MathDB

1

Existence of lattice points on circle

\mathcal{C}

be the circumcircle of the square with vertices

(0, 0), (0, 1978), (1978, 0), (1978, 1978)

in the Cartesian plane. Prove that

\mathcal{C}

contains no other point for which both coordinates are integers.

51

1

Finding relations among angles given condition

Find the relations among the angles of the triangle

ABC

whose altitude

AH

and median

AM

satisfy

\angle BAH =\angle CAM

.

50

1

Maximal length in a varying tetrahedron

A variable tetrahedron

ABCD

has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal

(BC = DA, CA = DB, AB = DC)

; and the vertices

A,B,C

lie respectively on three fixed spheres with the same center

P

and radii

3, 4, 12

. What is the maximal length of

PD

?

49

1

Four points in space and segments to form a triangle

Let

A,B,C,D

be four arbitrary distinct points in space.

(a)

Prove that using the segments

AB +CD, AC +BD

and

AD +BC

, it is always possible to construct a triangle

T

that is non-degenerate and has no obtuse angle.

(b)

What should these four points satisfy in order for the triangle

T

to be right-angled?

47

1

Proving an identity and P is a polynomial.

Given the expression

P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right],

prove:

(a) P_n(x)

satisfies the identity

P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0.

(b) P_n(x)

is a polynomial in

x

of degree

n.

45

1

Proving an inequality involving powers

If

r > s >0

and

a > b > c

, prove that

a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.

44

1

Prove that c/a+c/b>=2 if C=60 degrees in triangle ABC

In

ABC

with

\angle C = 60^{\circ}

, prove that

\frac{c}{a} + \frac{c}{b} \ge2.

43

1

Writing k/p^2 as sum of two unit fractions

If

p

is a prime greater than

3

, show that at least one of the numbers

\frac{3}{p^2} , \frac{4}{p^2} , \cdots, \frac{p-2}{p^2}

is expressible in the form

\frac{1}{x} + \frac{1}{y}

, where

x

and

y

are positive integers.

40

1

Proving a combinatorial identity and finding expression

If

C^p_n=\frac{n!}{p!(n-p)!} (p \ge 1)

, prove the identity

C^p_n=C^{p-1}_{n-1} + C^{p-1}_{n-2} + \cdots + C^{p-1}_{p} + C^{p-1}_{p-1}

and then evaluate the sum

S = 1\cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 97 \cdot 98 \cdot 99.

39

1

Prove that A+B+1 is a perfect square

A

is a

2m

-digit positive integer each of whose digits is

1

.

B

is an

m

-digit positive integer each of whose digits is

4

. Prove that

A+B +1

is a perfect square.

38

1

Construct a chord which is trisected by given radii.

Given a circle, construct a chord that is trisected by two given noncollinear radii.

36

1

Jumping by fifteens and finding the number of remaining.

The integers

1

through

1000

are located on the circumference of a circle in natural order. Starting with

1

, every fifteenth number (i.e.,

1, 16, 31, \cdots

) is marked. The marking is continued until an already marked number is reached. How many of the numbers will be left unmarked?

35

1

Existence of constant for concave functions

A sequence

(a_n)_0^N

of real numbers is called concave if

2a_n\ge a_{n-1} + a_{n+1}

for all integers

n, 1 \le n \le N - 1

.

(a)

Prove that there exists a constant

C >0

such that

\left(\displaystyle\sum_{n=0}^{N}a_n\right)^2\ge C(N - 1)\displaystyle\sum_{n=0}^{N}a_n^2\:\:\:\:\:(1)

for all concave positive sequences

(a_n)^N_0

(b)

Prove that

(1)

holds with

C = \frac{3}{4}

and that this constant is best possible.

37

1

Simplifying a logarithmic expression

Simplify

\frac{1}{\log_a(abc)}+\frac{1}{\log_b(abc)}+\frac{1}{\log_c(abc)},

where

a, b, c

are positive real numbers.

