SMO 2015 Junior Round 1

Among the five numbers $\dfrac59,\dfrac47,\dfrac35,\dfrac6{11}$ and $\dfrac{13}{21}$, which one has the smallest value?

Adrian, Billy, Christopher, David and Eric are the five starters of a school's basketball team. Two among the five shoot with their left hand while the rest shoot with their right hand. Among the five, only two are more than 1.8 metres in height. Adrian and Billy shoot with the same hand, but Christopher and David shoot with different hands. Billy and Christopher are respectively the shortest and tallest member of the team, while Adrian and David have the same height. Who is more than 1.8 metres tall and shoots with his left hand?

How many ways are there to arrange 3 identical blue balls and 2 identical red balls in a row if the two red balls must always be next to each other?

If $a,b$ and $c$ are positive real numbers such that $$\frac a{a+b}=\frac{a+b}{a+b+c}=\frac c{b+c},$$ then $\dfrac ab$ equals

In the figure below, each distinct letter represents a unique digit such that the arithmetic sum holds. What is the digit represented by the letter B? $$\begin{array}{rrrrr}&M&A&T&H\\+&M&A&T&H\\\hline H&A&B&I&T\end{array}$$

Find the minimum value of the function $2015-\dfrac{10}{x^2-4x+5}$.

It is known that 99900009 is the product of four consecutive odd numbers. Find the sum of squares of these four odd numbers.

The lengths of the sides of a triangle are $x^2$, $22-x$ and $x-2$. The total number of possible integer values of $x$ is

Find the value of $\left(4\sqrt{4+2\sqrt{3}}-\sqrt{49+8\sqrt{3}}\right)^2$.

If $x$ and $y$ satisfy the equation $2x^2+3y^2=4x$, the maximum value of $10x+6y^2$ is

In a shop, the price of a particular type of toy is a whole number greater than 100 dollars. The total sales of this type of toy on a particular Saturday and Sunday were 1518 dollars and 2346 dollars respectively. Find the total number of toys sold on these two days.

A quadrilateral $ABCD$ has perpendicular diagonals $AC$ and $BD$ with lengths 8 and 10 respectively. Find the area of the quadrilateral.

Find the smallest positive integer that is divisible by every integer from 1 to 12.

Results of a school wide vote for the president of the student council showed that two-fifths of the vote went to Alice, five-twelfths of the vote went to Bobby and the remaining 33 votes went to Charlie. If every student voted, how many students were there in the school?

Evaluate the sum $\left\lfloor\dfrac{2^2}3\right\rfloor+\left\lfloor\dfrac{3^2}4\right\rfloor+\left\lfloor\dfrac{4^2}5\right\rfloor+\cdots+\left\lfloor\dfrac{99^2}{100}\right\rfloor$.

A quiz was given to a class in which one quarter of the students are male. The class average score on the quiz was 16.5. Excluding three male students whose total score was 21, the average score of all the other students would be 17 while the average score of all the other male students would be 13.25. Find the average score of female students.

If $N=1001^4$, find the sum of all the digits of $N$.

In the diagram below, a rectangle $ABCD$ is inscribed in a circle and $E$ is a point on the circumference of the circle. Given that $|AE|^2+|BE|^2+|CE|^2+|DE|^2=450$ and $|AB|\times|BC|=108$, find the perimeter of the rectangle.

Find the largest prime factor of 999936.

Find the value of $p+q$, where $p$ and $q$ are two positive integers such that $p$ and $q$ have no common factor larger than 1 and $$\frac1{20}+\frac1{30}+\frac1{42}+\frac1{56}+\frac1{72}+\frac1{90}=\frac pq.$$