2007 Issue 366

  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).




Mathematics & Youth: 2007 Issue 366
2007 Issue 366
Mathematics & Youth
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