We solve problems

Problems Geometry Plane geometry Area Area (other)

Let \(E\) and \(F\) be the midpoints of the sides \(BC\) and \(AD\) of the parallelogram \(ABCD\). Find the area of the quadrilateral formed by the lines \(AE, ED, BF\) and \(FC\), if it is known that the area \(ABCD\) is equal to \(S\).

Problem #PRU-57006

Problems Geometry Plane geometry Quadrilateral Circumscribed quadrilaterals

13–16 4.0

Prove that a convex quadrilateral \(ICEF\) can contain a circle if and only if \(IC+EH = CE+IF\).

Problem #PRU-57165

Problems Geometry Plane geometry Locus Loci with nonzero area

13–15 2.0

Let \(O\) be the center of the rectangle \(ABCD\). Find the geometric points of \(M\) for which \(AM \geq OM, BM \geq OM\), \(CM \geq OM\), and \(DM \geq OM\).

Problem #PRU-57194

Problems Geometry Plane geometry Constructions Constructions (other)

12–15 3.0

Three segments whose lengths are equal to \(a, b\) and \(c\) are given. Construct a segment of length: a) \(ab/c\); b) \(\sqrt {ab}\).

Problem #PRU-30656

Problems Algebra and arithmetic Algebraic equations and systems of equations

13–15 2.0

Solve the equations \(x^2 = 14 + y^2\) in integers.

Problem #PRU-60659

Problems Algebra and arithmetic Algebraic equations and systems of equations

14–16 3.0

Solve the equations in integers:

a) \(3x^2 + 5y^2 = 345\);

b) \(1 + x + x^2 + x^3 = 2^y\).

Problem #PRU-65299

Problems Combinatorics

14–16 3.0

In honor of the March 8 holiday, a competition of performances was organized. Two performances reached the final. \(N\) students of the 5th grade played in the first one and \(n\) students of the 4th grade played in the second one. The performance was attended by \(2n\) mothers of all \(2n\) students. The best performance is chosen by a vote of the mothers. It is known that half of the mothers vote honestly, i.e. for the performance that was truly better and the mothers of the other half in any case vote for the performance in which their child participates.

a) Find the probability of the best performance winning by a majority of votes.

b) The same question but this time more than two performances made it to the final.

Problem #PRU-5001

Problems Cryptography Number theory

12–15 3.0

The meeting of the secret agents took place in the green house.

Considering the numbers in the windows of the green house, what should be drawn in the empty frame?

Problem #PRU-5002

Problems Cryptography Number theory

12–15 2.0

Find one way to encrypt letters of Latin alphabet as sequences of \(0\)s and \(1\)s, each letter corresponds to a sequence of five symbols.

Problem #PRU-5003

Problems Cryptography Number theory

12–15 3.0

Pinoccio keeps his Golden Key in the safe that is locked with a numerical password. For secure storage of the Key he replaced some digits in the password by letters (in such a way that different letters substitute different digits). After replacement Pinoccio got the password \(QUANTISED17\). Honest John found out that:
• the number \(QUANTISED\) is divisible by all integers less than 17, and
• the difference \(QUA-NTI\) is divisible by \(7\).
Could he find the password?