54

1

Existence of mutually perpendicular planes

Let

p, q

and

r

be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes

\alpha, \beta

, and

\gamma

, containing

p, q

, and

r

respectively, that are perpendicular to each other

(\alpha\perp\beta, \beta\perp\gamma, \gamma\perp \alpha).

33

1

Convex functions and logarithms

A sequence

(a_n)^{\infty}_0

of real numbers is called convex if

2a_n\le a_{n-1}+a_{n+1}

for all positive integers

n

. Let

(b_n)^{\infty}_0

be a sequence of positive numbers and assume that the sequence

(\alpha^nb_n)^{\infty}_0

is convex for any choice of

\alpha > 0

. Prove that the sequence

(\log b_n)^{\infty}_0

is convex.

8

1

Proving inequalities of areas given inequality of sides

For two given triangles

A_1A_2A_3

and

B_1B_2B_3

with areas

\Delta_A

and

\Delta_B

, respectively,

A_iA_k \ge B_iB_k, i, k = 1, 2, 3

. Prove that

\Delta_A \ge \Delta_B

if the triangle

A_1A_2A_3

is not obtuse-angled.

6

1

Sides of triangle being polynomial functions

Prove that for all

X > 1

, there exists a triangle whose sides have lengths

P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1

, and

P_3(X) = X^4-1

. Prove that all these triangles have the same greatest angle and calculate it.

42

1

Five concyclic points and intersection on circle.

A,B,C,D,E

are points on a circle

O

with radius equal to

r

. Chords

AB

and

DE

are parallel to each other and have length equal to

x

. Diagonals

AC,AD,BE, CE

are drawn. If segment

XY

on

O

meets

AC

at

X

and

EC

at

Y

, prove that lines

BX

and

DY

meet at

Z

on the circle.

31

1

P'(x)=nQ(x) for polynomials P, Q

Let the polynomials

P(x) = x^n + a_{n-1}x^{n-1 }+ \cdots + a_1x + a_0,

Q(x) = x^m + b_{m-1}x^{m-1} + \cdots + b_1x + b_0,

be given satisfying the identity

P(x)^2 = (x^2 - 1)Q(x)^2 + 1

. Prove the identity

P'(x) = nQ(x).

29

1

Finding two functions satisfying conditions

Given a nonconstant function

f : \mathbb{R}^+ \longrightarrow\mathbb{R}

such that

f(xy) = f(x)f(y)

for any

x, y > 0

, find functions

c, s : \mathbb{R}^+ \longrightarrow \mathbb{R}

that satisfy

c\left(\frac{x}{y}\right) = c(x)c(y)-s(x)s(y)

for all

x, y > 0

and

c(x)+s(x) = f(x)

for all

x > 0

.

28

1

Two functions and their properties.

Let

c, s

be real functions defined on

\mathbb{R}\setminus\{0\}

that are nonconstant on any interval and satisfy

c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{ for any }x \neq 0, y \neq 0

Prove that:

(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x)

for any

x = 0

, and also

c(1) = 1, s(1) = s(-1) = 0

;

(b) c

and

s

are either both even or both odd functions (a function

f

is even if

f(x) = f(-x)

for all

x

, and odd if

f(x) = -f(-x)

for all

x

). Find functions

c, s

that also satisfy

c(x) + s(x) = x^n

for all

x

, where

n

is a given positive integer.

25

1

Two real roots of magnitude less than 1

Consider a polynomial

P(x) = ax^2 + bx + c

with

a > 0

that has two real roots

x_1, x_2

. Prove that the absolute values of both roots are less than or equal to

1

if and only if

a + b + c \ge 0, a -b + c \ge 0

, and

a - c \ge 0

.

21

1

Proving a tangency with square, incircle and parallels.

A circle touches the sides

AB,BC, CD,DA

of a square at points

K,L,M,N

respectively, and

BU, KV

are parallel lines such that

U

is on

DM

and

V

on

DN

. Prove that

UV

touches the circle.

13

1

Seeing two satellites A and B at distance 2r

The satellites

A

and

B

circle the Earth in the equatorial plane at altitude

h

. They are separated by distance

2r

, where

r

is the radius of the Earth. For which

h

can they be seen in mutually perpendicular directions from some point on the equator?

