- The worksheet has 50 items, grouped into five topics and numbered continuously.
- For the multiple-choice items, circle the letter of the only correct answer.
- For the items with parts a) and b), write the complete solution in your notebook.
I. Natural numbers. Divisibility. Fractions (grades 5 and 6)
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The result of the calculation $2^{3} + 5 \cdot (18 - 3 \cdot 4):3$ is equal to:
- a) $14$
- b) $18$
- c) $26$
- d) $42$
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Consider the numbers $a = 2^{30}$, $b = 3^{20}$ and $c = 5^{10}$. The increasing order of these numbers is:
- a) $c < a < b$
- b) $a < b < c$
- c) $c < b < a$
- d) $b < a < c$
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The least common multiple of the numbers $36$ and $48$ is:
- a) $12$
- b) $96$
- c) $144$
- d) $1728$
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A T-shirt costs $80$ lei. After a $15\%$ discount, the price of the T-shirt becomes:
- a) $65$ lei
- b) $68$ lei
- c) $12$ lei
- d) $72$ lei
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The digit $x$ for which the number $\overline{25x}$ is divisible by both $3$ and $5$ is:
- a) $0$
- b) $2$
- c) $5$
- d) $8$
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The irreducible fraction equal to $\frac{45}{60}$ is:
- a) $\frac{9}{12}$
- b) $\frac{3}{4}$
- c) $\frac{15}{20}$
- d) $\frac{4}{3}$
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Consider the sets $A = \left\{ x \in {\mathbb{N}}\ |\ 2 \leq x < 7 \right\}$ and $B = \left\{ x \in {\mathbb{N}}\ |\ x\text{ is a divisor of }12 \right\}$. The set $A \cap B$ is:
- a) $\left\{ 2,\ 3,\ 4,\ 6 \right\}$
- b) $\left\{ 1,\ 2,\ 3,\ 4,\ 6 \right\}$
- c) $\left\{ 2,\ 3,\ 4,\ 5,\ 6 \right\}$
- d) $\left\{ 2,\ 4,\ 6 \right\}$
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A student read a book in three days. On the first day, the student read $\frac{1}{4}$ of the total number of pages, on the second day $\frac{2}{5}$ of the remaining pages, and on the third day the last $90$ pages.
a) Check whether this book can have $160$ pages.
b) Find the number of pages in the book.
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Between $300$ and $400$ students take part in a contest. If the students are put in groups of $12$, of $15$ or of $18$, each time $5$ students are left without a group.
a) Show that the least common multiple of the numbers $12$, $15$ and $18$ is $180$.
b) Find the number of students who take part in the contest.
II. Integers, rational and real numbers (grades 6 and 7)
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The result of the calculation $( - 2)^{3} - ( - 12):( - 3) + 4 \cdot ( - 1)^{2027}$ is equal to:
- a) $- 16$
- b) $- 8$
- c) $0$
- d) $- 4$
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The smallest of the numbers $- \frac{3}{4}$, $- 0,7$, $- \frac{4}{5}$ and $- \text{0,(7)}$ is:
- a) $- \frac{3}{4}$
- b) $- 0,7$
- c) $- \frac{4}{5}$
- d) $- \text{0,(7)}$
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The value of the expression $\left( \frac{2}{3} \right)^{- 2} \cdot \left( \frac{3}{2} \right)^{- 1} - 0,5$ is:
- a) $1$
- b) $0$
- c) $2$
- d) $- 1$
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The result of the calculation $\sqrt{48} - \sqrt{27} + \sqrt{12}$ is equal to:
- a) $3\sqrt{3}$
- b) $\sqrt{33}$
- c) $9\sqrt{3}$
- d) $5\sqrt{3}$
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Of the numbers $\sqrt{16}$, $\sqrt{0,25}$, $\sqrt{8}$ and $- \frac{7}{3}$, the irrational number is:
- a) $\sqrt{16}$
- b) $\sqrt{0,25}$
- c) $\sqrt{8}$
- d) $- \frac{7}{3}$
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The number $\sqrt{58}$ lies between the consecutive integers:
- a) $6$ and $7$
- b) $7$ and $8$
- c) $8$ and $9$
- d) $29$ and $30$
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Consider the real numbers $a = \frac{6}{\sqrt{3}} + \sqrt{75} - 2\sqrt{27}$ and $b = \left| \sqrt{3} - 2 \right| + \left| 1 - \sqrt{3} \right|$.
a) Show that $a = \sqrt{3}$.
b) Show that $b = 1$ and calculate the geometric mean of the numbers $3a$ and $4ab$.
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Consider the real number $N = \frac{10}{\sqrt{5}} - \sqrt{20} + \frac{\sqrt{45}}{3}$.
a) Show that $N = \sqrt{5}$.
b) Show that $2,2 < N < 2,3$.
