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

## $show=home 1. Find the last digit of the following number $A=1^{2015}+2^{2015}+3^{2015}+\ldots+2014^{2015}+2015^{2015}.$ 2. Given a right trapezoid$ABCD$($A$and$B$are right angles) with$AD<BC$and$AC\perp BD$. Prove that $AC^{2}+BD^{2}=3AB^{2}+CD^{2}.$ 3. Does there exist a pair of positive integers$(a,b)$such that both the equation$x^{2}+2ax-b-2=0$and the equation$x^{2}+bx-a=0$have integer solutions?. 4. Given a right triangle$ABC$with the right angle$A$such that $BC^{2}=2BC\cdot AC+4AC^{2}.$ Find the angle$\widehat{ABC}$. 5. Solve the equation $\sqrt{3x-2}-\sqrt{2x+1}=\sqrt{x-3}.$ 6. Solve the system of equations $\begin{cases} x+\sqrt{x^{2}+9} & =\sqrt{3^{y+4}}\\ y+\sqrt{y^{2}+9} & =\sqrt{3^{z+4}}\\ z+\sqrt{z^{2}+9} & =\sqrt{3^{x+4}} \end{cases}.$ 7. Given an acute triangle$ABC$with angles measured in radian. Prove that $\sin A+\sin B+\sin C>\frac{5}{2}-\frac{A^{2}+B^{2}+C^{2}}{\pi^{2}}.$ 8. Given a tetrahedron$ABCD$inscribed in a sphere of radius$1$. Assume that the product of the lengths of its sides is equal to$\frac{512}{27}$. Compute the lengths of its sides. 9. Let$a,b,c$be the lengths of three sides of a triangle. Prove that $\frac{3(a^{2}+b^{2}+c^{2})}{(a+b+c)^{2}}+\frac{ab+bc+ca}{a^{2}+b^{2}+c^{2}}\leq2.$ 10. Find the number of the triples$(a,b,c)$of integers in the interval$[1,2015]$such that$a^{3}+b^{3}+c^{3}$is divisible by$9$?. 11. Let$a,b,c$be real numbers such that$a^{2}+b^{2}+c^{2}=1$. Find the minimum value of the expression $P=|6a^{3}+bc|+|6b^{3}+ca|+|6c^{3}+ab|.$ 12. Given a triangle$ABC$with$(O)$and$H$respectively are the circumcircle and the orthocenter of the triangle. Let$P$be an arbitrary point on$OH$. Let$A_{0},B_{0},C_{0}$respectively be the intersections between$AH,BC,CH$and$BC,CA,AB$. Suppose that$A_{1},B_{1},C_{1}$respectively be the second intersections between$AP,BP,CP$and$(O)$. Let$A_{2},B_{2},C_{2}$respectively be the reflection points of$A_{1},B_{1},C_{1}$through$A_{0},B_{0},C_{0}$. Prove that$H,A_{2},B_{2},C_{2}$belong to a circle with center is on$OH$. ##$type=three$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: 2015 Issue 462
2015 Issue 462
Mathematics & Youth