12

1

x^3+a^3x^2+b^3x+c^3=0

The equation

x^3 + ax^2 + bx + c = 0

has three (not necessarily distinct) real roots

t, u, v

. For which

a, b, c

do the numbers

t^3, u^3, v^3

satisfy the equation

x^3 + a^3x^2 + b^3x + c^3 = 0

?

11

1

Find all n s.t. m<n, n<1978, (m, n)=1 implies m is prime.

Find all natural numbers

n < 1978

with the following property: If

m

is a natural number,

1 < m < n

, and

(m, n) = 1

(i.e.,

m

and

n

are relatively prime), then

m

is a prime number.

20

1

Circle and perpendicular radii.

Let

O

be the center of a circle. Let

OU,OV

be perpendicular radii of the circle. The chord

PQ

passes through the midpoint

M

of

UV

. Let

W

be a point such that

PM = PW

, where

U, V,M,W

are collinear. Let

R

be a point such that

PR = MQ

, where

R

lies on the line

PW

. Prove that

MR = UV

.
Alternative version: A circle

S

is given with center

O

and radius

r

. Let

M

be a point whose distance from

O

is

\frac{r}{\sqrt{2}}

. Let

PMQ

be a chord of

S

. The point

N

is defined by

\overrightarrow{PN} =\overrightarrow{MQ}

. Let

R

be the reflection of

N

by the line through

P

that is parallel to

OM

. Prove that

MR =\sqrt{2}r

.

15

1

Existence of multiple of n with no digit 1

Prove that for every positive integer

n

coprime to

10

there exists a multiple of

n

that does not contain the digit

1

in its decimal representation.

14

1

Existence of solution for function given inequality

Let

p(x, y)

and

q(x, y)

be polynomials in two variables such that for

x \ge 0, y \ge 0

the following conditions hold:

(i) p(x, y)

and

q(x, y)

are increasing functions of

x

for every fixed

y

.

(ii) p(x, y)

is an increasing and

q(x)

is a decreasing function of

y

for every fixed

x

.

(iii) p(x, 0) = q(x, 0)

for every

x

and

p(0, 0) = 0

. Show that the simultaneous equations

p(x, y) = a, q(x, y) = b

have a unique solution in the set

x \ge 0, y \ge 0

for all

a, b

satisfying

0 \le b \le a

but lack a solution in the same set if

a < b

.

27

1

Sixth digit after decimal point

Determine the sixth number after the decimal point in the number

(\sqrt{1978} +\lfloor\sqrt{1978}\rfloor)^{20}

18

1

Inequality on number of lattice points

Given a natural number

n

, prove that the number

M(n)

of points with integer coordinates inside the circle

(O(0, 0),\sqrt{n})

satisfies

\pi n - 5\sqrt{n} + 1<M(n) < \pi n+ 4\sqrt{n} + 1

3

1

x^2-2x[x]+x-alpha=0

Find all numbers

\alpha

for which the equation

x^2 - 2x[x] + x -\alpha = 0

has two nonnegative roots. (

[x]

denotes the largest integer less than or equal to x.)

10

1

n divides p-q where p, q are primes

Show that for any natural number

n

there exist two prime numbers

p

and

q, p \neq q

, such that

n

divides their difference.

5

1

Existence of point P, A', B', C' satisfying conditions

Prove that for any triangle

ABC

there exists a point P in the plane of the triangle and three points

A' , B'

, and

C'

on the lines

BC, AC

, and

AB

respectively such that

AB \cdot PC'= AC \cdot PB'= BC \cdot PA'= 0.3M^2,

where

M = max\{AB,AC,BC\}

.

