C

1103059505

Level: 
C
Let \( ABCDEFGH \) be a cube and \( X \) be the midpoint of its edge \( AE \). What is the cross-section of the cube if we slice it with a plane \( BGX \)?
a quadrilateral \( BGPX \) with \( P \) being the midpoint of the edge \( EH \)
a quadrilateral \( BGHX \)
a triangle \( BGX \)
a quadrilateral \( BGPX \) with \( P \) being the midpoint of the edge \( DH \)

1103059504

Level: 
C
Let \( ABCDEFGH \) be a cube with \( K \), and \( L \) being the midpoints of its edges \( AE \) and \( CG \), and let \( M \) be the centre of its face \( ABFE \). What is the mutual position of planes \( BCE \), \( ADF \), and \( KLM \)?
three mutually intersecting planes which share one common line
three mutually intersecting planes which share one common point
two planes are parallel with the third intersecting them in distinct parallel lines

1103059503

Level: 
C
Let ABCDEFGH be a cube. What is the mutual position of planes \( ECG \), \( BDF \), and \( ABH \)?
three mutually intersecting planes which share one common point
three mutually intersecting planes which share one common line
two planes are parallel with the third intersecting them in distinct parallel lines

1103059502

Level: 
C
Let \( ABCDV \) be a rectangle based-pyramid, where \( V \) is its apex and \( K \), \( L \), and \( M \) are the midpoints of its edges \( AD \), \( BC \), and \( CV \) respectively. What is the mutual position of planes \( BVK \) and \( DLM \)?
distinct parallel planes
identical planes
intersecting planes

1103021613

Level: 
C
A circle is inscribed in a rhombus \( ABCD \). The touching points of the circle and the rhombus divide each side into two parts that are \( 12\,\mathrm{dm} \) and \( 25\,\mathrm{dm} \) long. (See the picture.) Find the measure of the angle \( CAB \). Round the result to two decimal places.
\( 34.72^{\circ} \)
\( 43.85^{\circ} \)
\( 46.15^{\circ} \)
\( 23.14^{\circ} \)

1103021612

Level: 
C
Consider two circles: the circle \( k \) with centre \( S_1 \) and radius \( 3\,\mathrm{cm} \), and the circle \( n \) with centre \( S_2 \) and radius \( 8\,\mathrm{cm} \). The distance between \( S_1 \) and \( S_2 \) is \( 22\,\mathrm{cm} \). Common internal tangents of the circles intersect at point \( A \). Calculate the distance of the point \( A \) from the centre \( S_1 \). (See the picture.)
\( 6\,\mathrm{cm} \)
\( 16\,\mathrm{cm} \)
\( 11\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)

1103021608

Level: 
C
Consider a circle \( k \) with radius \( 2.5\,\mathrm{cm} \). In the circle is inscribed a convex quadrilateral \( ABCD \) so that the diagonal \( AC \) is the diameter of the circle, the length of \( BC \) is \( \sqrt{21}\,\mathrm{cm} \), and the length of \( DC \) is \( 4\,\mathrm{cm} \). What is the length of the shortest side of this quadrilateral? (See the picture.)
\( 2\,\mathrm{cm} \)
\( 3\,\mathrm{cm} \)
\( \sqrt5\,\mathrm{cm} \)
\( 2.5\,\mathrm{cm} \)

1103021605

Level: 
C
A circle of radius \( 22\,\mathrm{cm} \) is inscribed into the rhombus \( ABCD \). Calculate the measure of the angle \( CAB \) if the length of the rhombus side is \( 90\,\mathrm{cm} \). (See the picture.) Round the result to two decimal places.
\( 14.63^{\circ} \)
\( 29.27^{\circ} \)
\( 30.37^{\circ} \)
\( 28.30^{\circ} \)

1103021604

Level: 
C
Calculate the radius of a circle inscribed into the rhombus \( ABCD \) if the length of its side is \( 10\,\mathrm{cm} \) and the measure of the angle \( DAB \) is \( 40^{\circ} \). (See the picture.) Round the result to two decimal places.
\( 3.21\,\mathrm{cm} \)
\( 1.71\,\mathrm{cm} \)
\( 3.83\,\mathrm{cm} \)
\( 6.42\,\mathrm{cm} \)

1103021602

Level: 
C
The side of an equilateral triangle is \( 6\,\mathrm{cm} \) long. Find the area of the annulus between the incircle and circumcircle of the given triangle. (See the picture.)
\( 9\pi\,\mathrm{cm}^2 \)
\( 6\pi\,\mathrm{cm}^2 \)
\( 12\pi\,\mathrm{cm}^2 \)
\( 8\pi\,\mathrm{cm}^2 \)