IMAS 2017–2018 Junior Round 2

Among all the expressions listed below, how many are negative numbers? $(1000-1)^1$, $(1000-2)^2$, $\ldots$, $(1000-n)^n$, $\ldots$, $(1000-2018)^{2018}$.

In convex quadrilateral $ABCD$, bisectors of $\angle DAB$ and $\angle ABC$ intersect at $E$, bisectors of $\angle BCD$ and $\angle CDA$ intersect at $F$, as shown in the figure below. If $\angle AEB=80^\circ$, what is the angle measure, in degrees, of $\angle DFC$?

Two numbers $m$ and $n$, which may be equal, are taken from the set 1, 2, 3, 4, 5, 6, 7, 8 and 9. Which number below is not a possible value of $10(m+n)-mn$?

If $a$ and $b$ are real numbers, which of the following expressions below must be non-negative?

The product of the sum and arithmetic mean of $n$ integers is 2018. Which of the following statements below is true?

Rotate an equilateral triangle inscribed in a circle 40 degrees clockwise and counter-clockwise, as shown in the figure below. How many triangles are there in the figure?

Consider a four-digit number $\overline{abcd}$ where $a$ and $d$ are both non-zero. If the last two digits in the sum of $\overline{abcd}$ and $\overline{dcba}$ are 58, what is the maximum possible value of $\overline{abcd}$?

A rectangle is divided into 12 unit squares such that 10 are white and 2 are black, as shown in the figure below. To form a centrally symmetric picture by adding some white squares but no black squares, what is the least number of white squares needed?

A three-digit number is said to be "lucky" if it is divisible by 6 and by swapping its last two digits will give a number divisible by 6. How many "lucky" numbers are there?

Find the value of $x$ such that both $x$ and $\sqrt{2017-99\sqrt{x}}$ are integers.

In the figure below, quadrilaterals $ABCD$ and $ABCE$ are both isosceles trapezoids, where $AB\parallel CE$ and $BC\parallel AD$. If $AC=DE$, what is the measure, in degrees, of $\angle ABC$?

Place $\sqrt1$, $\sqrt2$, $\sqrt3$, $\ldots$, $\sqrt{100}$ into several groups such that the sum of each group is not more than 10. Find the least number of groups needed to attain this kind of an arrangement?

There is a sequence of five positive integers. Each number right after the first term is at least twice the number before it. If the sum of the five numbers is 2018, what is the least possible value of the last number?

Let $a$, $b$, $c$ and $d$ be four positive integers such that $\frac{b}{a}$, $\frac{c}{b}$, $\frac{d}{c}$ are simplified fractions and $\frac{b}{a}+\frac{c}{b}+\frac{d}{c}$ is an integer. Prove that $d\ge a-1$.

In the figure below, $ABC$ is a right isosceles triangle where $AB=AC$. Let $D$ be an exterior point such that $BD=\sqrt2 AD$. Prove that $\angle ADC+\angle BDC=45^\circ$.