02/13 Russian National Mathematical Competitions

[Pertsel Vladimir]

The second competition -- Moscow, 1962.

form
8 013 014 015 016 017
9 018 019 020 021 017
10 022 023 024 025 026

013.
Given points A',B',C',D', on the continuation of the [AB],[BC],[CD],[DA]
sides of the convex quadrangle ABCD, such, that the following pairs of
vectors are equal: [BB'[=[AB[, [CC['=[BC[, [DD'=CD[, [AA'[=[DA[.
Prove that the quadrangle A'B'C'D' area is five times more than the
quadrangle ABCD area.

014.
Given the circumference s and the straight line l, passing through the
centre O of s. Another circumference s' passes through the point O and
has its centre on the l. Describe the set of the points M, where the
common tangent of s and s' touches s'.

015.
Given positive numbers a_1, a_2, ..., a_99, a_100. It is known, that
a_1 > a_0; a_2 = 3a_1 - 2a_0,
a_3 = 3a_2 - 2a_1, ..., a_100 = 3a_99 - 2a _98.
Prove that a_100 > 2^99.

016.
Prove that there are no integers a,b,c,d such that the polynomial
ax^3 + bx^2 + cx + d equals 1 at x=19, and equals 2 at x=62.

017.
Given a nxn table, where n is odd. There is either 1 or -1 in its every
field. A product of the numbers in the column is written under every
column. A product of the numbers in the row is written to the right of
every row.
Prove that the sum of 2n products doesn't equal to 0.

018.
Given two sides of the triangle.
Build that triangle, if medians to those sides are orthogonal.

019.
Given a quartet of positive numbers a,b,c,d, and is known, that abcd=1.
Prove that a^2 + b^2 + c^2 + d^2 + ab + ac + ad + bc + bd + dc >= 10

020.
Given right pentagon ABCDE. M is an arbitrary point inside ABCDE or on
its side. Let the distances |MA|, |MB|, ... , |ME| be renumerated and
denoted with r_1 <= r_2 <= r_3 <= r_4 <= r_5.
Find all the positions of the M, giving r_3 the minimal possible value.
Find all the positions of the M, giving r_3 the maximal possible value.

021.
Given 1962 -digit number. It is dividable by 9. Let x be the sum of its
digits. Let the sum of the digits of x be y. Let the sum of the digits
of y be z. Find z.

022.
The M point is a middle of a isosceles triangle base [AC]. [MH] is
orthogonal to [BC] side. Point P is the middle of the segment [MH].
Prove that [AH] is orthogonal to [BP].

023.
What maximal area can have a triangle if its sides a,b,c satisfy
inequality 0 <= a <= 1 <= b <= 2 <= c <= 3?

024.
Given x,y,z, three different integers.
Prove that ( (x-y)^5 + (y-z)^5 + (z-x)^5 ) is
dividable by 5(x-y)(y-z)(z-x).

025.
Given a_0, a_1, ... , a_n. It is known that a_0 = a_n = 0;
a_{k-1} - 2a_k + a_{k+1} >=0 at all k = 1, 2, ... , k-1.
Prove that all the numbers are nonnegative.

026.
Given positive numbers a_1, a_2, ..., a_m; b_1, b_2, ..., b_n;
and is known that a_1 + a_2 + ... + a_m = b_1 + b_2 + ... + b_n.
Prove that You can fill an empty table with m rows and n columns with
no more than (m+n-1) positive number in such a way, that for all i,j
the sum of the numbers in the i-th row will equal to a_i, and the sum
of the numbers in the j-th column --- to b_j.

The third competition -- Moscow, 1963.

form
8 027 028 029a 030 031a
9 032 033 034 031b 028
10 035 036 037 029b 028
11 038 028 039 040 029b

027.
Given 5 circumferences, every four of them have a common point.
Prove that there exist a point that belongs to all five circumferences.

028.
Eight men had participated in the chess tournament. (Each meets each;
draws are allowed, giving 1/2 of pont; winner gets 1.) Everyone has
different number of points. The second one has got as many points as
the the four weakest together.
What was the result of the play between the third prizer and
the chess-player that have occupied the seventh place?

029.
a) Each diagonal of the quadrangle halves its area.
Prove that it is a parallelogram.
b) Three main diagonals of the hexagon halve its area.
Prove that they intersect in one point.

030.
Natural numbers a and b are relatively prime. Prove that the greatest
common divisor of (a+b) and (a^2+b^2) is either 1 or 2.

031.
Given two fixed points A and B .The point M runs along the
circumference containing A and B. K is the middle of the segment [MB].
[KP] is a perpendicular to the line (MA).
a) Prove that all the possible lines (KP) pass through one point.
b) Find the set of all the possible points P.

