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Outside an arbitrary triangle ABC, triangles ADB and BCE

Let d \geq1 be an integer that is not the square of an integer.

A fair coin is tossed repeatedly until there is a run of an odd

Given the vertex A and the centroid M of a triangle ABC,

In a Cartesian coordinate system, the circle C1 has center

Consider the binomial coefficients

Given that x1+x2+x3 = y1+y2+y3 = x1y1+x2y2+x3y3 = 0,

A sequence (an)N

In a plane we are given two distinct points A, B and two lines

Let k, m, and n be positive integers such that m+k + 1 is

Integers a1, a2, . . . , an satisfy |ak| = 1 and

Given real numbers xi (i = 1, 2, . . . , 4x + 2) such that

Is it possible to partition 3-dimensional Euclidean space into

Let f(x) be a polynomial with integer coefficients. Prove that

Let a, b, and c be real numbers such that

Under the conditions x1, x2 > 0, x1y1 > z2

Denote by xn(p) the multiplicity of the prime p in the canonical

Let a and b be arbitrary integers. Prove that if k is an integer

Let F be the correspondence associating with every point P =

The function f(n) is defined on the nonnegative integers n by:

A boy at point A wants to get water at a circular lake and

An infinite increasing sequence of positive integers nj (j =

Let n be an integer greater than 1. In the Cartesian coordinate

Prove that if x, y, z are real numbers such that x2+y2+z2 = 2,

Let P1(x), P2(x), . . . , Pn(x) be polynomials with real coefficients.

Determine the sum of all positive integers whose digits (in base

Prove that for any triangle with sides a, b, c and area P the

In a triangle ABC, let H be its orthocenter, O its circumcenter,

Let a and b be integers. Prove that 2a2−1

Let the polynomials

If

The plane is divided into equal squares by parallel lines; i.e.,

The equation x3 + ax2 + bx + c = 0 has three (not necessarily

On the circle with center O and radius 1 the point A0 is

Find all real numbers \lambda such that the equation

Find the integer solutions of the equation

In the plane 4000 points are given such that each line passes

Find the last eight digits of the binary development of 271986.

Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers.

A set of n standard dice are shaken and randomly placed in a

Prove that in any parallelepiped the sum of the lengths of the

Let f be a function from the real numbers to the real numbers

Let ABC be a triangle, O its circumcenter, S its centroid, and

An arithmetic function is a real-valued function whose do-

If a1, a2, . . . , an denote the lengths of the sides of an arbitrary

Find all natural numbers n < 1978 with the following property:

An (n2 +n+1) imes(n2 +n+1) matrix of zeros and ones is given.

On a line a set of segments is given of total length less than

A turtle runs away from an UFO with a speed of 0.2 m/s. The

Does there exist a number lpha (0 < lpha < 1) such that there is an

Let ABC and A′B′C′ be any two coplanar triangles. Let L be

We are given a circle K with center S and radius 1 and a square

It is well known that the binomial coefficients

Seventeen cities are served by four airlines. It is noted that

Three faces of a tetrahedron are right triangles, while the fourth

The opposite sides of the reentrant hexagon AFBDCE in-

For a finite set E of cardinality n \geq3, let f(n) denote the

Let n be a positive integer, n \geq2, and consider the polynomial

A solid right circular cylinder with height h and base-radius

With the medians of an acute-angled triangle another triangle is

Consider the number lpha obtained by writing one after another

Let a1, a2, . . . , an (n \geq2) be a sequence of integers. Show that

Let ABC and DEF be acute-angled triangles. Write d = EF,

Let m and n be natural numbers with m > n. Prove that

In a triangle ABC, the incircle touches the sides BC, CA, AB

Let \Gammai, i = 0, 1, 2, . . ., be a circle of radius ri inscribed in an

The circle k and its diameter AB are given. Find the locus of

In a plane a finite number of equal circles are given. These circles

Prove that for all x1, x2, . . . , xn \inR the following inequality

Let S \subset[0, 1] be a set of 5 points with {0, 1} \subsetS. The graph

Given the equation

Prove that in a Euclidean plane there are infinitely many

Let L denote the set of all lattice points of the plane (points

A regular pentagon A1A2A3A4A5 with side length s is given.

In a plane a circle with radius R and center w and a line 
ambda

Let a cube of side 1 be given. Prove that there exists a point

We match sets M of points in the coordinate plane to sets M∗

Let a and b be two natural numbers that have an equal number

Prove that the volume V and the lateral area S of a right circular

In the Martian language every finite sequence of letters of

Assume that two parallelograms P, P ′ of equal areas have sides

In the triangle ABC, let D, E, and F be the midpoints of the

Let r > 1 be a real number, and let n be the largest integer

Given n real numbers 0 < t1 \leqt2 \leq\cdot \cdot \cdot \leqtn < 1, prove that

Show that if the sides a, b, c of a triangle satisfy the equation

(1) Start with a white balls and b black balls.

Let A, B, C be points on the sides B1C1, C1A1, A1B1 of a

Find the radius of the circle circumscribed about the isosceles

In an urn there are n balls numbered 1, 2, . . . , n. They are

One hundred convex polygons are placed on a square with edge

Given a triangle, prove that the points of intersection of three

Let Sn = {1, . . . , n} and let f be a function that maps every

Let ABCD be a convex quadrilateral whose diagonals AC and

Let g(x) be a fixed polynomial and define f(x) by f(x) =

Find x such that

Given a triangle ABC, let R be the radius of its circumcir-

Find the greatest number c such that for all natural numbers

Suppose that 1985 points are given inside a unit cube. Show

There are a board with 2n\cdot2n (= 4n2) squares and 4n2−1 cards

Find the largest integer not exceeding $1992