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MATHEMATICS COMPETITIONS

 

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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,bR and a2+b2+16=8a+6b. Show that
    a) 104a+3b40.
    b) 7b24a.
  • 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+y2z. Find the minimum value of the expression
    P=xy+z+yx+zx+y2z.
  • Show the inequality
    (x+yxy)2020+(y+zyz)2020+(z+xzx)2020>2101031009
    where x,y,z are different numbers.
  • Solve the system of equations
    x2x3x2020x1=x313x1=x323x2=x320193x2019=x320203x2020
  • 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+98cos2xsin22y)+cos4x.
  • Find all functions f:RR satisfying
    f(2f(x)+2y)=x+f(2f(y)+x),x,yR.
  • There are n (n2) 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
    A=34+89+1516++999910000A=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+\ldots+\frac{9999}{10000}
    with the numbers 9898 and 9999.
  • Let ABCA B C be a triangle with ABC^=30\widehat{A B C}=30^{\circ}, ACB^=20\widehat{A C B}=20^{\circ}. The perpendicular bisector of ACA C cuts BCB C at EE and cuts the ray BAB A at FF. Prove that AF=EFA F=E F and AC=BEA C=B E.
  • Find the intergers x,yx, y satisfying
    x+2yx2+y2=720.\frac{x+2 y}{x^{2}+y^{2}}=\frac{7}{20}.
  • Prove that
    a+b2ab+(ab)2(3a+b)(a+3b)8(a+b)(a2+6ab+b2)\frac{a+b}{2} \geq \sqrt{a b}+\frac{(a-b)^{2}(3 a+b)(a+3 b)}{8(a+b)\left(a^{2}+6 a b+b^{2}\right)}
    for positive numbers a,ba, b.
  • Solve the inequation
    (x1)x22x+54xx2+12(x+1)(x-1) \sqrt{x^{2}-2 x+5}-4 x \sqrt{x^{2}+1} \geq 2(x+1)
  • On a line, take three distinct points AA, BB, CC in this order. Draw the tangents ADA D, AEA E to the circle with diameter BCB C (DD and EE are the touching points). Draw DHCED H \perp C E at HH. Let PP be the midpoint of DHD H. The line CPC P cuts again the circle at QQ. Prove that the circle passing through the three points AA, DD, QQ is tangent to the line ACA C.
  • MM is a point on the incircle of triangle ABCA B C. Let KK, HH, JJ be respectively the orthogonal projections of MM on the lines ABA B, BCB C, CAC A. Determine the position of MM so that the sum MK+MH+MJM K+M H+M J.
    a) attains its greatest value
    b) attains its least value
  • Determine all sequences of positive integers (xn)\left(x_{n}\right) (n=1,2,3,)(n=1,2,3, \ldots) satisfying
    x1=1,x2>1,xn+2=1+xn+14xn,n=1,2,3,x_{1}=1,\, x_{2}>1,\quad x_{n+2}=\frac{1+x_{n+1}^{4}}{x_{n}},\,\forall n=1,2,3, \ldots
  • Solve the system of equations
    {log2(1+3cosx)=log3(siny)+2log2(1+3siny)=log3(cosx)+2\begin{cases}\log _{2}(1+3 \cos x) &=\log _{3}(\sin y)+2 \\ \log _{2}(1+3 \sin y) &=\log _{3}(\cos x)+2\end{cases}
  • Let nn be a given positive integer. Find the least number t=t(n)t=t(n) such that for all real numbers x1,x2,,xnx_{1}, x_{2}, \ldots, x_{n}, holds the following inequality
    k=1n(x1+x2++xk)2t(x12+x22++xn2)\sum_{k=1}^{n}\left(x_{1}+x_{2}+\ldots+x_{k}\right)^{2} \leq t\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right)
  • Prove that for arbitrary triangle ABCA B C,
    sinAcosB+sinBcosC+sinCcosA334.\sin A \cos B+\sin B \cos C+\sin C \cos A \leq \frac{3 \sqrt{3}}{4}.
    When does equality occur?
  • In space, let be given nn distinct points A1,A2,,AnA_{1}, A_{2}, \ldots, A_{n} (n2)(n \geq 2) and nn points K1,K2,,KnK_{1}, K_{2}, \ldots, K_{n} (KiK_{i} does not coincide with AiA_{i} for every i=1,2,,n)i=1,2, \ldots, n). Prove that there exist nn spheres (with positive radii) SiS_{i} (i=1,2,,n)(i=1,2, \ldots, n) satisfying simultaneously the following two conditions
    a) the spheres do not intersect each others,
    b) the products PAi/SiPKi/SiP_{A_{i} / S_{i}} \cdot P_{K_{i} / S_{i}} are negative numbers for all i=1,2,,ni=1,2, \ldots, n, where PP denotes the power of a point with respect to a sphere.

PROBLEM OF THE WEEK | ЗАДАЧА НА НЕДЕЛАТА

    PW 150.  Prove that the sum  Sn=k=0n1Cnk   has a finite limit when n tends to infinity and find this limit (Cnk are binomial coefficients).