Let $ABC$ be an arbitrary triangle. Arbitrary points $C_1$, $A_1$ and $B_1$ are chosen on the sides $AB$, $BC$ and $CA$ respectively. Let $P_1$, $P_2$ and $P_3$ denote the areas of triangles $AC_1B_1$, $BC_1A_1$ and $CA_1B_1$ respectively, and let $P$ denote the area of triangle $ABC$. Prove that $\sqrt{P_1} + \sqrt{P_2} + \sqrt{P_3} \le \frac{3}{2}\sqrt{P}$.
A grass path surrounds a rectangular asphalt playground with dimensions $36$ dm and $12$ m (as in the figure). The width $d$ of the path equals one quarter of the semi-perimeter of the playground. How many metres of wire are needed to fence the grass path from the playground, if it is wound around with $5$ rows of wire? How many posts are needed to fasten the fence, if the distance between any two neighbouring posts is $6$ dm?
Fourth-grade students took a mathematics test containing 20 problems. For each correctly solved problem 5 points are earned, and for each incorrectly solved or unsolved problem 3 points are lost.
a) How many problems did Ivan solve, if he earned 76 points?
b) What is the maximum number of problems a student may get wrong if he wants to be sure of getting a grade of 5? The minimum number of points needed for grade 5 is the solution of the equation $8245:x=97$.
The students of class $IV^{a}$ are members of the ecological, the literary or the mathematical club. Five students are members of all three clubs, and nine students are members of exactly two clubs. The ecological and literary clubs have 8 students in common, and the same number of students is in both the literary and the mathematical club. Also, 20 students are members of only one club, namely 5 each in the ecological and the mathematical club. How many students are there in class $IV^{a}$?
Dimitar has two square pieces of cardboard with sides $3$ cm and $4$ cm. Can he form a square from them by cutting, without discarding any material? If the answer is yes, what is the side length of that square?
A 100-denar banknote must be changed into coins of 2 and 5 denars, with 32 coins in total. If such a change exists, how many coins of 2 denars and how many coins of 5 denars are used?
A rectangular metal plate has mass $10$ g. It must be divided (cut) into three pieces of integer mass. With the resulting three pieces one must be able to measure, in grams, every mass from $1$ to $10$ g that is a natural number. What must the mass of each of the pieces be?
Two segments $\overline{AB}$ and $\overline{CD}$ of equal lengths lie on the same line so that $\frac{1}{4}$ of their lengths is common to both. Determine the length of these segments if the distance between their midpoints is $6$ cm.
The twelve knights at a round table must choose a two-member delegation for a visit to King Arthur. In how many ways can this be done so that the two-member delegation does not contain knights who sit next to each other at the round table?
Given the fraction $\frac{57}{71}$. Which number must be subtracted from the numerator and added to the denominator so that the value of the fraction after reduction is $\frac{1}{3}$?
In triangle $ABC$, $\angle ACB = 40^{\circ}$. The bisectors of the interior and exterior angle at the vertex $C$ intersect the line $AB$ at the points $D$ and $E$, so that $\triangle CDE$ is isosceles. Determine the angles of triangle $ABC$.
In $\triangle ABC$, $\overline{AC}=5 \text{ cm}$ and $\overline{BC}=9 \text{ cm}$. From the midpoint of the side $AB$ a perpendicular is erected which intersects the side $BC$ at the point $E$. The point $E$ is joined to the vertex $A$. Calculate the perimeter of $\triangle AEC$.
In a square $3 \times 3$ table, Dimitar may write the numbers $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$. Can Dimitar write one of these numbers in each cell so that the sums of the numbers in the three rows, the three columns and the two diagonals are all different from each other?
**The point $M$ is the midpoint of the side $BC$ of the square $ABCD$.** The point $S$ lies inside the square and is equidistant from the points $A$, $D$ and $M$. The side length of the square is $a = 40 \text{ cm}$. Calculate the perimeter and the area of the quadrilateral $ABMS$.
We have two candles of different lengths and thicknesses. The longer and thinner candle burns out completely in $3{,}5$ hours, and the shorter and thicker one in $5$ hours. The candles were lit at the same time, and after $2$ hours of burning their lengths were equal. By what percent is the thinner candle longer than the thicker one?
In triangle $ABC$ the angle $\angle BAC = 70^\circ$ and the angle $\angle ABC = 50^\circ$. The point $M$ lies inside $\triangle ABC$ such that $\angle MAC = \angle MCA = 40^\circ$. Determine the angles $\angle AMB$ and $\angle BMC$.
