NEWS | ИНФОРМАЦИИ
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ARTICLES | СТАТИИ
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USEFUL INFORMATION | КОРИСНИ ИНФОРМАЦИИ
SELECTED PROBLEMS | ИЗБРАНИ ЗАДАЧИ
- Let be two natural numbers satisfying
Prove that is a perfect square.
- Let be a triangle with Take the point on the side such that . Caculate the measure of
- Find all positive integers greater than 1 so that divisible by .
- Prove that
where are positive numbers satisfying the condition . When does equality occur?
- Let be given two circles , with centers , with distinct radii, externally touching each other at a point . Let be a tangent to at a point let be a tangent to at a point so that the points are on the same side with respect to the line Let be the point on be the point on so that the lines are perpendicular to . The line cuts again at the line cuts againt at . The line cuts at . Prove that the lines , and are concurrent.
- Let be a set consisting of 43 distinct positive integers not exceeding For each subset of let be the product of its elements. Prove that there exist two disjoint substs and of such that is the cube of a natural numbers.
- Find the greast value of the expression
where are distinct real numbers satisfying the conditions and
- a) Let be a polynomial of degree with leading coefficient . Suppose that has distinct roots all not equal to zero. Prove that
b) Does there exist a polynomial of degree with leading coefficient such that has distinct roots all not equal to zero, satisfying the condition
- Let , , be the altitudes and be the orthocenter of an acute triangle Let , be respectively the points of intersection of and and of and . Prove that the line passing through perpendicular to the line passes through the circumcenter of triangle .
- Find the measures of the sides of all triangles satisfying the conditions : these measures are whole numbers, the perimeter and the area are expressed by equal numbers.
- Find the least value of the expression
where are positive numbers satisfying the condition . - Solve the system of equations
- Let be a parallelogram. Take a point on the side , a point on the side . Let be the point of intersection of and , be the point of intersection of and . Prove that passes through a fixed point when and move respectively on and .
- Let be a convex quadrilateral. Prove that
- Find all natural numbers such that in the coordinate plane, there exists a regular -polygon all vertices of which have integral coordinates.
- The sequence of numbers is defined by
Prove that the sequence has limit and find the limit. - Find the greatest value of the expression
( is a given number), where the numbers satisfy the conditions and . - In a triangle , let , , be respectively the measures of the medians issued from , , , let , , be respectively the radii of the escribed circles in the angles , , . Prove that
When does equality occur? - Let and be respectively the orthocenter and the circumcenter of the triangle of a tetrahedron such that , , are orthogonal each to the others. Prove that
where , , are cosinus of the angles of triangle .
PROBLEM OF THE WEEK | ЗАДАЧА НА НЕДЕЛАТА
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