MathDB

1

Prove that the functional equations are equivalent

f(x + y) = f(x) + f(y),

\text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) (x, y \in \mathbb R) are equivalent.

24

1

Line segment PQ is tangent to circle C

A circle

C

with center

O

on base

BC

of an isosceles triangle

ABC

is tangent to the equal sides

AB,AC

. If point

P

on

AB

and point

Q

on

AC

are selected such that

PB \times CQ = (\frac{BC}{2})^2

, prove that line segment

PQ

is tangent to circle

C

, and prove the converse.

23

1

Find n such that 2^8 +2^{11} +2^n is a perfect square

Find all natural numbers

n

for which

2^8 +2^{11} +2^n

is a perfect square.

21

1

Number of solutions of two systems

Let

N

be the number of integral solutions of the equation

x^2 - y^2 = z^3 - t^3

satisfying the condition

0 \leq x, y, z, t \leq 10^6

, and let

M

be the number of integral solutions of the equation

x^2 - y^2 = z^3 - t^3 + 1

satisfying the condition

0 \leq x, y, z, t \leq 10^6

. Prove that

N >M.

20

1

Find maximum of the sum

Given the integer

n > 1

and the real number

a > 0

determine the maximum of

\sum_{i=1}^{n-1} x_i x_{i+1}

taken over all nonnegative numbers

x_i

with sum

a.

19

1

Find the smallest integer m for which b_m > a_{100}.

Consider the sequences

(a_n), (b_n)

defined by a_1=3, b_1=100 , a_{n+1}=3^{a_n} , b_{n+1}=100^{b_n} Find the smallest integer

m

for which

b_m > a_{100}.

18

1

Prove that there exist fewer than 2^m such positive integers

Let

m

positive integers

a_1, \dots , a_m

be given. Prove that there exist fewer than

2^m

positive integers

b_1, \dots , b_n

such that all sums of distinct

b_k

’s are distinct and all

a_i \ (i \leq m)

occur among them.

17

1

Determine the angles of triangle PQR

Inside an equilateral triangle

ABC

one constructs points

P, Q

and

R

such that

\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.

Determine the angles of triangle

PQR.

16

1

All 16 members of F have exactly one element in common

Let

K

denote the set

\{a, b, c, d, e\}

.

F

is a collection of

16

different subsets of

K

, and it is known that any three members of

F

have at least one element in common. Show that all

16

members of

F

have exactly one element in common.

14

1

Bases of logarithms

Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

13

1

Show that 20/60 < sin 20 < 21/60

Show that

\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.

12

1

S-family over R

Let

R

be a set of exactly

6

elements. A set

F

of subsets of

R

is called an

S

-family over

R

if and only if it satisfies the following three conditions: (i) For no two sets

X, Y

in

F

is

X \subseteq Y

;
(ii) For any three sets

X, Y,Z

in

F

,

X \cup Y \cup Z \neq R,

(iii)

\bigcup_{X \in F} X = R

11

1

Find max and/or min P

Given real numbers

x_1, x_2, \dots , x_n \ (n \geq 2)

, with

x_i \geq \frac 1n \ (i = 1, 2, \dots, n)

and with

x_1^2+x_2^2+\cdots+x_n^2 = 1

, find whether the product

P = x_1x_2x_3 \cdots x_n

has a greatest and/or least value and if so, give these values.

10

1

Show the fact for any vectors a,b in Euclidean space

Show that for any vectors

a, b

in Euclidean space,

|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2

Remark. Here

\times

denotes the vector product.

8

1

Find f(x) for every rational x

For all rational

x

satisfying

0 \leq x < 1

, the functions

f

is defined by f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases} Given that

x = 0.b_1b_2b_3 \cdots

is the binary representation of

x

, find, with proof,

f(x)

.

6

1

Find the real values of p

Find the real values of

p

for which the equation

\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}

in

x

has exactly two real distinct roots.(

\sqrt t

means the positive square root of

t

).

5

1

Find the maximal cardinality

Let

n \geq 2

be an integer. Find the maximal cardinality of a set

M

of pairs

(j, k)

of integers,

1 \leq j < k \leq n

, with the following property: If

(j, k) \in M

, then

(k,m) \not \in M

for any

m.

4

1

Prism which has the upper and inferior basis the pentagons

We consider a prism which has the upper and inferior basis the pentagons:

A_{1}A_{2}A_{3}A_{4}A_{5}

and

B_{1}B_{2}B_{3}B_{4}B_{5}

. Each of the sides of the two pentagons and the segments

A_{i}B_{j}

with

i,j=1,\ldots,5

is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

3

1

Find all polynomials f(x) such that f(x)f(2x^2)=f(2x^3 + x)

Find all polynomials

f(x)

with real coefficients for which

f(x)f(2x^2) = f(2x^3 + x).

2

1

Find the probablity - a bag containing 5 pairs of socks

From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative?

1

Nice problem - Prove that we can dissect into lozenges

Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

1979 IMO Shortlist

Subcontests

Prove that the functional equations are equivalent

Line segment PQ is tangent to circle C

Find n such that 2^8 +2^{11} +2^n is a perfect square

Number of solutions of two systems

Find maximum of the sum

Find the smallest integer m for which b_m > a_{100}.

Prove that there exist fewer than 2^m such positive integers

Determine the angles of triangle PQR

All 16 members of F have exactly one element in common

Bases of logarithms

Show that 20/60 < sin 20 < 21/60

S-family over R

Find max and/or min P

Show the fact for any vectors a,b in Euclidean space

Find f(x) for every rational x

Find the real values of p

Find the maximal cardinality

Prism which has the upper and inferior basis the pentagons

Find all polynomials f(x) such that f(x)f(2x^2)=f(2x^3 + x)

Find the probablity - a bag containing 5 pairs of socks

Nice problem - Prove that we can dissect into lozenges

1979 IMO Shortlist

Subcontests

Prove that the functional equations are equivalent

Line segment PQ is tangent to circle C

Find n such that 2^8 +2^{11} +2^n is a perfect square

Number of solutions of two systems

Find maximum of the sum

Find the smallest integer m for which b_m &gt; a_{100}.

Prove that there exist fewer than 2^m such positive integers

Determine the angles of triangle PQR

All 16 members of F have exactly one element in common

Bases of logarithms

Show that 20/60 &lt; sin 20 &lt; 21/60

S-family over R

Find max and/or min P

Show the fact for any vectors a,b in Euclidean space

Find f(x) for every rational x

Find the real values of p

Find the maximal cardinality

Prism which has the upper and inferior basis the pentagons

Find all polynomials f(x) such that f(x)f(2x^2)=f(2x^3 + x)

Find the probablity - a bag containing 5 pairs of socks

Nice problem - Prove that we can dissect into lozenges

Find the smallest integer m for which b_m > a_{100}.

Show that 20/60 < sin 20 < 21/60