set contained in the plane and let sets

A_1, A_2,\cdots,A_m

B_1, B_2,\cdots, B_n

be its images under parallel translations. Let us suppose that

A_1\cup A_2 \cup \cdots\cup A_m \subset B_1 \cup B_2 \cup\cdots\cup B_n

and that the sets

A_1, A_2,\cdots,A_m

are disjoint. Prove that

m \le n

1

Find sequence of natural numbers satisfying conditions

Find a sequence of natural numbers

a_i

a_i = \displaystyle\sum_{r=1}^{i+4} d_r

d_r \neq d_s

r \neq s

d_r

a_i

1

Prove that x_i is either 0 or 1

n > 1

x_i \in \mathbb{R}

i = 1,\cdots, n

S_k = x_1^k+ x^k_2+\cdots+ x^k_n

k \ge 1

S_1 = S_2 =\cdots= S_{n+1}

x_i \in \{0, 1\}

i = 1, 2,\cdots, n.

1

Given two sequences. Show that they have finite common terms

(a_n)_{n\ge 1}

(b_n)_{n\ge 1}

be two sequences of natural numbers such that

a_{n+1} = na_n + 1, b_{n+1} = nb_n - 1

n\ge 1

. Show that these two sequences can have only a finite number of terms in common.

1

Find measure of angle CDE

ABC

be an isosceles triangle,

AB = AC, \angle A = 20^{\circ}

D

AB

E

AC

\angle ACD = 20^{\circ}

\angle ABE = 30^{\circ}

. What is the measure of the angle

\angle CDE

1

Sets forming partition of N iff k is irrational

(MOR 1)

[x]

the greatest integer not exceeding

x

k > 1

, define two sequences:

a_n(k) = [nk]\text{ and } b_n(k) =\left[\frac{nk}{k - 1}\right]

A(k) = \{a_n(k) : n \in\mathbb{N}\}

B(k) = \{b_n(k) : n \in \mathbb{N}\}

A(k)

B(k)

form a partition of

\mathbb{N}

k

1

Constructing polynomials and proving limit

f_1(x) = x^3+a_1x^2+b_1x+c_1 = 0

be an equation with three positive roots

\alpha>\beta>\gamma > 0

. From the equation

f_1(x) = 0

, one constructs the equation

f_2(x) = x^3 +a_2x^2 +b_2x+c_2 = x(x+b_1)^2 -(a_1x+c_1)^2 = 0

. Continuing this process, we get equations

f_3,\cdots, f_n

\lim_{n\to\infty}\sqrt[2^{n-1}]{-a_n} = \alpha

1

Prove that all terms of number triangle are integers

A “number triangle”

(t_{n, k}) (0 \le k \le n)

t_{n,0} = t_{n,n} = 1 (n \ge 0),

t_{n+1,m} =(2 -\sqrt{3})^mt_{n,m} +(2 +\sqrt{3})^{n-m+1}t_{n,m-1} (1 \le m \le n) Prove that all

t_{n,m}

1

Locus of cylinder container's vertical end

A cylindrical container has height

6 cm

4 cm

. It rests on a circular hoop, also of radius

4 cm

, fixed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points. The cylinder is now moved so that each of its circular rims still touches the hoop in two points. Find with proof the locus of one of the cylinder’s vertical ends.

1

Prove existence of points on sides of triangle

c

be the inscribed circle of the triangle

ABC

d

a line tangent to

c

which does not pass through the vertices of triangle

ABC

. Prove the existence of points

A_1,B_1, C_1

, respectively, on the lines

BC,CA,AB

satisfying the following two properties:

(i)

AA_1,BB_1

CC_1

(ii)

AA_1,BB_1

CC_1

d

respectively at points

A' ,B'

C'

\frac{A'A_1}{A' A}=\frac{B'B_1}{B 'B}=\frac{C'C_1}{C'C}

1

Given equation regarding areas, prove triangle is isosceles

ABC

be a triangle with interior angle bisectors

AA_1, BB_1, CC_1

I

\sigma[IA_1B] + \sigma[IB_1C] + \sigma[IC_1A] = \frac{1}{2}\sigma[ABC]

\sigma[ABC]

denotes the area of

ABC

ABC

1

Smallest integer n for f(n)=5 given an identity

f(n)

is defined on the nonnegative integers

n

f(0) = 0, f(1) = 1

f(n) = f\left(n -\frac{1}{2}m(m - 1)\right)-f\left(\frac{1}{2}m(m+ 1)-n\right)

