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  • Let a,ba, b be two natural numbers satisfying
    2006a2+a=2007b2+b.2006 a^{2}+a=2007 b^{2}+b.
    Prove that aba-b is a perfect square.
  • Let ABCA B C be a triangle with BAC^=90,ABC^\widehat{B A C}=90^{\circ}, \widehat{A B C} =60.=60^{\circ} . Take the point MM on the side BCB C such that AB+BM=AC+CMA B+B M=A C+C M. Caculate the measure of CAM^\widehat{C A M}
  • Find all positive integers x,yx, y greater than 1 so that 2xy12 x y-1 divisible by (x1)(y1)(x-1)(y-1).
  • Prove that
    a4b2a+b+b4c2b+c+c42c+a1\frac{a^{4} b}{2 a+b}+\frac{b^{4} c}{2 b+c}+\frac{c^{4}}{2 c+a} \geq 1
    where a,b,ca, b, c are positive numbers satisfying the condition ab+bc+ca3abca b+b c+c a \leq 3 a b c. When does equality occur?
  • Let be given two circles (O1)\left(O_{1}\right), (O2)\left(O_{2}\right) with centers O1O_{1}, O2O_{2} with distinct radii, externally touching each other at a point TT. Let O1AO_{1} A be a tangent to (O2)\left(O_{2}\right) at a point A,A, let O2BO_{2} B be a tangent to (O1)\left(O_{1}\right) at a point BB so that the points A,BA, B are on the same side with respect to the line O1O2.O_{1} O_{2} . Let HH be the point on O1A,KO_{1} A, K be the point on O2BO_{2} B so that the lines BH,AKB H, A K are perpendicular to O1O2O_{1} O_{2}. The line THT H cuts (O1)\left(O_{1}\right) again at E,E, the line TKT K cuts (O2)\left(O_{2}\right) againt at FF. The line EFE F cuts ABA B at SS. Prove that the lines O1AO_{1} A, O2BO_{2} B and TST S are concurrent.
  • Let SS be a set consisting of 43 distinct positive integers not exceeding 100.100 . For each subset XX of SS let tXt_{X} be the product of its elements. Prove that there exist two disjoint substs AA and BB of SS such that tAtB2t_{A} t_{B}^{2} is the cube of a natural numbers.
  • Find the greast value of the expression
    ac+bd+ca+dbabcd(ab+cd)2\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}-\frac{a b c d}{(a b+c d)^{2}}
    where a,b,cda, b, c d are distinct real numbers satisfying the conditions ab+bc+cd+da=4\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4 and ac=bda c=b d
  • a) Let f(x)f(x) be a polynomial of degree nn with leading coefficient aa. Suppose that f(x)f(x) has nn distinct roots x1,x2,,xnx_{1}, x_{2}, \ldots, x_{n} all not equal to zero. Prove that
    (1)n1ax1x2xnk=1n1xk=k=1n1xk2f(xk).\frac{(-1)^{n-1}}{a x_{1} x_{2} \ldots x_{n}} \sum_{k=1}^{n} \frac{1}{x_{k}}=\sum_{k=1}^{n} \frac{1}{x_{k}^{2} f^{\prime}\left(x_{k}\right)}.
    b) Does there exist a polynomial f(x)f(x) of degree n,n, with leading coefficient a=1,a=1, such that f(x)f(x) has nn distinct roots x1,x2,,xn,x_{1}, x_{2}, \ldots, x_{n}, all not equal to zero, satisfying the condition
    1x1f(x1)+1x2f(x2)++1xnf(xn)+1x1x2xn=0?\frac{1}{x_{1} f^{\prime}\left(x_{1}\right)}+\frac{1}{x_{2} f^{\prime}\left(x_{2}\right)}+\ldots+\frac{1}{x_{n} f^{\prime}\left(x_{n}\right)}+\frac{1}{x_{1} x_{2} \ldots x_{n}}=0 ?
