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SELECTED PROBLEMS | ИЗБРАНИ ЗАДАЧИ

  • Find the last two digit numbers of 39999−299993^{9999}-2^{9999}.
  • Compare the sum (consisting of n+1n+1 terms)
    Sn=2101+1+221012+1+…+2n+11012′+1S_{n}=\frac{2}{101+1}+\frac{2^{2}}{101^{2}+1}+\ldots+\frac{2^{n+1}}{101^{2 \prime}+1}
    with 0,020,02.
  • Let aa, bb and cc be positive numbers such that ab+bc+ca=1a b+b c+c a=1. Prove the inequality
    1ab+1bc+1ca≥3+1a2+1+1b2+1+1c2+1.\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a} \geq 3+\sqrt{\frac{1}{a^{2}}+1}+\sqrt{\frac{1}{b^{2}}+1}+\sqrt{\frac{1}{c^{2}}+1}.
    When does equality occur?
  • Solve the equation
    7x2−22x+28+7x2+8x+13+31x2+14x+4=33(x+2).\sqrt{7 x^{2}-22 x+28}+\sqrt{7 x^{2}+8 x+13}+\sqrt{31 x^{2}+14 x+4}=3 \sqrt{3}(x+2).
  • Let ABCDA B C D be a rectangular with AB<BCA B<B C. Let MM be a point, different from AA and BB, on the half-circle with ABA B as its diameter and on the same side with CDCD. MAMA and MBMB meet CDCD at PP and QQ respectively. MCM C and MDM D meet ABA B at EE and FF respectively. Find the position of the point MM on the half-circle such that the sum PQ+EFP Q+E F is smallest possible. Calculate this smallest value.
  • Find all pairs of positive integers aa, bb such that q2−r=2007,q^{2}-r=2007, where qq and rr are respectively the quotient and the remainder obtained when dividing a2+b2a^{2}+b^{2} by a+ba+b
  • Consider the equation
    ax3−x2+bx−1=0a x^{3}-x^{2}+b x-1=0
    where a,ba, b are real numbers, a≠0a \neq 0 and a≠ba \neq b such that all of its roots are positive real numbers. Find the smallest value of
    P=5a2−3ab+2a2(b−a).P=\frac{5 a^{2}-3 a b+2}{a^{2}(b-a)}.
  • Choose five points AA, BB, CC, DD and EE on a sphere with radius RR such that
    BAC^=CAD^=DAE^=EAB^=23BAD^=23CAE^.\widehat{B A C}=\widehat{C A D} =\widehat{D A E}=\widehat{E A B}=\frac{2}{3} \widehat{B A D}=\frac{2}{3} \widehat{C A E}.
    Prove the inequality
    AB+AC+AD+AE≤42R.A B+A C+A D+A E \leq 4 \sqrt{2} R.
  • Suppose that MM, NN and PP are three points lying respectively on the edges ABA B, BCB C. CAC A of a triangle ABCA B C such that
    AM+BN+CP=MR+NC+PA.A M+B N+C P=M R+N C+P A.
    Prove the inequality SMNP≤14SABCS_{M N P} \leq \dfrac{1}{4} S_{A B C}.
  • Find the limit
    limn→+∞2−2⋅2−2+2⋯2−2+2+…+2⏟n signsor railiad \lim_{n \rightarrow+\infty}\sqrt{2-\sqrt{2}} \cdot \sqrt{2-\sqrt{2+\sqrt{2}}} \cdots \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}}_{n \text { signsor railiad }}
  • Denote by [x][x] the largest integer not exceeding xx and write {x}=x−[x].\{x\}=x-[x] . Find the limit limn→+∞{(7+43)n }\displaystyle \lim _{n \rightarrow+\infty}(7+4 \sqrt{3})^{n}}}
  • Let nn be a positive integer and 2n+22 n+2 real numbers a,b,a1,a2,…,an,b1,b2,…,bna, b, a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n} such that ai≠0(i=1,2,…,n)a_{i} \neq 0(i=1,2, \ldots, n) and the function
    F(x)=∑i=1naix+bi−(ax+b)F(x)=\sum_{i=1}^{n} \sqrt{a_{i} x+b_{i}}-(a x+b)
    satisfies the following property: There exists distinct real numbers α,β\alpha, \beta such that F(α)=F(β)=0F(\alpha)=F(\beta)=0 Prove that α\alpha and β\beta are the only real solutions of the equation F(x)=0
  • There are three bells in the laboratory. The first bell rings every 44 minutes, the second every 1212 minutes, the third every 1616 minutes. The three bells ring simultaneously at 7h 307^{\text {h }} 30 in the morning.
    a) At what next time do the three bells ring simultaneously?
    b) From 7h307^{\mathrm{h}} 30 to 11h3011^{\mathrm{h}} 30 p.m., how many times do only two bells ring simultaneously?