III. Ratios and proportions. Equations, inequalities, systems. Organizing data (grades 6 and 7)
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The number of elements of the set $A = \left\{ x \in {\mathbb{Z}}\ |\ \text{−3} < 2x + 1 \leq 7 \right\}$ is:
- a) $4$
- b) $5$
- c) $6$
- d) $7$
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Six workers finish a job in $12$ days. Working at the same pace, eight workers finish the same job in:
- a) $16$ days
- b) $9$ days
- c) $10$ days
- d) $8$ days
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A number is chosen at random from the set $\left\{ 1,\ 2,\ 3,\ \ldots,\ 20 \right\}$. The probability that the chosen number is prime is:
- a) $\frac{2}{5}$
- b) $\frac{9}{20}$
- c) $\frac{1}{2}$
- d) $\frac{3}{10}$
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The solution of the equation $3(x - 2) + 5 = 2x + 7$ is:
- a) $x = 8$
- b) $x = 6$
- c) $x = 2$
- d) $x = - 8$
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The pair of real numbers $(x,\ y)$ that is the solution of the system $\left\{ \begin{array}{r} x + y = 7 \\ x - y = 1 \end{array} \right.$ is:
- a) $(3,\ 4)$
- b) $(4,\ 3)$
- c) $(5,\ 2)$
- d) $(6,\ 1)$
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The natural numbers $a$, $b$ and $c$ are directly proportional to the numbers $2$, $3$ and $5$.
a) Show that if $a = 8$, then $b + c = 32$.
b) Find the numbers $a$, $b$ and $c$, given that their arithmetic mean is $20$.
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At a stationery shop, $3$ notebooks and $2$ pens cost $27$ lei together, and $2$ notebooks and $3$ pens cost $23$ lei together. All the notebooks have the same price and all the pens have the same price.
a) Can a notebook cost $6$ lei? Justify your answer.
b) Find the price of a notebook and the price of a pen.
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The table below shows the math averages of the $25$ students in a class.
Average $6$ $7$ $8$ $9$ $10$ Number of students $2$ $6$ $9$ $5$ $3$ a) Find what percentage of the students in the class have an average of at least $8$.
b) Calculate the class average in math (the weighted arithmetic mean of the students' averages).
IV. Units of measurement. Angles. Triangles (grades 5 and 6)
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An aquarium in the shape of a rectangular cuboid has the dimensions $60\text{ cm}$, $30\text{ cm}$ and $40\text{ cm}$. The capacity of the aquarium is:
- a) $7,2$ liters
- b) $72$ liters
- c) $720$ liters
- d) $130$ liters
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A square has a perimeter of $3,6\text{ dm}$. The side of the square has a length of:
- a) $9\text{ cm}$
- b) $0,9\text{ cm}$
- c) $90\text{ cm}$
- d) $1,8\text{ dm}$
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The supplement of an angle that measures $47{^\circ}35'$ measures:
- a) $42{^\circ}25'$
- b) $132{^\circ}25'$
- c) $133{^\circ}35'$
- d) $132{^\circ}65'$
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The lines $AB$ and $CD$ intersect at point $O$. If $m(\sphericalangle AOC) = 38{^\circ}$, then $m(\sphericalangle AOD)$ is equal to:
- a) $38{^\circ}$
- b) $52{^\circ}$
- c) $142{^\circ}$
- d) $152{^\circ}$
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The parallel lines $a$ and $b$ are cut by the transversal $c$. One of the angles formed by the lines $a$ and $c$ measures $65{^\circ}$. The measure of an obtuse angle formed by the lines $b$ and $c$ is:
- a) $65{^\circ}$
- b) $115{^\circ}$
- c) $125{^\circ}$
- d) $25{^\circ}$
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In triangle $ABC$, $m(\sphericalangle A) = 50{^\circ}$ and $m(\sphericalangle B) = 70{^\circ}$. The measure of the exterior angle of the triangle at vertex $C$ is:
- a) $60{^\circ}$
- b) $120{^\circ}$
- c) $110{^\circ}$
- d) $130{^\circ}$
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Triangle $ABC$ is isosceles with base $BC$, and $m(\sphericalangle A) = 40{^\circ}$. The measure of angle $B$ is:
- a) $40{^\circ}$
- b) $70{^\circ}$
- c) $110{^\circ}$
- d) $140{^\circ}$
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A garden has the shape of a rectangle with a length of $24\text{ m}$ and a width of $15\text{ m}$.
a) Calculate the length of the fence needed to enclose the garden.
b) A square flower bed with a side of $6\text{ m}$ is made in the garden, and grass is sown on the rest of the area. One bag of seed is enough for $40\text{ m}^{2}$. Find the minimum number of bags needed.
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Consider the acute angle $xOy$. The points $A$ and $C$ are taken on the ray $Ox$, and the points $B$ and $D$ on the ray $Oy$, so that $OA = OB$, $OC = OD$ and $OA < OC$. The lines $AD$ and $BC$ intersect at point $M$.
a) Prove that $\Delta AOD \equiv \Delta BOC$.
b) Prove that the ray $OM$ is the bisector of angle $xOy$.
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Triangle $ABC$ is a right triangle with the right angle at $A$, $m(\sphericalangle B) = 30{^\circ}$ and $BC = 12\text{ cm}$. Point $M$ is the midpoint of segment $BC$, and $AD\bot BC$, $D \in BC$.
a) Calculate the length of segment $AC$.
b) Prove that triangle $AMC$ is equilateral and calculate the length of segment $DM$.
