IMAS 2018–2019 Junior — Round 2

Question 1

Which of the following statement is false?

If $a$ divides $b$ and $k$ is an integer, then $a$ divides $kb$.
If $a$ divides $b$ and $b$ divides $c$, then $a$ divides $c$.
If $a=bc$, and $b,c$ are positive integers, then $a$ is divisible by $b$ or $c$.
If $b$ divides $a$ and $c$ divides $a$, then $bc$ divides $a$.
If $p\mid bc$, then $p\mid b$ or $p\mid c$, where $p$ is a prime number, $b$ and $c$ are integers.

Question 2

How many positive integers from 1 to 2019 can be expressed as $n^3-3n^2+2n$, where $n$ is an positive integer?

Question 3

The perimeter of an isosceles triangle is known to be 32 cm and length of each side is an integer, in cm. How many different non-identical such triangles are there?

Given four distinct non-zero digits $a,b,c$ and $d$, if $\overline{ab}+\overline{cd}=\overline{dc}+\overline{ba}$, then this expression is called a palindrome expression and the sum of the two numbers $\overline{ab}+\overline{cd}$ is called a palindrome sum. For example, $53+46=64+35=99$. What is the minimum possible value of a palindrome sum?

Question 5

In the figure below, the area of the trapezoid $ABCD$ is 100 cm$^2$, the area of parallelogram $ABEF$ is 40 cm$^2$ and $CD=10$ cm. What is the length, in cm, of $AB$?

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Question 6

Four identical chess pieces are to be placed into a $4\times4$ chess board that is colored black and white alternately, as shown in the figure below. You can place at most one chess piece on each square. All chess pieces must be placed in squares of the same color and no two pieces are on the same row or on the same column. In how many different ways can the chess pieces be placed?

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Question 7

It is known that for any $x\ne\pm\frac12$, then $\frac{a}{x+\frac12}+\frac{b}{x-\frac12}=\frac{24x+4}{4x^2-1}$. What is the value of $a+b$?

Question 8

In the figure below, $ABCD$ is a square and the point $G$ lies on the side $CD$. Now, flip triangle $BCG$ along $BG$ to get a new triangle $BEG$. If $\angle CBG=32^\circ$, then what is the size, in degrees, of $\angle DAE$?

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Question 9

How many ordered triples $(a,b,c)$ of integers that satisfy the equation $|ab|+|bc|+|ca|=9$?

Question 10

If $x+y=\sqrt{4z-1}$, $y+z=\sqrt{4x-1}$ and $z+x=\sqrt{4y-1}$, where $x,y$ and $z$ are real numbers, then what is value of $x+y+z$?

Question 11

Place 9 distinct positive integers into each of the unit squares of the $3\times3$ square below, with one number in each unit square, such that the sum of the numbers in every $2\times2$ square is 50. What is the minimum possible value of sum of the 9 integers?

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Question 12

The three side lengths of an acute triangle are consecutive integers, in cm, and it is known that one altitude on one side is 12 cm. What is the area, in cm$^2$, of this triangle?

Question 13

Arrange all positive integers less than 30 and not divisible by 3 in an increasing order, and compute the sum of the reciprocals of product of every three consecutive numbers, that is $S=\frac1{1\times2\times4}+\frac1{2\times4\times5}+\cdots+\frac1{26\times28\times29}$. Now, if we reduce $S$ into its simplest form, then what would be the value of the numerator?

Question 14

In the figure below, a convex quadrilateral $ABCD$ is inscribed in circle $O$. Points $E$ and $F$ are on segments $AB$ and $AD$ respectively such that $BE=CD$ and $DF=BC$. If point $M$ is the midpoint of $EF$, then prove that $BM\perp DM$.

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Question 15

A robot can generate a set of digit codes according to user’s reasonable instructions. Wayne gives out the following commands:
(1) Each code is a four-digit number (nonzero for the left-most digit).
(2) Every two codes in the set have identical digits at no more than two corresponding positions.
Find the maximum number of codes in a set the robot can generate.