HANOI OPEN MATHEMATICAL OLYMPIAD 2017
Junior level
- 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
- 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
- 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
- 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
- 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
- Find all triples of positive integers $(m,n,p)$ such that $$2^mp^2+27=q^3,$$ and $p$ is a prime.
- Determine the two last digits of number $$Q=2^{2017}+2017^2.$$
- Determine all real solutions $x,y,z$ of the following system of equations $$\begin{cases} x^3-3x &=4-y \\ 2y^3-6y &=6-z \\ 3z^3-9z &=8-x. \end{cases}$$
- Prove that every equilateral triangle of area $1$ can be covered by $5$ arbitrary equilateral triangles which have the total area of $2$.
- 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.
- 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.
- Does there exist a sequence of $2017$ consecutive integers which contains exactly $17$ primes?
- Let $a,b,c$ be the side-lengths of triangle $ABC$ which $a+b+c=12$. Determine the smallest value of $$M=\frac{a}{b+c-a}+\frac{4b}{c+a-b}+\frac{9a}{a+b-c}.$$
- 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?
- Show that an arbitrary quadrilateral can be devided into $9$ isosceles triangles.
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