MathDB

1

0273 combo geo 2nd edition Round 7 p3

P

can be partitioned into

27

parallelograms. Prove that it can also be partitioned into

21

parallelograms.

7.2

1

0272 number theory 2nd edition Round 7 p2

Find all positive integers n with the property that

n^3 - 1

is a perfect square.

7.1

1

0271 combinatorics 2nd edition Round 7 p1

Fifty students take part in a mathematical competition where a set of

8

problems is given (same set to each participant). The final result showed that a total of

171

correct solutions were obtained. Prove that there are

3

of the given problems that have been correctly solved by the same

3

students.

6.1

1

0261 parity with floor function 2nd edition Round 6 p1

Determine the parity of the positive integer

N

, where

N = \lfloor \frac{2002!}{2001 \cdot2003} \rfloor.

6.3

1

0263 combinatorics 2nd edition Round 6 p3

At a party there were some couples attending. As they arrive each person gets to talk with all the other persons which are already in the room. During the party, after all the guests arrive, groups of persons form, such that no two persons forming a couple belong to the same group, and for each two persons that do not form a couple, there is one and only one group to which both belong. Find the number of couples attending the party, knowing that there are less groups than persons at the party.

6.2

1

0262 geometry 2nd edition Round 6 p2

A triangle

ABC

is located in a cartesian plane

\pi

and has a perimeter of

3 + 2\sqrt3

. It is known that the triangle

ABC

has the property that any triangle in the plane

\pi

, congruent with it, contains inside or on the boundary at least one lattice point (a point with both coordinates integers). Prove that the triangle

ABC

is equilateral.

5.3

1

0253 permutation 2nd edition Round 5 p2

Let

n \ge 3

and

\sigma \in S_n

a permutation of the first

n

positive integers. Prove that the numbers

\sigma (1), 2\sigma (2), 3\sigma(3), ... , n\sigma (n)

cannot form an arithmetic, nor a geometric progression.

5.2

1

0252 number theory 2nd edition Round 5 p2

Let S be the set of positive integers

n

for which

\frac{3}{n}

cannot be written as the sum of two rational numbers of the form

\frac{1}{k}

, where

k

is a positive integer. Prove that

S

cannot be written as the union of finitely many arithmetic progressions.

5.1

1

0251 combo geo 2nd edition Round 5 p1

For which positive integers

n \ge 4

one can find n points in the plane, no three collinear, such that for each triangle formed with three of the

n

points which are on the convex hull, exactly one of the

n - 3

remaining points belongs to its interior.

4.3

1

0243 combinatorics 2nd edition Round 4 p3

In a country there are

100

cities, some of which are connected by roads. For each four cities there are at least two roads between them. Also, there is no path that passes through each city exactly one time. Prove that one can choose two cities among those

100

, such that each of the

98

remaining cities would be connected by a road with at least one of the two chosen cities.

4.2

1

0242 translations of equilateral 2nd edition Round 4 p2

Given is a finite set of points

M

and an equilateral triangle

\Delta

in the plane. It is known that for any subset

M' \subset M

, which has no more than

9

points, can be covered by two translations of the triangle

\Delta

. Prove that the entire set

M

can be covered by two translations of

\Delta

.

4.1

1

0241 functional 2nd edition Round 4 p1

The real polynomial

f \in R[X]

has an odd degree and it is given that

f

is co-prime with

g(x) = x^2 - x - 1

and

f(x^2 - 1) = f(x)f(-x), \forall x \in R.

Prove that

f

has at least two complex non-real roots.

3.2

1

0232 geometry 2nd edition Round 3 p2

Let

ABC

be a triangle with altitudes

AD, BE, CF

. Choose the points

A_1, B_1, C_1

on the lines

AD, BE, CF

respectively, such that

\frac{AA_1}{AD}= \frac{BB_1}{BE}= \frac{CC_1}{CF} = k.

Find all values of

k

such that the triangle

A_1B_1C_1

is similar to the triangle

ABC

for all triangles

ABC

.

