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2018 Issue 490

  1. The natural number $a$ is coprime with $210$. Dividing $a$ by $210$ we get the remainder $r$ satisfying $1<r<120$. Prove that $r$ is prime.
  2. Given non-zero numbers $a,b,c,d$ satisfying $b^{2}=ac$, $c^{2}=bd$, $b^{3}+27c^{3}+8d^{3}\ne0$. Show that \[\frac{a}{d}=\frac{a^{3}+27b^{3}+8c^{3}}{b^{3}+27c^{3}+8d^{3}}.\]
  3. Find all natural solutions of the equation \[3xyz-5yz+3x+3z=5.\]
  4. Given a half circle with the center $O$, the diameter $AB$, and the radius $OD$ perpendicular to $AB$. A point $C$ is moving on the arc $BD$. The line $AC$ intersects $OD$ at $M$. Prove that the circumcenter $I$ of the triangle $DMC$ always belongs to a fixed line.
  5. Let $x,y$ be real numbers such that $x^{3}+y^{3}=2$. Find the minimum value of the expression \[P=x^{2}+y^{2}+\frac{9}{x+y}.\]
  6. Given positive numbers $a,b,c$ satisfying $abc=1$. Prove that \[\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+a^{5}+b^{2}}\leq1.\]
  7. Find all positive integers $a,b$ such that \[\sqrt{8+\sqrt{32+\sqrt{768}}}=a\cos\frac{\pi}{b}.\]
  8. Given a triangle $ABC$. Let $(K)$ be the circle passing through $A$, $C$ and is tangent to $AB$ and let $(L)$ be the circle passing through $A$, $B$ and is tangent to $AC$. Assume that $(K)$ intersects $(L)$ at another point $D$ which is different from $A$. Assume that $AK$, $AL$ respectively intersect $DB$, $DC$ at $E$ and $F$. Let $M$, $N$ respectively be the midpoints of $BE$, $CF$. Prove that $A$, $M$, $N$ are colinear.
  9. Given real numbers $a,b,c$ such that \[2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})\geq a^{4}+b^{4}+c^{4}.\] Prove that \[|b+c-a|+|c+a-b|+|a+b-c|+|a+b+c|=2(|a|+|b|+|c|).\]
  10. Find all triples of positive integers $(a,b,c)$ such that \[2^{a}+5^{b}=7^{c}.\]
  11. The sequence $(u_{n})$ is determined as follows \[u_{1}=14,\,u_{2}=20,\,u_{3}=32,\quad u_{n+2}=4u_{n+1}-8u_{n}+8u_{n-1},\,\forall n\geq2. \] Show that $u_{2018}=5\cdot2^{2018}$.
  12. Given a triangle $ABC$ with $(O)$ is the circumcircle and $I$ is the incenter. Let $D$ be the second intersection of $AI$ and $(O)$. Let $E$ be the intersection between $BC$ and the line pasing through $I$ and perpecdicular to $AI$. Assume that $K$, $L$ respectively are the intersections between $BC$, $DE$ and the line passing through $I$ and perpendicular to $OI$. Prove that $KI=KL$.

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Mathematics & Youth: 2018 Issue 490
2018 Issue 490
Mathematics & Youth
https://www.molympiad.org/2018/05/mathematics-and-youth-magazine-problems_18.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2018/05/mathematics-and-youth-magazine-problems_18.html
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