032.
Given equilateral triangle with the side l.
What is the minimal length d of a brush (segment), that will paint all
the triangle, if its ends are moving along the sides of the triangle.

033.
A chess-board 6x6 is tiled with the 2x1 dominos.
Prove that You can cut the board onto two parts by
a straight line that does not cut dominos.

034.
Given n different positive numbers a_1, a_2, ... , a_n.
We construct all the possible sums (from 1 to n terms).
Prove that among those sums there are at least n(n+1)/2 different ones.

035.
Given a triangle ABC. We build two angle bisectors in the corners A and
B. Than we build two lines parallel to those ones through the point C.
D and E are intersections of those lines with the bisectors. It happens,
that (DE) line is parallel to (AB).
Prove that the triangle is isosceles.

036.
Given the endless arithmetic progression with the positive integer
members. One of those is an exact square. Prove that the progression
contain the infinite number of the exact squares.

037.
Given right 45-angle. Can You mark its corners with the digits
{0,1,...,9} in such a way, that for every pair of digits there would be
a side with both ends marked with those digits?

038.
Find such real p,q,a,b, that for all x helds an equality:
(2x-1)^20 - (ax+b)^20 = (x^2+px+q)^10.

039.
On the ends of the diameter two "1"s are written. Each of the
semicircles is divided onto two parts and the sum of the numbers of its
ends (i.e. "2") is written at the middle point. Then every of the four
arcs is halved and in its middle the sum of the numbers on its ends is
written.
Find the total sum of the numbers on the circumference after n steps.

040.
Given an isosceles triangle. Find the set of the points inside the
triangle such, that the distance from that point to the base equals to
the geometric mean of the distances to the sides.

The 4-th competition -- Moscow, 1964.

form
8 041 042 043 044 045a
9 041 046 047 048 049
10 050 051 045ab 052 053
11 054 055 052 053 054

041.
The two heights in the triangle are not less than the respective sides.
Find the angles.

042.
Prove that for no natural m a number m(m + 1) is a power of an integer.

043.
Given 1000000000 first natural numbers. We change each number with the
sum of its digits an repeat this procedure until there will remain
1000000000 one digit numbers. Is there more 1-s or 2-s?

044.
Given an arbitrary set of 2k+1 integers a_1, a_2, ... , a_{2k+1}.
We make a new set:
a_1 + a_2 a_2 + a_3 a_{2k} + a_{2k+1} a_{2k+1} + a_1
---------, ---------, ... , -----------------, --------------,
2 2 2 2
and a new one, according to the same rule, and so on...
Prove that if we obtain integers only, the initial set consisted of
equal integers only.

045.
a) Given a convex hexagon ABCDEF with all the equal angles.
Prove that |AB|-|DE| = |EF|-|BC| = |CD|-|FA|.
b) The opposite problem: Prove that it is possible to build a convex
hexagon with equal angles of six segments a_1, a_2, ... , a_6, whose
lengths satisfy the condition a_1 - a_4 = a_5 - a_2 = a_3 - a_6.

046.
Find integer solutions (x,y) of the equation (1964 times "sqrt"):
sqrt(x + sqrt(x + sqrt( ....(x + sqrt(x))....))) = y.

047.
Four perpendiculars are drawn from the vertices of a convex quadrangle
to its diagonals.
Prove that their bases make a quadrangle similar to the given one.

048.
Find all the natural n such that n! is not dividable by n^2.

049.
A honeybug crawls along the honeycombs with the unite length of their
hexagons. He has moved from the node A to the node B along the shortest
possible trajectory.
Prove that the half of his way he moved in one direction.

050.
The quadrangle ABCD is outscribed around the circle with the
centre O. Prove that the sum of AOB and COD angles equals 180 degrees.

051.
Given natural a,b,n. It is known, that for every natural k (k<>b) the
number (a - k^n) is dividable by (b-k). Prove that a = b^n.

052.
Given an expression x_1 : x_2 : ... : x_n (: means division).
We can put the braces as we want. How many expressions can we obtain?

053.
We have to divide a cube onto k nonoverlapping tetrahedrons.
For what smallest k is it possible?

054.
Find the smallest exact square with last digit not 0, such that after
deleting its last two digits we shall obtain another exact square.

055.
Let ABCD be an outscribed trapezoid; E is a point of its diagonals
intersection; r_1, r_2, r_3, r_4 -- the radiuses of the circles
inscribed in the triangles ABE, BCE, CDE, DAE respectively.
1 1 1 1
Prove that --- + --- = --- + ---.
r_1 r_3 r_2 r_4

--
____/|
\ o.O| Vladimir A. Pertsel
=(_)= E-mail: vold...@wisdom.weizmann.ac.il
U Disclaimer: This posting represents the poster's views only.