# New exercises and problems in Mathematics

February 1998

## New exercises for practice in February 1998

**C. 493.** Find all perfect squares with the following property:
if we divide them by 11, the (partial) quotient is a prime number and
the remainder is 4.

**C. 494.** Write the numbers 1,2,...,*n* in a row, in
increasing order. In the next row, write the same numbers in decreasing
order. Consider the difference of any two numbers, one of which is lying
below the other. Determine the sum of the absolute values of all such
differences.

**C. 495.** Prove that if the base angles of a trapezoid are not
congruent, then the diagonal starting at the vertex of the smaller angle
is longer than the other diagonal.

**C. 496.** The diagonals of a regular hexagonal prism have lengths
12 and 13. Calculate the volume of the polyhedron.

## New exercieses in February 1998

**Gy. 3182.** The non-negative real numbers *a,b,c,d* add up
to 1. Prove inequality .

**Gy. 3183.** Some people, who know each other very well, are
sitting at a round table. Some of them always tell the truth, but the
others always lie. Each of them claims that he is veracious, but the
*k*th person sitting on his right hand side is a liar. How many
people can be there around the table?

**Gy. 3184.** In America, temperature is measured in Fahrenheit
degrees. This is a linear scale in which the melting point of ice is
taken as 32 degrees (32 ^{o}F) and the boiling point of water
is taken as 212 ^{o}F. Someone gives us the temperature in
Fahrenheit degrees, rounded to the nearest integer degree. We then
convert it to the Celsius scale and round it to the nearest integer
degree. Express, in centigrades, the maximum possible deviation of the
temperature calculated this way from the actual temperature.

**Gy. 3185.** The numbers 1,2,...,*n* are assigned, in some
order, to the vertices
*A*_{1}, *A*_{2}, ..., *A _{n}*
of a regular

*n*-gon.

- a) Prove that the sum of the absolute values of the differences
between adjacent numbers is at least 2

*n*-2.

- b) Find the number of arrangements in which the above sum is exactly
2

*n*-2.

**Gy. 3186.** Construct a triangle, given the lengths of an altitude
and a median, starting at the same vertex of the triangle, and the
distance between that vertex and the orthocentre of the triangle.

**Gy. 3187.** Ann and Bob agreed to have a date somewhen between 5
and 5:30 p.m. Assuming that they arrive within the specified interval,
find the probability that no one has to wait more than 10 minutes for
the other.

**Gy. 3188.** Prove that every tetrahedron has a vertex such that
the edges starting at that vertex can be rearranged to form a triangle.

**Gy. 3189.** Rotate a cube of unit edge by 60^{o} about
one of its diagonals. Calculate the volume of the intersection of the
rotated cube with the original one.

## New problems in February 1998

**F. 3214.** Aladdin walked all over the equator in such a way
that in each moment he either was moving to the west or was moving to
the east or applied some magic trick to get to the opposite point of the
Earth. We know that he travelled at most 19000 km's alltogether during
his westward moves. Prove that there was a moment when the difference
between the distances he had covered moving to the east and moving to
the west, respectively, was at least half of the length of the equator.

**F. 3215.** We are given real numbers
*a*_{1}, *a*_{2}, ..., *a _{n}*
such that each of them is greater than a positive number

*k*. Prove inequality .

**F. 3216.** Decide, whether inequality
holds for every natural number
*n*.

**F. 3217.** Let
*s*_{1}, *s*_{2}, *s*_{3}
denote the lengths of the medians in a triangle, and let
*d*_{1}, *d*_{2} and *d*_{3}
denote the distances of a point *P* from the medians, respectively.
Prove that one of the products *s*_{1}*d*_{1},
*s*_{2}*d*_{2}
*s*_{3}*d*_{3}
equals the sum of the other two.

**F. 3218.** We are given a triangle
*A*_{1}*A*_{2}*A*_{3}
and a point *P* inside the triangle. Let *B _{i}*,
for

*i*=1,2,3, denote the foot of the perpendicular from

*P*to line

*A*

_{i}A_{i+1}(indices are taken modulo 3). In a similar manner, we can obtain a triangle

*C*

_{1}

*C*

_{2}

*C*

_{3}from triangle

*B*

_{1}

*B*

_{2}

*B*

_{3}, and then triangle

*D*

_{1}

*D*

_{2}

*D*

_{3}from triangle

*C*

_{1}

*C*

_{2}

*C*

_{3}. Prove that triangle

*D*

_{1}

*D*

_{2}

*D*

_{3}is similar to triangle

*A*

_{1}

*A*

_{2}

*A*

_{3}.

**F. 3219.** Each face of a regular dodecahedron is coloured with
one of four colours, red, blue, yellow and green, such that adjacent
faces have different colours. Find the number of such edges for which
one of the two faces incident to the edge is blue and the other one is
green.

## New advanced problems in February 1998

**N. 163.** Consider words of length 3*n* obtained from a
three letter alphabet **A**,**B**,**C** according to the
folllowing rules. Firstly, each letter has to be used exactly *n*
times. Secondly, for each , the
*k*th letter `**A**' has to precede the *k*th letter
`**B**', which, in turn, has to precede the *k*th letter
`**C**'. Prove that the number of such words is
.

**N. 164.** Is there any number in the Fibonacci sequence whose six
last digits (in decimal system) are all 9?

**N. 165.** Sets *A*_{0}, *A*_{1},
*A*_{2}, ..., *A _{k}* are subsets, each of
size 2

*n*, of a set of 4

*n*elements. Prove that there can be found two of them whose intersection has at least elements.

**N. 166.** We are given four points on a line in the following
order: *A,B,C,D*. Moreover, we know that *AB*=*CD*. Is it
possible to construct the midpoint of segment *BC*, if we are only
allowed to use a straightedge?