HOMO TST - Nguyen Du Specialised High School (second round)

[Gin Mellkior]

Daklak Education and Training Service
Nguyen Du Specialised High School

Team Selection Test

(Second round)

Hanoi Open Mathematical Olympicad 2013

Multiple Choice Questions

Question 1: The remainder of the division of $109^{345}$ to $14$ is:
A) $12$
B) $1$
C) $4$
D) $5$
E) None of the above

Question 2: Which is largest positive integer $n$ satisfying the inequality $$\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+ ...+\frac{1}{n\left ( n+1 \right )\left ( n+2 \right )}$$
A) $6$
B) $7$
C) $8$
D) $9$
E) None of the above

Question 3: A student has three different Mathematics books, two different English books, and four different Science books. The number of ways that the books can be arranged on a shelf, if all books of the same subject are kept together is:
A) $288$
B) $1728$
C) $1260$
D) $1544$
E) None of above

Question 4: How many triples $\left ( a;b;c \right )$ where $a$, $b$, $c\in \left \{ 1;2;3;4;5;6 \right \}$ such that the number $abc-\left ( 5-a \right )\left ( 5-b \right )\left ( 5-c \right )$ is divisible by $5$.
A) $10$
B) $80$
C) $45$
D) $90$
E) None of above

Question 5: Find the number of positive integer triples $\left ( x;y;z \right )$ satsfying the equations $x+y+z=12$.
A) $165$
B) $78$
C) $66$
D) $55$
E) None of above

Short Questions

Question 6: Find all polynomials $P\left ( x \right )$ satisfying the equation $P\left ( x \right )-P\left ( 2x+1 \right )=3x^{2}+4x+1$.

Question 7: Show that there do not exist four successive those product is $2013$ less than a perfect square.

Question 8: Let $f\left ( x \right )$ be a function such that $f\left ( x-1 \right )-3f\left ( \frac{x-1}{1-2x} \right )=1-2x$. Find the value of $f\left ( 2013 \right )$.

Question 9: Solve the system equations $x^{2}+y^{2}+\frac{2xy}{x+y}=1$ abd $\sqrt{x+y}=x^{2}-y$ in real numbers $x$, $y$.

Question 10: Let $ABC$ be a triangle, $AB\neq AC$. Assume that the internal bisector of angle $\widehat{ACB}$ bisects also the angle formed by the altitude and the median emanating from vertex $C$. Show that $ABC$ is a right triangle.

Tính chất đường đối trung

Question 11: Calculate the sum of all divisors of the form $2^{x}.3^{y}$ (with $x$, $y>0$) of the number $N=19^{88}-1$.

Question 12: Let $ABCD$ be a parallelogram and the points $M$, $N$ such that $C\in \left ( AM \right )$ and $D\in \left ( BN \right )$. The lines $NA$ and $NC$ intersect the lines $MB$ and $MD$ in the points $E$, $F$, $G$, $H$. Prove that $EFGH$ is cyclic iff $ABD$ is a rhombus.

Question 13: Consider real numbers $a$, $b$ such that all roots of the equation $ax^{3}-x^{2}+bx-1=0$ are real an positive. Determine the smallest possible value of the following expression $$P=\frac{5a^{2}-3ab+2}{a^{2}\left ( b-a \right )}$$

Question 14: Prove that for any $a$, $b$, $c>0$ satisfy $ab+bc+ca=3$, we have that $a^{3}+b^{3}+c^{3}+6abc\geq 9$.

Question 15: The real - valued function $f$ is defined for $0\leq x\leq 1$, and satisfies $f\left ( 0 \right )=0$, $f\left ( 1 \right )=1$, and $\frac{1}{2}\leq \frac{f\left ( z \right )-f\left ( y \right )}{f\left ( y \right )-f\left ( x \right )}\leq 2, \ \ \forall 0<x<y<z<1$ with $z-y=y-z$. Prove that $\frac{1}{7}\leq f\left ( \frac{1}{3} \right )\leq \frac{4}{7}$.

-END-

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