  • Let ADA D, BEB E, CFC F be the altitudes and HH be the orthocenter of an acute triangle ABC.A B C . Let MM, NN be respectively the points of intersection of DED E and CFC F and of DED E and BEB E. Prove that the line passing through AA perpendicular to the line MNM N passes through the circumcenter of triangle BHCB H C.
  • 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
    (1+a+1+b)(1+c+1+d)(\sqrt{1+a}+\sqrt{1+b})(\sqrt{1+c}+\sqrt{1+d})
    where a,b,c,da, b, c, d are positive numbers satisfying the condition abcd=1a b c d=1.
  • Solve the system of equations
    {x+y+z=3(1+x)(1+y)(1+z)=(1+xyz3)3\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z} &=3 \\ (1+x)(1+y)(1+z) &=(1+\sqrt[3]{x y z})^{3}\end{cases}
  • Let ABCDA B C D be a parallelogram. Take a point MM on the side ABA B, a point NN on the side CDC D. Let PP be the point of intersection of ANA N and DMD M, QQ be the point of intersection of BNB N and CMC M. Prove that PQP Q passes through a fixed point when MM and NN move respectively on ABA B and CDC D.
  • Let ABCDA B C D be a convex quadrilateral. Prove that
    min{AB,BC,CD,DA}AC2+BD22max{AB,BC,CD,DA}\min \{A B, B C, C D, D A\} \leq \frac{\sqrt{A C^{2}+B D^{2}}}{2} \leq  \max \{A B, B C, C D, D A\}
  • Find all natural numbers n3n \geq 3 such that in the coordinate plane, there exists a regular nn-polygon all vertices of which have integral coordinates.
  • The sequence of numbers (un)\left(u_{n}\right) (n=1,2,3,)(n=1,2,3, \ldots) is defined by
    un=k=1n(1)k1k,n=1,2,3.u_{n}=\sum_{k=1}^{n} \frac{(-1)^{k-1}}{k},\,\forall n=1,2,3 \ldots.
    Prove that the sequence has limit and find the limit.
  • Find the greatest value of the expression
    x13+x23++xn3x14x24x14x_{1}^{3}+x_{2}^{3}+\ldots+x_{n}^{3}-x_{1}^{4}-x_{2}^{4}-\ldots-x_{1}^{4}
    (nn is a given number), where the numbers xix_{i} (i=1,2,,n)(i=1,2, \ldots, n) satisfy the conditions 0xi10 \leq x_{i} \leq 1 (i=1,2,,n)(i=1,2, \ldots, n) and x1+x2++xn=1x_{1}+x_{2}+\ldots+x_{n}=1.
  • In a triangle ABCA B C, let mum_{u}, mbm_{b}, mcm_{c} be respectively the measures of the medians issued from AA, BB, CC, let rar_{a}, rbr_{b}, rcr_{c} be respectively the radii of the escribed circles in the angles AA, BB, CC. Prove that
    ra2+rb2+rc2ma2+mb2+mc2.r_{a}^{2}+r_{b}^{2}+r_{c}^{2} \geq m_{a}^{2}+m_{b}^{2}+m_{c}^{2}.
    When does equality occur?
  • Let HH and OO be respectively the orthocenter and the circumcenter of the triangle ABCA B C of a tetrahedron SABCS A B C such that SAS A, SBS B, SCS C are orthogonal each to the others. Prove that
    OH2SH2+2=14cosAcosBcosC\frac{O H^{2}}{S H^{2}}+2=\frac{1}{4 \cos A \cdot \cos B \cdot \cos C}
    where cosA\cos A, cosB\cos B, cosC\cos C are cosinus of the angles of triangle ABCA B C.

PROBLEM OF THE WEEK | ЗАДАЧА НА НЕДЕЛАТА

    PW 148. Let be given an isosceles, right triangle ABC with base BC. Find the locus of points M satisfying the condition

    MB2MC2=2MA2