2021 Issue 528

  1. Assume that $A$ is a natural number which has two prime factors $p$ and $q$ only. Let $S$ be the sum of all positive factors of $A$. Show that $S < 2 A$.
  2. Denote $a_{n}$ the integer which is closest to $\sqrt{n}$ $(n \in \mathbb{N}^{*})$, for example $$\sqrt{1}=1=a_{1} ; \quad \sqrt{2} \approx 1,4 \Rightarrow a_{2}=1 ; \quad \sqrt{3} \approx 1,7 \Rightarrow a_{3}=2 ; \ldots.$$ Compute $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{2020}+\frac{1}{2021}.$$
  3. Find all $4$-tuples of positive integers $(x ; y, z, t)$ satisfying $$\begin{cases}x+y+z+t=54 \\ \dfrac{x z}{(x+y)(z+t)} =\dfrac{23}{104} \\ x>z, x+y>z+t \end{cases}.$$
  4. Given an acute triangle $A B C$ ($CA < CB$) with the orthocenter $H$. Let $D$ be the second intersection between $B C$ and the circumcircle of $A B H$. Denote $P$ the intersection between $D H$ and $A C$. Let $M$ be the second intersection between $A H$ and the circumcircle of $A P D$; and $N$ the second intersection between the circumcircle of $A B C$ and the circumcircle of $A P D$. Show that three points $C, M$, and $N$ are collinear.
  5. Given positive numbers $a, b, c$ satisfying $4 b^{2}+c^{2}=a c$. Find the minimum value of the expression $$P=\frac{2 a}{a+2 b}+\frac{b}{b+c}+\frac{c}{c+a} .$$
  6. Solve the equation $$\sin 3 x-\cos 3 x+\sin x+\cos x=\frac{1}{\sin 3 x+\cos x}-\frac{1}{\cos 3 x-\sin x} .$$
  7. Given positive numbers $a, b, c$ satisfying $a^{2}+b^{2}+c^{2} \geq 2(a b+b c+c a)$. Find the minimum value of the expression $$P=a+b+c+\frac{8}{a b c} .$$
  8. Given two circles $(O ; R)$ and $\left(O^{\prime} ; R^{\prime}\right)\left(R>R^{\prime}\right)$ which is internally tangent to each other at $A$. Let $M$ be a point on $\left(O^{\prime} ; R^{\prime}\right)(M \neq A) .$ The tangent of $\left(O^{\prime} ; R^{\prime}\right)$ at $M$ intersects $(O ; R)$ at $P$ and $Q .$ The circumcircle of $O^{\prime} P Q$ intersects $\left(O^{\prime} ; R^{\prime}\right)$ at $B$ and $C$. Show that $A B M C$ is a harmonic quadrilateral, i.e. a cyclic quadrilateral of which the products of opposite sides are equal.
  9. Given a triangle $A B C$ and $x, y, z$ positive numbers. Show that $$\frac{x}{y+z}(1+\cos A)+\frac{y}{z+x}(1+\cos B)+\frac{z}{x+y}(1+\cos C) \geq \frac{\sqrt{3}}{2}(\sin A+\sin B+\sin C) .$$
  10. Two boxes contain $25$ small white and black balls. From each box, pick randomly $1$ ball. Find the probability that we get $2$ balls with different colors assuming that the box with more balls has more black balls and the probability to get two black balls is $0,42$.
  11. Find all functions $f: \mathbb{R} \rightarrow(0+\infty)$ satisfying $$\frac{1}{2015} \leq\left(\frac{f(x)}{f(r)}\right)^{\frac{1}{(x-r)^{2}}} \leq 2015, \forall x \in \mathbb{R}, \forall r \in \mathbb{Q}, x \neq r$$
  12. Outside a triangle $A B C$, draw pairwise similar triangles $B C P$, $A C Q$, $A B R$, and $B A S$. Let $K$, $L$ respectively be the midpoints of $B C$ and $C A .$ Prove that two triangles $R P K$ and $S Q L$ have the same area.




Mathematics & Youth: 2021 Issue 528
2021 Issue 528
Mathematics & Youth
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