C

1003047804

Level: 
C
The pianist wanted to learn a new composition in $3$ weeks ($21$ days). He decided to learn the same number of bars (measures) a day. In the end, however, he met his plan only on the first day. Every other day, he managed to learn by one bar fewer than the previous day. Find how many bars did he learn on the $15$th day knowing that he was able to learn a total of $462$ measures (in $21$ days).
$18$
$22$
$32$
$15$
$20$

1003047803

Level: 
C
In Duolingo application, every user earns so-called lingots (=virtual currency), providing he or she learns at least $10$ minutes for $10$ consecutive days. For the first $10$ days a user earns $1$ lingot, $2$ lingots for the next ten-day period, $3$ lingots for the following $10$ days (i.e. for the first $30$ days a user earns $6$ lingots), etc. Find the smallest number of days in which a user can accumulate $1000$ lingots.
$450$
$45$
$440$
$44$
$430$

1003047801

Level: 
C
A certain type of bamboo grows $1.3\,\mathrm{m}$ per day during the vegetation period. Concerning the fact that it reached a height of $30\,\mathrm{m}$ after twenty days of regular growth, how tall was it at the beginning of the first day?
$4\,\mathrm{m}$
$5.3\,\mathrm{m}$
$2.7\,\mathrm{m}$
$10\,\mathrm{m}$
$4.3\,\mathrm{m}$

1003233607

Level: 
C
Determine the relative position of three planes: \begin{align*} \alpha\colon\ &2x+y+9z-18=0, \\ \beta\colon\ &x+3y+2z+16=0, \\ \gamma\colon\ &x+2y+3z+6=0. \end{align*}
Planes $\alpha$, $\beta$ and $\gamma$ intersect in a straight line.
Each of the two planes are intersecting and the lines of intersection are three different lines parallel to each other.
All three planes intersect at just one point.

1003233605

Level: 
C
We are given skew lines $p$ and $q$. \begin{align*} p\colon x&= 1-t, & q\colon x&= 1-2s, \\ y&= 1+t, & y&=s, \\ z&= 3+2t;\ t\in\mathbb{R}, & z&= 3+3s;\ s\in\mathbb{R}. \end{align*} Find parametric equations of a straight line $r$, that is intersecting both lines $p$ and $q$ and lying in the plane $x+2y-z+2=0$.
$\begin{aligned} r\colon x&=-1+2m, \\ y&=3-3m, \\ z&=7-4m;\ m\in\mathbb{R} \end{aligned}$
$\begin{aligned} r\colon x&=-1+m, \\ y&=3+3m, \\ z&=7-m;\ m\in\mathbb{R} \end{aligned}$
$\begin{aligned} r\colon x&=-1+3m, \\ y&=3+2m, \\ z&=7+5m;\ m\in\mathbb{R} \end{aligned}$
$\begin{aligned} r\colon x&=-1+m, \\ y&=3-m, \\ z&=7+m;\ m\in\mathbb{R} \end{aligned}$

1103233603

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find the angle between its faces and round the number to the nearest minute.
$70^{\circ}32'$
$54^{\circ}44'$
$45^{\circ}$
$51^{\circ}4'$

1103233602

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find the distance between the opposite edges of this tetrahedron.\[ \] Hint: A tetrahedron’s opposite edges lie on skew lines. Their distance is the same as the distance of the midpoint of one edge from the opposite edge.
$1$
$\sqrt3$
$\frac{\sqrt3}2$
$\frac{\sqrt5}2$