NIMO « Kumpulan Soal Kompetisi

1 A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$ , with 0 a,b > 0″ />. Rectangle $M$ has vertices $(0,0)$ , $(x,0)$ , $(x,y)$ , and $(0,y)$ , while rectangle $N$ has vertices $(x,y)$ , $(x,b)$ , $(a,b)$ , and $(a,y)$ . What is the ratio of the area of $M$ to that of $N$ ?

2 Two sequences $\{a_i\}$ and $\{b_i\}$ are defined as follows: $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$ and $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots$ . If both sequences are defined with $i$ ranging across the natural numbers, how many numbers belong to both sequences?

3 Billy and Bobby are located at points $A$ and $B$ , respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$ ; the second time they meet, they are located 10 units from point $B$ . Find all possible values for the distance between $A$ and $B$ .

4 In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for $b+e+h$ ?

$\begin{tabular}{cccc} &a&b&c \\ &d&e&f \\ + & g&h&i \\ \hline 1&6&6&5 \end{ta...$

5 We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are $\sqrt{3}$ units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, they both explode. Given that no explosions take place, how many possible configurations of on/off light bulbs exist?

6 Circle $\odot O$ with diameter $\overline{AB}$ has chord $\overline{CD}$ drawn such that $\overline{AB}$ is perpendicular to $\overline{CD}$ at $P$ . Another circle $\odot A$ is drawn, sharing chord $\overline{CD}$ . A point $Q$ on minor arc $\overline{CD}$ of $\odot A$ is chosen so that $\text{m} \angle AQP + \text{m} \angle QPB = 60^\circ$ . Line $l$ is tangent to $\odot A$ through $Q$ and a point $X$ on $l$ is chosen such that $PX=BX$ . If $PQ = 13$ and $BQ = 35$ , find $QX$ .

7 The number $\left (2+2^{96} \right )!$ has $2^{93}$ trailing zeroes when expressed in base $B$ .

a) Find the minimum possible $B$ .
b) Find the maximum possible $B$ .
c) Find the total number of possible $B$ .

8 Define $f(x)$ to be the nearest integer to $x$ , with the greater integer chosen if two integers are tied for being the nearest. For example, $f(2.3) = 2$ , $f(2.5) = 3$ , and $f(2.7) = 3$ . Define $[A]$ to be the area of region $A$ . Define region $R_n$ , for each positive integer $n$ , to be the region on the Cartesian plane which satisfies the inequality <img title=”f(|x|) + f(|y|) < n” src=”http://data.artofproblemsolving.com/images/latex/d/c/3/dc30d44d8951d826bd03abacc3ea16d5410bdb23.gif” alt=”f(|x|) + f(|y|) . We pick an arbitrary point $O$ on the perimeter of $R_n$ , and mark every two units around the perimeter with another point. Region $S_{nO}$ is defined by connecting these points in order.

a) Prove that the perimeter of $R_n$ is always congruent to $4 \pmod{8}$ .
b) Prove that $[S_{nO}]$ is constant for any $O$ .
c) Prove that $[R_n] + [S_{nO}] = (2n-1)^2$ .

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