2

1

(x + 2x^2 +...+ nx^n)^2 = a_2x^2 + a_3x^3 +...+ a_{2n}x^{2n}

If

f(x) = (x + 2x^2 +\cdots+ nx^n)^2 = a_2x^2 + a_3x^3 + \cdots+ a_{2n}x^{2n},

prove that

a_{n+1} + a_{n+2} + \cdots + a_{2n} =\dbinom{n + 1}{2}\frac{5n^2 + 5n + 2}{12}

22

1

x+y is a divisor of x^2+y^2

Let

x

and

y

be two integers not equal to

0

such that

x+y

is a divisor of

x^2+y^2

. And let

\frac{x^2+y^2}{x+y}

be a divisor of

1978

. Prove that

x = y

.
German IMO Selection Test 1979, problem 2

1978 IMO Longlists

Subcontests

Existence of lattice points on circle

Finding relations among angles given condition

Maximal length in a varying tetrahedron

Four points in space and segments to form a triangle

Proving an identity and P is a polynomial.

Proving an inequality involving powers

Prove that c/a+c/b>=2 if C=60 degrees in triangle ABC

Writing k/p^2 as sum of two unit fractions

Proving a combinatorial identity and finding expression

Prove that A+B+1 is a perfect square

Construct a chord which is trisected by given radii.

Jumping by fifteens and finding the number of remaining.

Existence of constant for concave functions

Simplifying a logarithmic expression

Existence of mutually perpendicular planes

Convex functions and logarithms

Proving inequalities of areas given inequality of sides

Sides of triangle being polynomial functions

Five concyclic points and intersection on circle.

P'(x)=nQ(x) for polynomials P, Q

Finding two functions satisfying conditions

Two functions and their properties.

Two real roots of magnitude less than 1

Proving a tangency with square, incircle and parallels.

Seeing two satellites A and B at distance 2r

x^3+a^3x^2+b^3x+c^3=0

Find all n s.t. m<n, n<1978, (m, n)=1 implies m is prime.

Circle and perpendicular radii.

Existence of multiple of n with no digit 1

Existence of solution for function given inequality

Sixth digit after decimal point

Inequality on number of lattice points

x^2-2x[x]+x-alpha=0

n divides p-q where p, q are primes

Existence of point P, A', B', C' satisfying conditions

(x + 2x^2 +...+ nx^n)^2 = a_2x^2 + a_3x^3 +...+ a_{2n}x^{2n}

x+y is a divisor of x^2+y^2

1978 IMO Longlists

Subcontests

Existence of lattice points on circle

Finding relations among angles given condition

Maximal length in a varying tetrahedron

Four points in space and segments to form a triangle

Proving an identity and P is a polynomial.

Proving an inequality involving powers

Prove that c/a+c/b&gt;=2 if C=60 degrees in triangle ABC

Writing k/p^2 as sum of two unit fractions

Proving a combinatorial identity and finding expression

Prove that A+B+1 is a perfect square

Construct a chord which is trisected by given radii.

Jumping by fifteens and finding the number of remaining.

Existence of constant for concave functions

Simplifying a logarithmic expression

Existence of mutually perpendicular planes

Convex functions and logarithms

Proving inequalities of areas given inequality of sides

Sides of triangle being polynomial functions

Five concyclic points and intersection on circle.

P'(x)=nQ(x) for polynomials P, Q

Finding two functions satisfying conditions

Two functions and their properties.

Two real roots of magnitude less than 1

Proving a tangency with square, incircle and parallels.

Seeing two satellites A and B at distance 2r

x^3+a^3x^2+b^3x+c^3=0

Find all n s.t. m&lt;n, n&lt;1978, (m, n)=1 implies m is prime.

Circle and perpendicular radii.

Existence of multiple of n with no digit 1

Existence of solution for function given inequality

Sixth digit after decimal point

Inequality on number of lattice points

x^2-2x[x]+x-alpha=0

n divides p-q where p, q are primes

Existence of point P, A', B', C' satisfying conditions

(x + 2x^2 +...+ nx^n)^2 = a_2x^2 + a_3x^3 +...+ a_{2n}x^{2n}

x+y is a divisor of x^2+y^2

Prove that c/a+c/b>=2 if C=60 degrees in triangle ABC

Find all n s.t. m<n, n<1978, (m, n)=1 implies m is prime.