Question bank SMO 2008

How many zeroes does the product of $25^5$, $150^4$ and $2008^3$ end with?

Given that $\sqrt{2x+y}+\sqrt{x^2-9}=0$, find the value(s) of $y-x$.

The number of integers between 208 and 2008 ending with 1 is

The remainder when $7^{2008}+9^{2008}$ is divided by $64$ is

John has two $20$-cent coins and three $50$-cent coins in his pocket. He takes two coins out of his pocket, at random, one after the other without replacement. What is the probability that the total value of the two coins taken out is $70$ cents?

In the following sum, $O$ represents the digit $0$. $A,B,X,Y$ represent distinct digits. How many possible digits can $A$ be?

$$\begin{array}{r}
\mathtt{AOOBAOOB}\\
+\mathtt{BOOABOOA}\\\hline
\mathtt{XXOXYXOXX}
\end{array}$$

The least integer that is greater than $(2+\sqrt3)^2$ is

Let $x,y,z$ be non-negative numbers. Suppose $x+y=10$ and $y+z=8$. Let $S=x+z$. What is the sum of the maximum and the minimum value of $S$?

How many integer solutions $(x,y,z)$ are there to the equation $xyz=2008$?

The last two digits of $9^{2008}$ are

Find the remainder when $x^{2008}+2008x+2008$ is divided by $x+1$.

Find the maximum value of $\sqrt{x-144}+\sqrt{722-x}$.

Five identical rectangles of area $8\ \mathrm{cm}^2$ are arranged into a large rectangle as shown.



Find the perimeter of this large rectangle.

60 students were interviewed. Of these, 33 liked swimming and 36 liked soccer. Find the greatest possible number of students who neither liked swimming nor soccer.

As shown in the picture, the knight can move to any of the indicated squares of the $8\times8$ chessboard in one move. If the knight starts from the position shown, find the number of possible landing positions after $20$ consecutive moves.

Given that $\alpha+\beta=17$ and $\alpha\beta=70$, find the value of $|\alpha-\beta|$.

Evaluate, in simplest form, $$\sqrt{2008+2007\sqrt{2008+2007\sqrt{2008+2007\sqrt{\cdots}}}}.$$

Find the sum of all the positive integers less than 999 that are divisible by 15.

A brand of orange juice is available in shop $A$ and shop $B$ at an original price of $2.00$ dollars per bottle. Shop $A$ offers “buy $4$, get $1$ free”, and shop $B$ gives a $15\%$ discount if one buys $4$ bottles or more. Find the minimum cost, in cents, to buy $13$ bottles of the orange juice.

Anna randomly picked five integers from the list $$53,62,66,68,71,82,89$$ and discovered that their average is an integer. If two of the integers she picked were $62$ and $89$, find the sum of the remaining three integers.