3.3

1

0233 number theory 2nd edition Round 3 p3

Prove that for every positive integer

m

there exists a positive integer N such that

S(2^n) > m

for every positive integer

n > N

, where by

S(x)

we denote the sum of digits of a positive integer

x

.

3.1

1

0231 functional inequality 2nd edition Round 3 p1

Determine all values of

a \in R

such that there exists a function

f : [0, 1] \to R

fulfilling the following inequality for all

x \ne y

:

|f(x) - f(y)| \ge a.

2.3

1

0223 geometry 2nd edition Round 2 p3

Prove that if two triangles are inscribed in the same circle, then their incircles are not strictly contained one into each other.

2.2

1

0222 number theory 2nd edition Round 2 p2

Let

\{a_n\}_{n\ge 0}

be a sequence of rational numbers given by

a_0 = a_1 = a_2 = a_3 = 1

and for all

n \ge 4

we have

a_{n-4}a_n = a_{n-3}a_{n-1} + a^2_{n-2}

. Prove that all the terms of the sequence are integers.

2.1

1

0221 inequalities 2nd edition Round 2 p1

Given are six reals

a, b, c, x, y, z

such that

(a + b + c)(x + y + z) = 3

and

(a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = 4

. Prove that

ax + by + cz \ge 0

.

1.3

1

0213 geometry 2nd edition Round 1 p3

Given are on a line three points

A, B, C

such that

AB = 1

and

BC = x

. Consider the circles

\Omega_a, \Omega_b

and

\Omega_c

which are tangent to the given line at the points

A, B, C

respectively, and such that

\Omega_b

is tangent externally with both

\Omega_a

and

\Omega_c

in points

M, N

respectively. Find all values of the radius of the circle

\Omega_b

for which the triangle

BMN

is isosceles.

1.2

1

0212 friendly permutations 2nd edition Round 1 p2

We call a permutation

\sigma

of the first

n

positive integers friendly if and only if the following conditions are fulfilled: (1)

\sigma(k + 1) \in \{2\sigma(k), 2\sigma(k) - 1, 2\sigma(k) - n, 2\sigma(k) - n - 1\}, \forall k \in \{1, 2, ..., n - 1\}

(2)

\sigma(1) \in \{2 \sigma(n), 2\sigma(n) - 1, 2\sigma(n) - n, 2\sigma(n) - n - 1\}

. Find all positive integers

n

for which there exists such a friendly permutation of the first

n

positive integers.

1.1

1

0211 inequalities 2nd edition Round 1 p1

Let

x, y, z

be positive numbers such that

xyz \le 2

and

\frac{1}{x^2}+ \frac{1}{y^2}+ \frac{1}{z^2}< k

, for some real

k \ge 2

. Find all values of

k

such that the conditions above imply that there exist a triangle having the side-lengths

x, y, z

.

MathLinks Contest 2nd

Subcontests

0273 combo geo 2nd edition Round 7 p3

0272 number theory 2nd edition Round 7 p2

0271 combinatorics 2nd edition Round 7 p1

0261 parity with floor function 2nd edition Round 6 p1

0263 combinatorics 2nd edition Round 6 p3

0262 geometry 2nd edition Round 6 p2

0253 permutation 2nd edition Round 5 p2

0252 number theory 2nd edition Round 5 p2

0251 combo geo 2nd edition Round 5 p1

0243 combinatorics 2nd edition Round 4 p3

0242 translations of equilateral 2nd edition Round 4 p2

0241 functional 2nd edition Round 4 p1

0232 geometry 2nd edition Round 3 p2

0233 number theory 2nd edition Round 3 p3

0231 functional inequality 2nd edition Round 3 p1

0223 geometry 2nd edition Round 2 p3

0222 number theory 2nd edition Round 2 p2

0221 inequalities 2nd edition Round 2 p1

0213 geometry 2nd edition Round 1 p3

0212 friendly permutations 2nd edition Round 1 p2

0211 inequalities 2nd edition Round 1 p1