  • a) In the figure let BEC^=30∘\widehat{B E C}=30^{\circ}, ACD^=70∘\widehat{A C D}=70^{\circ}, CDE^=110∘\widehat{C D E}=110^{\circ} and BAC^=CED^=ABE^\widehat{B A C}=\widehat{C E D}=\widehat{A B E} and justify the answer.
    b) How many triangles are there in the figure? Write down these triangles.
  • Find the integer-solutions of the equation
    4(a−x)(x−b)+b−a=y24(a-x)(x-b)+b-a=y^{2}
    where a,ba, b are given integers, a>ba>b.
  • Prove that the equation
    (n+1)xn+2−3(n+2)xn+1+an+2=0(n+1) x^{n+2}-3(n+2) x^{n+1}+a^{n+2}=0
    (where nn is a given even positive integer and a>3a>3) has no solution.
  • Prove that
    ab+ba+3(a+b)a+b>6\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}+\frac{3(\sqrt{a}+\sqrt{b})}{\sqrt{a+b}}>6
  • Let ABCA B C be a triangle ; on the opposite rays of the rays BA,CAB A, C A take respectively the points E,FE, F (distinct from B,C)B, C) BFB F cuts CEC E at MM. Prove that
    MBMF+MCME≥2AB⋅ACAF⋅AE\frac{M B}{M F}+\frac{M C}{M E} \geq 2 \sqrt{\frac{A B \cdot A C}{A F \cdot A E}}
    When does equality occur?
  • Let be given a convex quadrilateral ABCDA B C D. ABA B cuts CDC D at EE, ADA D cuts BCB C at FF. The diagonals ACA C, BDB D intersect at OO. Let MM, NN, PP, QQ be respectively the midpoints of ABA B, BCB C, CDC D, DADA. CFCF cuts MPM P at HH, OEO E cuts NQN Q at KK. Prove that HKH K is parallel to EFE F.
  • Let x,y,px, y, p be integers such that p>1p>1 and x2002x^{2002}, y2003y^{2003} are divisible by pp. Prove that 1+x+y1+x+y is not divisible by pp.
  • Let a1,a2,…ana_{1}, a_{2}, \ldots a_{n} and b1,b2,…,bnb_{1}, b_{2}, \ldots, b_{n} be positive numbers. Prove that
    a) d1rb1r−1+c2b2r−1+…+anrbnr−1≥(a1+a2+…+an)r(b1+b2+…+bn)r−1\displaystyle \frac{d_{1}^{r}}{b_{1}^{r-1}}+\frac{c_{2}}{b_{2}^{r-1}}+\ldots+\frac{a_{n}^{r}}{b_{n}^{r-1}} \geq \frac{\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{r}}{\left(b_{1}+b_{2}+\ldots+b_{n}\right)^{r-1}} where rr is a rational number, r>1r>1;
    b) a1sb1+a2sb2+…+ansbn≥(a1+a2+…+an)sns−2(b1+b2+…+bn)\displaystyle \frac{a_{1}^{s}}{b_{1}}+\frac{a_{2}^{s}}{b_{2}}+\ldots+\frac{a_{n}^{s}}{b_{n}} \geq \frac{\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{s}}{n^{s-2}\left(b_{1}+b_{2}+\ldots+b_{n}\right)} where ss is a rational number, s≥2s \geq 2.
  • The sequence of numbers (vn)\left(v_{n}\right) is defined by
    v0=1,vn=−13+vn−1,∀n=1,2,3,…v_{0}=1,\quad v_{n}=\frac{-1}{3+v_{n-1}},\,\forall n=1,2,3, \ldots
    Prove that the sequence has a limit and find this limit.
  • MM is a point inside an acute triangle ABCA B C with BC=aB C=a, CA=bC A=b, AB=cA B=c. Let DD, EE, FF be respectively the orthogonal projections of MM on the sides BCB C, CAC A, ABA B. Find the greatest value of the expression
    P=a⋅ME⋅MF+b⋅MF⋅MD+c⋅MD⋅MEP=a \cdot M E \cdot M F + b \cdot M F \cdot M D+c \cdot M D \cdot M E
    and determine the position of MM where this expression assumes its greatest value. 
  • Let SAS_{A}, SBS_{B}, SCS_{C}, SDS_{D} be respectively the areas of the faces BCDB C D, CDAC D A, DABD A B, ABCA B C of a tetrahedron ABCDA B C D and RR be the radius of the circumscribed sphere of ABCDA B C D. Prove that
    R≥TSA+SB+SC+SDR \geq \frac{T}{S_{A}+S_{B}+S_{C}+S_{D}}
    where
    T2=AB2SASB+AC2SASC+AD2SASD+BC2SBSC+BD2SBSD+CD2SCSD.T^{2}=A B^{2} S_{A} S_{B}+A C^{2} S_{A} S_{C}+A D^{2} S_{A} S_{D}+B C^{2} S_{B} S_{C}+B D^{2} S_{B} S_{D}+C D^{2} S_{C} S_{D}.
    When does equality occur?

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).