MathNet 2015-part-06 — 191 problems (EN–VI)

A clothing store holds a two-week clearance sale. In the first week, a customer buying three items of clothing got the cheapest item for free. In the first week Janez bought a jacket, trousers and a sweater. For the jacket and the trousers he paid 115.01 euros, and thanks to this offer he saved $17.85 \%$ of the regular price of the clothes. In the second week there is an offer giving a $20 \%$ discount on all jackets. For the same purchase as in the first week, Janez would have paid 125 euros in the second week. Calculate the regular prices of the sweater, the trousers and the jacket.

Determine $m \in \mathbb{R}$ such that the minimum value of the function $f(x) = (m-1)x^{2} + (1-3m)x + 2m + 1$ equals $-1$.

Solve the equation $\frac{4^{\log_{3} x}}{6} = \frac{2^{\log_{3}(x+1)}}{3}$.

Find $x$ such that the sequence $x+2,\ 3x,\ x^{2}+5$ is geometric and its terms are integers. Find the common ratio and write down the sequence.

The function $f$ is given by $f(x)=A \cdot \cos \left(x+\frac{\pi}{6}\right)$. Determine $A \in \mathbb{R}$ such that $f\left(-\frac{\pi}{3}\right)=-\sqrt{3}$. For the function $f$ determined in this way, write down its range and sketch its graph.

From the letters of the word LJUBEZNIVOST we form words of length 4. Letters must not repeat.

a) How many different words can we form?

b) How many different words can we form if we use only vowels?

c) Calculate the probability that from randomly chosen letters we form the word NEBO.

d) Calculate the probability that from randomly chosen letters we form a word that begins and ends with a consonant.

(8 points)

Luka drove along a straight road over a mountain pass. At a horizontal distance of $x$ metres from the foot of the pass he was at an altitude of $h(x) = -\frac{x^{2}}{20000} + \frac{3x}{5} + 560$ metres (see figure).



a) At what altitude is the highest point of the pass?
b) Calculate the slope angle of the road, accurate to the minute, at the point where Luka was 4000 metres from the foot of the pass in the horizontal direction.

(8 points)

The value of $2 - (0 - (1 - 5))$ is
(A) 3
(B) 1
(C) 0
(D) -1
(E) -2

If $\frac{12}{x}$ is a natural number and $x$ is a natural number, how many possible values are there for $x$?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

If $\frac{20}{3} = \frac{120}{y}$ then $y$ equals
(A) 9
(B) 12
(C) 15
(D) 18
(E) 20

The last digit of $2011 \times 2013 \times 2015 - 2010 \times 2012 \times 2014$ is
(A) 0
(B) 1
(C) 2
(D) 4
(E) 5

If today is Thursday, what day of the week will it be in 150 days from now?
(A) Sunday
(B) Monday
(C) Tuesday
(D) Wednesday
(E) Thursday

It is said that you can only fold a piece of paper in half 7 times. Harold folds a sheet of paper in half 5 times and then makes a hole in the folded paper. How many holes does the sheet of paper have after it is unfolded again?
(A) 32
(B) 36
(C) 81
(D) 50
(E) 64

Alfred and four other people want to be in a group photograph. In how many different ways can they be arranged in a row with Alfred in the middle?
(A) 4
(B) 8
(C) 12
(D) 24
(E) 40

If $a \bullet b$ means the value of $ab + a + b$, and $5 \bullet x = 35$, the value of $x$ must be

(A) 5
(B) 7
(C) 9
(D) 12
(E) 15

$ABCD$ is a rectangle and $P$ is a point on $BC$. If the area of triangle $ABP$ is one third of the area of the rectangle, then the ratio $BP : PC$ is

(A) $5 : 2$
(B) $3 : 2$
(C) $2 : 1$
(D) $3 : 1$
(E) $9 : 4$

How many of the integers between $97$ and $199$ are multiples of $2$ or $3$?
(A) $33$ (B) $40$ (C) $55$ (D) $60$ (E) $68$

If $t$ toffees cost $c$ cents, the number of toffees that can be bought for $r$ rands is
(A) $\frac{100rc}{t}$ (B) $\frac{100rt}{c}$ (C) $\frac{100r}{ct}$ (D) $\frac{rt}{100c}$ (E) $\frac{100c}{rt}$

Human hair grows at a rate of about $1$ centimetre per month. This is equivalent to about how many millimetres every ten years?
(A) $12$
(B) $120$
(C) $1\,200$
(D) $12\,000$
(E) $120\,000$

One of the factors of $48$ is chosen at random. What is the probability that the chosen factor is NOT a multiple of $4$?

(A) $40\%$
(B) $30\%$
(C) $25\%$
(D) $20\%$
(E) $10\%$