brain

There are two circles in the plane. Let a point A be one

M is a subset of {1, 2, 3, . . ., 15} such that the product of

Let p be an odd prime and n a positive integer. In the

3b.(GBR 4) A sequence of polynomials Pm(x, y, z), m = 0, 1, 2, . . ., in

Determine for which positive integers k the set

Does there exist a function s: Q … such that if x

1a.(CZS 3) The positive integers x1, . . . , xn, n \geq3, satisfy x1 < x2 <

A plane cuts a right circular cone into two parts. The plane is

Prove that there are exactly

Let a1, a2, a3, . . . be any infinite increasing sequence of pos-

Given a, \theta \inR, m \inN, and P(x) = x2m −2|a|mxm cos \theta+a2m,

Let f(x) = xn where n is a fixed positive integer and x =

II 1 (POL 2) Let ai, bi be coprime positive integers for i = 1, 2, . . . , k,

Let ABCD be a convex quadrilateral for which the circle

Let f be a function that satisfies the following conditions:

Let f(n) be the least number of distinct points in the plane

Let n \geqk \geq0 be integers. The numbers c(n, k) are defined as

Let (Fn)n\geq1 be the Fibonacci sequence F1 = F2 = 1, Fn+2 =

Let a > b > c > d be positive integers and suppose

Let r \geq2 be a fixed positive integer, and let F be an infinite

Given nine points in space, no four of which are coplanar,

A number of signal lights are equally spaced along a one-way

Solve the following system of equations, in which a is a given

Let A, B, and C be noncollinear points. Prove that there is

Let dn be the last nonzero digit of the decimal representation

(a) Let n be a positive integer. Prove that there exist distinct

Ten points such that no three of them lie on a line are marked in

Prove the following assertion: The four altitudes of a tetrahe-

Let f, g, and a be polynomials with real coefficients, f and g

For a positive integer n, let f(n) denote the number of ways to

Prove that there exist infinitely many positive integers n such

Let ABC be an isosceles triangle with AC = BC, whose

In the convex pentagon ABCDE, the sides BC, CD, DE have

Let \Gamma1, \Gamma2, \Gamma3, \Gamma4 be distinct circles such that \Gamma1, \Gamma3 are

C5 (USS 5) The right triangles ABC and AB1C1 are similar and have

C3 (CAN 5) Show that

Find the integer represented by

The positive integers … and … are such that the numbers

A4 (BUL 2) Determine all real values of the parameter a for which the

Let p be a prime number. Prove that there exists a prime

Let ABCD be a convex quadrilateral such that AC =

Let S be any point on the circumscribed circle of rianglePQR. Then

Let f and g be two integer-valued functions defined on the set

Let x1, x2, . . . , xn be real numbers satisfying the conditions

Find all solutions … of the equation

In a given tetrahedron ABCD let K and L be the centers of

Prove that the sum of an odd number of unit vectors passing

Consider the n imes n array of nonnegative integers

A point M is chosen on the side AC of the triangle ABC in

Four integers are marked on a circle. At each step we simultaneously replace each number by the difference between this…

Let R+ be the set of all nonnegative real numbers. Given two

For all rational x satisfying 0 \leqx < 1, f is defined by

Prove that the sequence 2n −3 (n > 1) contains a subse-

Let n be a positive integer and let x1 \leqx2 \leq\cdot \cdot \cdot \leqxn be

We are given a positive integer … and a rectangular board … with dimensions …, …. The rectangle is divided into a grid…

Suppose that {x1, x2, . . . , xn} are positive integers for which

C8 (TUN 3) Let ABCD be a convex quadrilateral and draw regular tri-

Let n be a positive integer and let a, b be given real numbers.

Let M be an interior point of the tetrahedron ABCD. Prove

Let a1, a2, . . . , an be positive real numbers, n > 1. Denote by

Two circles Ω1 and Ω2 touch internally the circle Ωin

Determine the smallest natural number n having the following

Let a tetrahedron ABCD be inscribed in a sphere S. Find the

Prove that there is no positive integer n such that for k =

O is a point inside a convex quadrilateral ABCD of area

Let a, b, c, d be nonnegative real numbers such that ab + bc +

The incircle of ABC touches BC, CA, and AB at D, E, and

A function f from the set of positive integers N into itself is

Let p and q be relatively prime positive integers. A subset

Find all functions f : R oR satisfying

The diagonals of a quadrilateral ABCD are perpendicular:

Prove:

Let N be a positive integer. Two players A and B, taking

Prove that for each positive integer n, there exists a positive

Let A be a 101-element subset of the set S = {1, 2, . . . ,

Let S be a convex quadrilateral ABCD and O a point inside

Find all functions f defined on the positive real numbers

Let …, …, and … be positive real numbers such that ….

Let B be a point on a circle S1, and let A be a point distinct

Determine the smallest integer n \geq4 for which one can choose

Let ABC be a triangle for which there exists an interior

In the coordinate plane a rectangle with vertices (0, 0), (m, 0),

Let a be a positive integer and let {an} be defined by a0 = 0

Determine all the triples (a, b, c) of positive real numbers such

Prove that there exist infinitely many positive integers n

Given riangleABC with no side equal to another side, let G, K,

Let a, b, c be positive integers satisfying the conditions b > 2a

Let n be a positive integer and let (x1, . . . , xn), (y1, . . . , yn)

B1 (CAN 2)

Let {ak}\infty

Let D be an internal point on the side BC of a triangle ABC.

Prove that for any positive integers x, y, z with xy−z2 = 1 one

Let ABCD be a cyclic quadrilateral. Let E and F be variable

Let a1 \geq\cdot \cdot \cdot \geqan \geqan+1 = 0 be a sequence of real numbers.

For any positive integer n, let au(n) denote the number of its

What is the smallest positive integer t such that there exist

The sequence {an} of integers is defined by a1 = 2, a2 = 7,

Find all solutions in positive real numbers xi (i =

Prove that there exists a four-coloring of the set M =

Let a, b, c, d be four nonnegative numbers satisfying a+b+c+d =