### Chú ý

Nếu các bạn đăng kí thành viên mà không nhận được email kích hoạt thì hãy kiểm tra thùng thư rác (spam). Nếu không biết cách truy cập vào thùng thư rác thì các bạn chịu khó Google hoặc đăng câu hỏi vào mục Hướng dẫn - Trợ giúp để thành viên khác có thể hỗ trợ.      # HANOI OPEN MATHEMATICAL OLYMPIAD 2017 - Junior level

Chưa có bài trả lời

### #1 IMO20xx

IMO20xx

Đã gửi 05-03-2017 - 21:57

Junior level

1. Suppose $x_1,x_2,x_3$ are the roots of polynomial $$P(x)=x^3-6x^2+5x+12.$$ Then $|x_1|+|x_2|+|x_3|$ is

A.    $4$

B.     $6$

C.     $8$

D.    $14$

E.     None of the above

1. How many pairs of positive integers $(x,y)$ are there, those satisfy the identity $$2^x-y^2=1?$$

A.    $1$

B.     $2$

C.     $3$

D.    $4$

E.     None of the above

1. Suppose $n^2+4n+25$ is a perfect square. How many such non-negative integers $n$’s are there?

A.    $1$

B.     $2$

C.     $4$

D.    $6$

E.     None of the above

1. Put $$S=2^1+3^5+4^9+5^{13}+\cdots +505^{2013}+506^{2017}.$$ The last digit of $S$ is

A.    $1$

B.     $3$

C.     $5$

D.    $7$

E.     None of the above

1. Let $a,b,c$ be two-digit, three-digit, four-digit numbers respectively. Assume that the sum of all digits of numbers $a+b$, and the sum of all digits of number $b+c$ are equal to $2$. The largest value of $a+b+c$ is

A.    $1099$

B.     $2099$

C.     $1199$

D.    $2199$

E.     None of the above

1. Find all triples of positive integers $(m,n,p)$ such that $$2^mp^2+27=q^3,$$ and $p$ is a prime.
2. Determine the two last digits of number $$Q=2^{2017}+2017^2.$$
3. Determine all real solutions $x,y,z$ of the following system of equations $$\begin{cases} x^3-3x &=4-y \\ 2y^3-6y &=6-z \\ 3z^3-9z &=8-x. \end{cases}$$
4. Prove that every equilateral triangle of area $1$ can be covered by $5$ arbitrary equilateral triangles which have the total area of $2$.
5. Find all non-negative integers $a,b,c$ such that the roots of equations $$x^2-2ax+b=0,\quad (1)$$ $$x^2-2bx+c=0,\quad (2)$$ $$x^2-2cx+a=0.\quad (3)$$ are non-negative integers.
6. Let $S$ denote a square of the side-length $7$ and let $8$ squares of the side-length $3$ be given. Show that $S$ can be covered by those $8$ small squares.
7. Does there exist a sequence of $2017$ consecutive integers which contains exactly $17$ primes?
8. Let $a,b,c$ be the side-lengths of triangle $ABC$ which $a+b+c=12$. Determine the smallest value of $$M=\frac{a}{b+c-a}+\frac{4b}{c+a-b}+\frac{9a}{a+b-c}.$$
9. Given trapezoid $ABCD$ which bases $AB\parallel CD$ ($AB<CD$). Let $O$ be the intersection of $AC$ and $BD$. Two straight lines from $D$ and $C$ are perpendicular to $AC$ and $BD$ intersect at $E$, i.e. $CE\perp BD$ and $DE\perp AC$. By analogy, $AF\perp BD$ and $BF\perp AC$. Are three points $E,O,F$ collinear?
10. Show that an arbitrary quadrilateral can be devided into $9$ isosceles triangles.

END OF PAPER

#### 1 người đang xem chủ đề

0 thành viên, 1 khách, 0 thành viên ẩn danh