Langson's Examination for gifted students of mathematics in English 2014

No calculators are allowed

 

MULTIPLE CHOICE QUESTIONS

Question 1. What is the value of positive $n$ if

$$\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+...+\frac{1}{\sqrt{n}+\sqrt{n+1}}=10$$

(A) $142$

(B) $143$

(C) $145$

(D) $146$

(E) $147$

Question 2. Let $x\geq 3$ be a positive real number. What is the smallest positive value of $S=x+\frac{1}{x}$.

(A) $2$

(B) $3$

(C) $\frac{10}{3}$

(D) $4$

(E) $\frac{13}{3}$

Question 3. What is the last digit of the integer $1!+2!+3!+...+2014!$ ?

(A) $3$

(B) $4$

(C) $5$

(D) $6$

(E) $8$

Question 4. How many pairs $(x;y)$ of nonnegative intergers satisfy the following equation?

$$106x^2+19y=2014$$

(A) $0$

(B) $1$

(C) $2$

(D) $3$

(E) $4$

Question 5. How many roots are there in the following equation?

$$(x-1)(x^2-2)(x^3-27)...(x^{2014}-2014^{2014})=0$$

(A) $3051$

(B) $2014$

(C) $3021$

(D) $4028$

(E) None of above

 SHORT ANSWER

Question 6. Given a function $f:\mathbb{R} \setminus  \{ 0 \} \to \mathbb{R}$ such that $f(x) + 2f\left( \frac{1}{x} \right) = x$. Find $f(2014)$.

Question 7.  Let $P = 2013^{4028} + 2014^{2013}$. Prove that $P$ is divisible by $5$.

Question 8. Show that $f(n) = n^4 +3n^2-2n+3$ is a composite number, for any $n > 1, n \in \mathbb{Z}$.

Question 9.  Find all pairs of the nonnegative integers $(x;y)$ such that $x^3+8x^2-6x+8=y^3$

Question 10. Let $O$ be the incenter of a triangle $ABC$. Given two points $M$ and $K$ on sides $AC$ and $BC$ respectively, so that $BK.BA = BO^2$ and $AM.AB=AO^2$. Prove that $M,K,O$ are collinear.

Question 11. Let $X$ be arbitrary point on the side $AB$ of a triangle $ABC (AB < AC)$. Suppose that $XD$ is the internal bisector of the triangle $BXC$ ($D$ lies on $BC$). Take a point $S$ on the extension of the $BC$ such that $\widehat{SXD}=90^o$. A line passing $S$ meets $AB,AC$ at $F,E$ respectively. Prove that three segments $AD,BE,CF$ are concurrent.

Question 12. Let $(O)$ be the circumcircle of an acute triangle $ABC$. Suppose that points $B,C$ and circle $(O)$ are fixed. Let $M$ be the midpoint of side $BC$ and $S$ be the midpoint of $AM$. The line $AO$ meets $(O)$ again at $N$. Let $P$ be the midpoint of the $MN$. Prove that the circumcircle of triangle $SOP$ always goes through a fixed point when point $A$ moves along circle $(O)$.

Question 13. Solve the following system in real numbers:

$$\left\{\begin{matrix}2x^2 +y^2 + y &=& 3xy +2x\\x^2+y^2&=&13\end{matrix}\right.$$

Question 14. Find the minimum value of $P = \frac{a^2}{b}+ \frac{b^2}{c}+ \frac{c^2}{a}$, in which $a,b,c$ are postive real numbers and $a+b+c=1$.

Question 15. Each of ten segment in longer than $37$ but shorter than $1014$. Prove that you can select three sides of a triangle among the segments.