\frac{1}{2}m(m - 1) < n \le \frac{1}{2}m(m+ 1), m \ge 2

. Find the smallest integer

n

f(n) = 5

1

Areas forming arithmetic or geometric progression

Assume that the bisecting plane of the dihedral angle at edge

AB

of the tetrahedron

ABCD

CD

E

S_1, S_2, S_3

, respectively the areas of the triangles

ABC, ABE

ABD

. Prove that no tetrahedron exists for which

S_1, S_2, S_3

(in this order) form an arithmetic or geometric progression.

1

Line with sum of distances from vertices of triangle minimum

In the plane of a given triangle

A_1A_2A_3

determine (with proof) a straight line

l

such that the sum of the distances from

A_1, A_2

A_3

l

is the least possible.

1

C^{2n}_{n} divides lcm(1, 2, ..., 2n)

Prove that for any natural number

n

\dbinom{2n}{n}

divides the least common multiple of the numbers

1, 2,\cdots, 2n -1, 2n

1

Identity involving GIF

For a real number

x

[x]

denote the greatest integer not exceeding

x

m \ge 3

\left[\frac{m(m+1)}{2(2m-1)}\right]=\left[\frac{m+1}{4}\right]

1

Fermat point and properties

Given a triangle

ABC

, three equilateral triangles

AEB, BFC

CGA

are constructed in the exterior of

ABC

(a) CE = AF = BG

(b) CE, AF

BG

have a common point.
I could not find a separate topic for this question and I need one. http://en.wikipedia.org/wiki/Fermat_point of course.

1

Colouring natural numbers using 15 colours

Decide whether it is possible to color the

1984

natural numbers

1, 2, 3, \cdots, 1984

15

colors so that no geometric sequence of length

3

of the same color exists.

1

Partitioning {1, 2, ... ,49} in three subsets

\{1, 2, \cdots, 49\}

is divided into three subsets. Prove that at least one of these subsets contains three different numbers

a, b, c

a + b = c

1

f(x + y)f(x - y) = (f(x)f(y))^2

Determine all continuous functions

f: \mathbb R \to \mathbb R

such that f(x + y)f(x - y) = (f(x)f(y))^2, \forall(x, y) \in\mathbb{R}^2.

1

Construct scalene triangle given a trigonometric equation.

Construct a scalene triangle such that

a(\tan B - \tan C) = b(\tan A - \tan C)

1

a_n <= a_{n+m} <= a_n + a_m

(a_n)_1^{\infty}

be a sequence such that

a_n \le a_{n+m} \le a_n + a_m

for all positive integers

n

m

\frac{a_n}{n}

n

approaches infinity.

1

539^n+m.132^n is divisible by 262417

Determine the smallest positive integer

m

529^n+m\cdot 132^n

is divisible by

262417

for all odd positive integers

n

1

Sum of edges of circumscribable tetrahedron.

A tetrahedron is inscribed in a sphere of radius

1

such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.

1

Perpendiculars from midpoints to incentral triangle

The circle inscribed in the triangle

A_1A_2A_3

is tangent to its sides

A_1A_2, A_2A_3, A_3A_1

T_1, T_2, T_3

, respectively. Denote by

M_1, M_2, M_3

the midpoints of the segments

A_2A_3, A_3A_1, A_1A_2

, respectively. Prove that the perpendiculars through the points

M_1, M_2, M_3

T_2T_3, T_3T_1, T_1T_2

meet at one point.

1

Proving geometric identities for special hexagon

The opposite sides of the reentrant hexagon

AFBDCE

intersect at the points

K,L,M

(as shown in the figure). It is given that

AL = AM = a, BM = BK = b

CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f

. http://imgur.com/LUFUh.png

(a)

a

and the three angles

\alpha, \beta

\gamma

at the vertices

A, B,

C,

respectively, satisfying the condition

\alpha+\beta+\gamma<180^{\circ}

, show that all the angles and sides of the hexagon are thereby uniquely determined.

(b)

\frac{1}{a}+\frac{1}{c}=\frac{1}{b}+\frac{1}{d}

Easier version of

(b)

(a + f)(b + d)(c + e)= (a + e)(b + f)(c + d)

1

2m gon with exactly one pair of parallel sides

Given a regular convex

2m

- sided polygon

P

, show that there is a

2m

\pi

with the same vertices as

P

(but in different order) such that

\pi

has exactly one pair of parallel sides.

