# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. The natural number$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$KI=KL$. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2018 Issue 490
2018 Issue 490
Mathematics & Youth