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SMO 2017 Junior Round 1

Question 1

Among the five numbers $\frac{23}{44}$, $\frac{24}{45}$, $\frac{25}{46}$, $\frac{26}{47}$ and $\frac{27}{48}$, which one has the smallest value?
  1. $\frac{23}{44}$
  2. $\frac{24}{45}$
  3. $\frac{25}{46}$
  4. $\frac{26}{47}$
  5. $\frac{27}{48}$

Question 2

Let $a$ and $b$ be real numbers satisfying $\frac1a\lt \frac1b\lt 0$. Which of the following is incorrect?
  1. $|a|\lt |b|$
  2. $a>b$
  3. $a+b\lt ab$
  4. $a^3>b^3$
  5. $a^2>b^2$

Question 3

How many ways can the letters of the word “IGLOO” be arranged?
  1. $4$
  2. $5$
  3. $30$
  4. $60$
  5. $120$

Question 4

Jenny and Mary received identical fruit baskets, each containing 3 apples, 4 oranges and 2 bananas. Assuming that both Jenny and Mary randomly picked a fruit from their own basket, what is the probability that they both picked an apple?
  1. $\frac19$
  2. $\frac16$
  3. $\frac13$
  4. $\frac23$
  5. None of the above

Question 5

A cylinder has base radius $r$ and height $7r$. If a sphere has the same surface area as the cylinder, find the ratio of the volume of the cylinder to the volume of the sphere.
  1. $\frac34$
  2. $\frac43$
  3. $\frac4{21}$
  4. $\frac{21}{32}$
  5. $7$

Question 6

Let $ABCD$ be a rectangular sheet of paper with $AB=6$ and $BC=8$. We can fold the paper along the crease line $EF$ so that point $C$ coincides with point $A$. Find the length of the resulting line segment $AF$.

Question illustration
  1. $\frac{25}4$
  2. $\frac{13}2$
  3. $\frac{27}4$
  4. $7$
  5. None of the above

Question 7

Given three consecutive positive integers, which of the following is a possible value for the difference of the squares of the largest and the smallest of these three integers?
  1. $91$
  2. $92$
  3. $93$
  4. $94$
  5. $95$

Question 8

You have 30 rods of length 5, 30 rods of length 17 and 30 rods of length 19. Using each rod at most once, how many non-congruent triangles can you form?
  1. $6$
  2. $7$
  3. $8$
  4. $9$
  5. $10$

Question 9

Let $a$ and $b$ be positive integers. If the highest common factor of $a$ and $b$ is 6 and the lowest common multiple of $a$ and $b$ is $2^3 3^4 5^6$, how many possible values are there for $a$?
  1. $2$
  2. $4$
  3. $8$
  4. $14$
  5. $16$

Question 10

If $x$ and $y$ are non-zero real numbers satisfying $x+y=2$ and $$\frac{(1-y)^2}{y}+\frac{(1-x)^2}{x}=-4,$$ find the value of $xy$.
  1. $1$
  2. $-1$
  3. $\frac12$
  4. $-\frac12$
  5. $\sqrt2$

Question 11

An $n$-sided polygon has two interior angles of sizes $94^\circ$ and $51^\circ$. The remaining interior angles are all equal in size. If $4\le n\le20$, determine the value of $n$.

Question 12

Find the number of multiples of 7 in the sequence 80, 81, 82, ..., 2016, 2017.

Question 13

A list of six positive integers has a unique mode of 4, median of 6 and mean of 8. Find the largest possible integer in the list.

Question 14

In the diagram, $EF$ is a diameter of the circle and $ABCD$ is a square with points $B$ and $C$ on $EF$ and points $A$ and $D$ on the circle. If $AB=17\sqrt5$, find the length of $EF$.

Question illustration

Question 15

Find the remainder when $$2017^1+2017^2+\cdots+2017^{2017}$$ is divided by 9.

Question 16

Assume that $$(x+2)^{2017}+(x+2)^{2016}+\cdots+(x+2)^{2000}=a_{2017}x^{2017}+a_{2016}x^{2016}+\cdots+a_1x+a_0.$$ Find the value of the following expression: $$(a_0-a_1)+(a_2-a_3)+(a_4-a_5)+\cdots+(a_{2016}-a_{2017}).$$

Question 17

If $x=\sqrt{2017}+1$, find the value of $$x^3-(2+\sqrt{2017})x^2+(1+2\sqrt{2017})x-\sqrt{2017}.$$

Question 18

Let $ABC$ be a triangle, $D$ be a point on $AC$ such that $AD=DC$ and $E$ be a point on $BC$ such that $BE=2EC$. Let $F$ be the intersection of $BD$ and $AE$. If the area of the triangle $ABC$ is 100, find the area of triangle $ADF$.

Question 19

Find the largest integer from 1 to 100 which has exactly 3 positive integer divisors. For example, the only positive divisors of 4 are 1, 2 and 4.

Question 20

Let $a,b$ and $c$ be positive integers such that $a^2+bc=257$ and $ab+bc=101$. Determine the value of $abc$.