NEWS | ИНФОРМАЦИИ
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ARTICLES | СТАТИИ
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USEFUL INFORMATION | КОРИСНИ ИНФОРМАЦИИ
SELECTED PROBLEMS | ИЗБРАНИ ЗАДАЧИ
- Find all 6-digit natural numbers which are both a perfect square and a cube.
- Given a triangle ABC with Aˆ=30∘, Bˆ=20∘. On the side AB choose the point D such that AD=BC. Find the value of the angle BCDˆ.
- Assume that a,b∈R and a2+b2+16=8a+6b. Show that
a) 10≤4a+3b≤40.
b) 7b≤24a.
- Given a half circle with the center O, the diameter BC. Choose a point G inside the half circle so that BGOˆ=135∘. The line which is perpendicular to GB at G intersects the half circle at A. The incircle I of ABC is tangent to BC, CA respectively at D and E. Show that G lies on ED.
- Suppose that x,y,z are positive numbers satisfying x+y≤2z. Find the minimum value of the expression
P=xy+z+yx+z−x+y2z.
- Show the inequality
(x+yx−y)2020+(y+zy−z)2020+(z+xz−x)2020>2101031009 where x,y,z are different numbers.
- Solve the system of equations
⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪x2x3…x2020x1=x31−3x1=x32−3x2…=x32019−3x2019=x32020−3x2020
- Given a right triangle ABC with the right angle A and the altitude AH. On the line segment AH choose a point I, the line CI intersects AB at E. On the side AC choose the point F such that EF is parallel to BC. The line which passes through F and is perpendicular to CE at N intersects BI at M. Let D be the intersection between AN and BC. Prove that four points M, N, D, C both lies on a circle.
- Let x, y be real numbers. Find the minimum value of the expression
P=sin4x(sin4y+cos4y+98cos2x⋅sin22y)+cos4x.
- Find all functions f:R→R satisfying
f(2f(x)+2y)=x+f(2f(y)+x),∀x,y∈R.
- There are n (n≥2) soccer teams attending a tournament. Each team will play with all other teams once. The winning team get 3 points, the losing team gets 0 point; and if the match ties, both teams get 1 point. After the tournament, we recognize that all teams got different total points. What is the possible minimal value for the difference between the team with the most points and the team with the least points?
- Given a triangle ABC with I is the center of the excircle relative to the vertex A. This circle is tangent to BC, CA, AB respectively at M,N,P. Let E be the intersection between MN and BI, and F be the intersection between MP and CI. The line BC intersects AE, AF respectively at G, D. Show that AI is parallel to the line passing through M and the center of the Euler circle of AGD
- Compare the value of the expression
with the numbers and . - Let be a triangle with , . The perpendicular bisector of cuts at and cuts the ray at . Prove that and .
- Find the intergers satisfying
- Prove that
for positive numbers . - Solve the inequation
- On a line, take three distinct points , , in this order. Draw the tangents , to the circle with diameter ( and are the touching points). Draw at . Let be the midpoint of . The line cuts again the circle at . Prove that the circle passing through the three points , , is tangent to the line .
- is a point on the incircle of triangle . Let , , be respectively the orthogonal projections of on the lines , , . Determine the position of so that the sum .
a) attains its greatest value b) attains its least value - Determine all sequences of positive integers satisfying
- Solve the system of equations
- Let be a given positive integer. Find the least number such that for all real numbers , holds the following inequality
- Prove that for arbitrary triangle ,
When does equality occur? - In space, let be given distinct points and points ( does not coincide with for every . Prove that there exist spheres (with positive radii) satisfying simultaneously the following two conditions
a) the spheres do not intersect each others, b) the products are negative numbers for all , where denotes the power of a point with respect to a sphere.
PROBLEM OF THE WEEK | ЗАДАЧА НА НЕДЕЛАТА
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