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## $show=home 1. Let$\left(a_{n}\right)$be$a_{1}=3$,$a_{2}=8$,$a_{3}=13$,$a_{4}=24$,$a_{5}=31$,$a_{6}=48, \ldots,$in general, $$a_{n+2}=\begin{cases}a_{n}+4 n+8 & \text { if } n \text { is odd } \\ a_{n}+4 n+6 & \text { if } n \text { is cven }\end{cases}.$$ a) Do the numbers$2007$and$2024$appear in this sequence?. b) Find the$2007$-th number in this sequence. 2. Let$ABC$be a triangle with$\widehat{B A C}=40^{\circ}$and$\widehat{A B C}=60^{\circ} .$Denote$D$and$E$are two points on$A B$and$A C$respectively such that$\widehat{D C B}=70^{\circ}$and$\widehat{E B C}=40^{\circ}$;$D C$and$E B$meets at a point$F$. Prove that$A F$and$B C$are orthogonal. 3. Let$x,y,z,t$and$u$be positive real numbers such that$x+y+z+t+u=4$. Find the smallest possible value of the following expression $$P=\frac{(x+y+z+t)(x+y+z)(x+y)}{x y z t u}.$$ 4. Solve for$x$$$2 \sqrt{x+1}+6 \sqrt{9-x^{2}}+6 \sqrt{(x+1)\left(9-x^{2}\right)}=38+10 x-2 x^{2}-x^{3}.$$ 5. Let$P A$and$P B$be two tangent lines through a point$P$outside a circle with center$O$.$OP$and$A B$meet at$M$. Draw a secant$C D$through$M$($C D$does not contain$O$). The tangent lines at$C$and$D$meet at$Q$. Find the measure of the angle$O P Q$. 6. Find all pairs of positive integers$(x; y)$such that $$x^{y}+y=y^{x}+x.$$ 7. Prove that if the equation$x^{3}+a x^{2}+b x+c=0$has three distinct real roots, then so is $$x^{3}+a x^{2}+\left(-a^{2}+4 b\right) x+a^{3}-4 a b+8 c=0.$$ 8. Let$I$denote the incenter of an acute triangle$A B C$. The incircle$(I)$touches$B C$,$C A$and$A B$at$D$,$E$and$F$respectively. The angle bisector of$B I C$meets$B C$at$M$.$A M$meets$E F$at$P$. Prove that $$P D \geq \frac{1}{2} \sqrt{4 D E \cdot D F-E F^{2}}.$$ 9. Let$a_{1}, a_{2}, \ldots, a_{2007}$be pairwise distinct integers, all greater than$1$such that$\displaystyle \sum_{i=1}^{20} a_{i}=2017035$. Could it be possible that the sum$\displaystyle \sum_{i=1}^{2007} a_{i}^{a_{i} a_{i}}$is a perfect square? 10. Find all polynornial with real coefficients$P(x)$such that $$P(P(x)+x)=P(x) P(x+1),\,\forall x \in \mathbb{R}.$$ 11. Let$a_{1}, a_{2}, a_{3}, a_{4}$and$a_{5}$be nonnegative real numbers whose sum equal$1$. Prove the inequality $$a_{2} a_{3} a_{4} a_{5}+a_{1} a_{3} a_{4} a_{5}+a_{1} a_{2} a_{3} a_{5}+a_{1} a_{2} a_{3} a_{4} \leq \frac{1}{256}+\frac{3275}{256} a_{1} a_{2} a_{3} a_{4} a_{5}.$$ When does equality occur? 12. Let$P$be the intersection of the diagonals of an inscribed quadrilateral$A B C D$. Prove that all four Euler lines of the triangles$PAB$,$PBC$,$PCD$and$PDA$intersect in a single point. (The Euler line of a triangle is the line connecting its centroid and its orthocenter). ##$type=three$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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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
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Mathematics & Youth: 2007 Issue 366
2007 Issue 366
Mathematics & Youth