2019 Issue 502

  1. Show that $$A=10^{10}+10^{10^1}+10^{10^2}+\ldots+10^{10^{10}}-5$$ is divisible by $7$.
  2. Given integers $a$, $b$ and $c$. Find the natural pumber $d$ so that $$|a-b|+|b-c|+|c-a|=2018^{d}+2019.$$
  3. Given positive oumbers $a, b, c$ such that $a^2+b^2+c^{2}=\dfrac{3}{4}$. Find the maximum value of the expression $$P=\left(1-\frac{1}{a}\right)\left(1-\frac{1}{b}\right)\left(1-\frac{1}{c}\right).$$
  4. Given a triangle $A B C$ and $M$ is inside the triangle so that $\widehat{A B M}=\widehat{A C M}$. Let $D$ be the symmetrical point of $M$ about the line $A C$. Draw the parallelogram $B M C E$. The ray $B M$ intersects $A C$ at $l$. The ray $D l$ intersects $B E$ at $F$. Show that the points $A$. $D$, $C$, $E$, $F$ are on the same circle.
  5. Solve the equation $$x^{6}-7 x^{2}+\sqrt{6}=0.$$
  6. Solve the system of equations $$\begin{cases}\log_{\frac{1}{4}} x &=\left(\dfrac{1}{4}\right)^{2}+1 \\ 128 x^{3} \sqrt{4 x^{2}+y^{2}} &=y^{3} \sqrt{64 x^{2}+y^{2}}\end{cases}.$$
  7. Suppose that $$(1+x+x^2+\ldots+x^{2018})^{2019} = a_0 +a_1 x +a_2 x^2 + \ldots +a_{4074342}x^{4074342}.$$ Prove that $$C_{2019}^0 a_{2019}-C_{2019}^1 a_{2018}+C_{2019}^{2} a_{2017}-C_{2019}^{3} a_{2016}+\ldots+C_{2019}^{2018} a_{1}-C_{2019}^{2019} a_{0}=-2019.$$
  8. Given an aculc triangle $A B C$ inscribed in a circle $(O)$. Let $H$ be the orthocenter of the triangle, $P$ the midpoint of the minor are $\widehat{B C}$. Assume that $E$ is the intersection between $B H$ and $A C$. $F$ is the intersection between $\mathrm{CH}$ and $A B$. Let $Q$, $R$ respectively be the second interscctions between $P E$, $P F$ and $(O)$. Let $K$ be the interscetion between $B Q$ and $C R$. Show that $K H \parallel A P$.
  9. Given non-ncgative numbers $a, b, c$ with at most one of them is equal to $0$. Prove that $$\frac{a^{3}}{b^{2}-b c+c^{2}}+\frac{b^{3}}{c^{2}-c a+a^{2}}+\frac{c^{3}}{a^{2}-a b+b^{2}} \geq a+b+c.$$
  10. Let $\displaystyle S_n = \sum_{k=2}^n k\cos\dfrac{\pi}{k}$. Find $\displaystyle \lim_{n \rightarrow+\infty} \dfrac{S_{n}}{n^{2}}$.
  11. There are $100$ marbles distributed into $k$ groups. A distribution is called "special" if any two groups have different numbers of marbles and if we divide any group into two smaller ones then among $k+1$ new groups there are groups with equal numbers of marbles. Find the maximum and minimum values for $k$ so that there exist corresponding special distributions. 
  12. Outside a cyclic quadrilateral $A B C D$, draw the squares $A B M N$, $B C P Q$, $C D R S$, $D A U V$. Let $B^{\prime}$ be the intersection between $P Q$ and $M N$, $D^{\prime}$ the intersection between $U V$ and $RS$. Show that the midpoint of $B^{\circ} D^{\circ}$ belongs to $B D$.




Mathematics & Youth: 2019 Issue 502
2019 Issue 502
Mathematics & Youth
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