Question bank HOMC 2010

Exam papers

20 questions

Question 1

Compare the numbers
$$P=\underbrace{888\ldots888}_{2010\text{ digits}}\times\underbrace{333\ldots333}_{2010\text{ digits}}$$
and
$$Q=\underbrace{444\ldots444}_{2010\text{ digits}}\times\underbrace{666\ldots667}_{2010\text{ digits}}.$$

$P=Q$
$P>Q$
$P\lt Q$

Question 2

The number of integers $n$ from the set $\{2000,2001,\ldots,2010\}$ such that $2^{2n}+2^n+5$ is divisible by $7$ is:

$0$
$1$
$2$
$3$
None of the above.

Question 3

The last five digits of the number $M=5^{2010}$ are:

$65625$
$45625$
$25625$
$15625$
None of the above.

Question 4

How many real numbers $a\in(1,9)$ are there such that $a-\dfrac1a$ is an integer?

$0$
$1$
$8$
$9$
None of the above.

Question 5

Each box in a $2\times2$ table can be colored black or white. How many different colorings of the table are there?

$4$
$8$
$16$
$32$
None of the above.

Question 6

The greatest integer less than $(2+\sqrt3)^5$ is:

$721$
$722$
$723$
$724$
None of the above.

Question 7

Determine all positive integers $a$ such that the equation $2x^2-30x+a=0$ has two prime roots, i.e. both roots are prime numbers.

Question 8

If $n$ and $n^3+2n^2+2n+4$ are both perfect squares, find $n$.

Question 9

Given a triangle $ABC$, let $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD\parallel AC$ and $ND\parallel AB$. If $S_{\triangle BMD}=9\text{ cm}^2$ and $S_{\triangle DNC}=25\text{ cm}^2$, compute $S_{\triangle AMN}$.

Question 10

Find the maximum value of $$M=\frac{x}{2x+y}+\frac{y}{2y+z}+\frac{z}{2z+x},\qquad x,y,z>0.$$

Question 1

The number of integers $n\in[2000,2010]$ such that $2^{2n}+2^n+5$ is divisible by $7$ is:

$0$
$1$
$2$
$3$
None of the above.

Question 2

The last five digits of the number $5^{2010}$ are:

$65625$
$45625$
$25625$
$15625$
None of the above.

Question 3

How many real numbers $a\in(1,9)$ are there such that $a-\dfrac1a$ is an integer?

$0$
$1$
$8$
$9$
None of the above.

Question 4

Each box in a $2\times2$ table can be colored black or white. How many different colorings of the table are there?

Question 5

Determine all positive integers $a$ such that the equation $2x^2-30x+a=0$ has two prime roots, i.e. both roots are prime numbers.

Question 6

Let $a,b$ be the roots of $x^2-px+q=0$ and let $c,d$ be the roots of $x^2-rx+s=0$, where $p,q,r,s$ are positive real numbers. Suppose that
$$M=\frac{2(abc+bcd+cda+dab)}{p^2+q^2+r^2+s^2}$$
is an integer. Determine $a,b,c,d$.

Question 7

Let $P$ be the common point of the three internal angle bisectors of a triangle $ABC$. The line passing through $P$ and perpendicular to $CP$ intersects $AC$ and $BC$ at $M$ and $N$, respectively. If $AP=3\text{ cm}$ and $BP=4\text{ cm}$, compute $\dfrac{AM}{BN}$.