V. Quadrilaterals. The circle. Similar triangles. Metric relations (grade 7)
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In parallelogram $ABCD$, $m(\sphericalangle A) = 3 \cdot m(\sphericalangle B)$. The measure of angle $C$ is:
- a) $45{^\circ}$
- b) $135{^\circ}$
- c) $60{^\circ}$
- d) $120{^\circ}$
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Rhombus $ABCD$ has the diagonals $AC = 10\text{ cm}$ and $BD = 24\text{ cm}$. The side of the rhombus has a length of:
- a) $13\text{ cm}$
- b) $17\text{ cm}$
- c) $26\text{ cm}$
- d) $12\text{ cm}$
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Trapezoid $ABCD$ has the bases $AB = 14\text{ cm}$ and $CD = 8\text{ cm}$. The length of the midline of the trapezoid is:
- a) $22\text{ cm}$
- b) $11\text{ cm}$
- c) $6\text{ cm}$
- d) $3\text{ cm}$
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In triangle $ABC$, segment $AM$ is a median, and $G$ is the centroid of the triangle. If $AM = 15\text{ cm}$, then the length of segment $AG$ is:
- a) $5\text{ cm}$
- b) $7,5\text{ cm}$
- c) $10\text{ cm}$
- d) $12\text{ cm}$
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The points $A$, $B$ and $C$ are on the circle with center $O$, and point $C$ is on the major arc $AB$. If $m(\sphericalangle AOB) = 110{^\circ}$, then $m(\sphericalangle ACB)$ is equal to:
- a) $110{^\circ}$
- b) $55{^\circ}$
- c) $70{^\circ}$
- d) $125{^\circ}$
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The wheel of a bicycle has a diameter of $70\text{ cm}$. The distance the bicycle travels when the wheel makes $100$ complete turns is (use $\pi \approx 3,14$):
- a) $219,8\text{ m}$
- b) $109,9\text{ m}$
- c) $21,98\text{ m}$
- d) $439,6\text{ m}$
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In triangle $ABC$, the points $M \in AB$ and $N \in AC$ are such that $MN \parallel BC$. If $AM = 4\text{ cm}$, $MB = 6\text{ cm}$ and $AN = 6\text{ cm}$, then the length of segment $NC$ is:
- a) $4\text{ cm}$
- b) $9\text{ cm}$
- c) $8\text{ cm}$
- d) $15\text{ cm}$
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The triangles $ABC$ and $DEF$ are similar ($\Delta ABC \sim \Delta DEF$), and $\frac{AB}{DE} = \frac{2}{3}$. If the area of triangle $DEF$ is $45\text{ cm}^{2}$, then the area of triangle $ABC$ is:
- a) $30\text{ cm}^{2}$
- b) $20\text{ cm}^{2}$
- c) $67,5\text{ cm}^{2}$
- d) $10\text{ cm}^{2}$
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In the right triangle $ABC$ with the right angle at $A$, $AD\bot BC$, $D \in BC$, $BD = 4\text{ cm}$ and $DC = 9\text{ cm}$. The length of segment $AD$ is:
- a) $6\text{ cm}$
- b) $6,5\text{ cm}$
- c) $13\text{ cm}$
- d) $36\text{ cm}$
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In the right triangle $ABC$ with the right angle at $A$, $AB = 6\text{ cm}$ and $BC = 10\text{ cm}$. The value of $\sin B$ is:
- a) $\frac{3}{5}$
- b) $\frac{4}{5}$
- c) $\frac{3}{4}$
- d) $\frac{4}{3}$
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Rectangle $ABCD$ has $AB = 16\text{ cm}$ and $BC = 12\text{ cm}$. The diagonals of the rectangle intersect at point $O$, and $M$ is the midpoint of side $BC$.
a) Calculate the length of the diagonal $AC$.
b) Calculate the perimeter of triangle $OMC$.
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The isosceles trapezoid $ABCD$ has the bases $AB = 18\text{ cm}$ and $CD = 6\text{ cm}$, and the legs $AD = BC = 6\sqrt{2}\text{ cm}$.
a) Calculate the area of trapezoid $ABCD$.
b) Prove that the lines $AD$ and $BC$ are perpendicular.
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A vertical pole casts a shadow $6\text{ m}$ long on the ground. At the same moment, a vertical stick $1,5\text{ m}$ long casts a shadow $2\text{ m}$ long.
a) Find the height of the pole.
b) Find the length of the shadow of a person $1,8\text{ m}$ tall at the same moment.
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In the right triangle $ABC$ with the right angle at $A$, $AB = 15\text{ cm}$, $AC = 20\text{ cm}$, and $AD\bot BC$, $D \in BC$.
a) Calculate the length of the hypotenuse $BC$ and the area of triangle $ABC$.
b) Calculate the lengths of the segments $AD$ and $BD$.
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The regular hexagon $ABCDEF$ is inscribed in the circle with center $O$ and radius $R = 8\text{ cm}$.
a) Calculate the perimeter and the area of the hexagon.
b) Calculate the length of segment $AC$ and the distance from point $O$ to the line $AB$.