1

Cirumradii of triangle and triangle formed by its medians

With the medians of an acute-angled triangle another triangle is constructed. If

R

R_m

are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that

R_m>\frac{5}{6}R

1

Finding natural numbers satisfying inequality

Prove that there exist distinct natural numbers

m_1,m_2, \cdots , m_k

satisfying the conditions

\pi^{-1984}<25-\left(\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}\right)<\pi^{-1960}

\pi

is the ratio between a circle and its diameter.

1

Inequality with a function on sets.

S_n = \{1, \cdots, n\}

f

be a function that maps every subset of

S_n

into a positive real number and satisfies the following condition: For all

A \subseteq S_n

x, y \in S_n, x \neq y, f(A \cup \{x\})f(A \cup \{y\}) \le f(A \cup \{x, y\})f(A)

. Prove that for all

A,B \subseteq S_n

the following inequality holds:

f(A) \cdot f(B) \le f(A \cup B) \cdot f(A \cap B)

1

Minimum of a function in isosceles triangle.

ABC

be an isosceles triangle with right angle at point

A

. Find the minimum of the function

F

F(M) = BM +CM-\sqrt{3}AM

1

Minimum of sum or difference of fifth powers

Consider all the sums of the form

\displaystyle\sum_{k=1}^{1985} e_kk^5=\pm 1^5\pm 2^5\pm\cdots\pm1985^5

e_k=\pm 1

. What is the smallest nonnegative value attained by a sum of this type?

1

Finding words in a new language.

In the Martian language every finite sequence of letters of the Latin alphabet letters is a word. The publisher “Martian Words” makes a collection of all words in many volumes. In the first volume there are only one-letter words, in the second, two-letter words, etc., and the numeration of the words in each of the volumes continues the numeration of the previous volume. Find the word whose numeration is equal to the sum of numerations of the words Prague, Olympiad, Mathematics.

1

Odd numbers of heads and a tail in a fair coin.

A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.

1

Number formed by 1, 9, 8, 4 dividing 1984!

a, b, c, d

be a permutation of the numbers

1, 9, 8,4

n = (10a + b)^{10c+d}

. Find the probability that

1984!

is divisible by

n.

1

Travelling between n cities using minimum planes.

One country has

n

cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of flights that the worker will have to use?

1

Filling a cuboid with smaller cuboids

2\times 2\times 12

box fixed in space is to be filled with twenty-four

1 \times 1 \times 2

bricks. In how many ways can this be done?

1

Probability of process ending after some number of steps.

(1)

a

white balls and

b

(2)

Draw one ball at random.

(3)

If the ball is white, then stop. Otherwise, add two black balls and go to step

2

S

be the number of draws before the process terminates. For the cases

a = b = 1

a = b = 2

a_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n

, and the expectation value of the number of balls drawn:

E(S) =\displaystyle\sum_{n\ge1} na_n.

1

Writing a fraction as sum of two unit fractions

\frac{3}{10}

can be written as the sum of two positive fractions with numerator

1

\frac{3}{10} =\frac{1}{5}+\frac{1}{10}

\frac{3}{10}=\frac{1}{4}+\frac{1}{20}

. There are the only two ways in which this can be done. In how many ways can

\frac{3}{1984}

be written as the sum of two positive fractions with numerator

1

? Is there a positive integer

n,

not divisible by

3

\frac{3}{n}

can be written as the sum of two positive fractions with numerator

1

1984

1

If P^n+Q^n+R^n = 0, prove that n < 3

P,Q,R

be the polynomials with real or complex coefficients such that at least one of them is not constant. If

P^n+Q^n+R^n = 0

n < 3.

1

System of equations

a,b,c

be positive numbers with

\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}

Prove that the system of equations

\sqrt{y-a}+\sqrt{z-a}=1

\sqrt{z-b}+\sqrt{x-b}=1

\sqrt{x-c}+\sqrt{y-c}=1

has exactly one solution

(x,y,z)

in real numbers. It was proposed by Poland. Have fun! :lol:

1

divisors of n - pigfly #2

1=d_1<d_2<....<d_k=n

be all different divisors of positive integer n written in ascending order. Determine all n such that:

d_6^{2} +d_7^{2} - 1=n