MathDB

I Soros Olympiad 1994-95 (Rus + Ukr)

Part of Soros Olympiad in Mathematics

Subcontests

(28)
11.6
1
11.3
2
10.4
2
10.3
2
10.2
2
10.1
2
10.8
1
10.7
1
9.8
1
11.10
1

property of points whose projections on edges of tetrahedron are coplanar

Given a tetrahedron A1A2A3A4A_1A_2A_3A_4 (not necessarily regulart). We shall call a point NN in space Serve point, if it's six projection points on the six edges of the tetrahedron lie on one plane. This plane we denote it by a(N)a (N) and call the Serve plane of the point NN. By BijB_{ij} denote, respectively, the midpoint of the edges A1AjA_1A_j, 1≤i<j≤41\le i <j \le 4. For each point MM, denote by MijM_{ij} the points symmetric to MM with respect to Bij,B_{ij}, 1≤i<j≤41\le i <j \le 4. Prove that if all points MijM_{ij} are Serve points, then the point MM belongs to all Serve planes a(Mij)a (M_{ij}), 1≤i<j≤41\le i <j \le 4.
11.8
2

(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},... (I Soros Olympiad 1994-95 R1 11.8)

A polynomial with rational coefficients is called integer, if it takes integer values ​​for all integer values ​​of the variable. For an integer polynomial PP, consider the sequence (−1)P(1),(−1)P(2),(−1)P(3),...(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...
a) Prove that this sequence is periodic, the period of which is some power of two (i.e. for some integer kk and for all natural ii, the ii-th and (i+2ki+2^k)th members of the sequence are equal).
b) Prove that for any periodic sequence consisting of (−1)(- 1) and 1 1 and with a period of some power of two, there exists a integer, polynomial P for which this sequence is (−1)P(1),(−1)P(2),(−1)P(3),...(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...

min length of arithm. progression (I Soros Olympiad 1994-99 Round 2 11.8)

Let's write down a segment of a series of integers from 00 to 19951995. Among the numbers written out, two have been crossed out. Let's consider the longest arithmetic progression contained among the remaining 19941994 numbers. Let KK be the length of the progression. Which two numbers must be crossed out so that the value of KK is the smallest?
11.7
2
11.5
3

f(x+y)&gt;= f(x)+ f(y) if f(x)/x increasing (I Soros Olympiad 1994-95 R1 11.5)

Function f(x)f(x). which is defined on the set of non-negative real numbers, acquires real values. It is known that f(0)≤0f(0)\le 0 and the function f(x)/xf(x)/x is increasing for x>0x>0. Prove that for arbitrary x≥0x\ge 0 and y≥0y\ge 0, holds the inequality f(x+y)≥f(x)+f(y)f(x+y)\ge f(x)+ f(y) .

f(x) + f(2x^2 - 1) = 2x + a (I Soros Olympiad 1994-99 Round 2 11.5)

Is there a function f(x)f(x) defined for all xx and such that for some aa and all xx holds the equality f(x)+f(2x2−1)=2x+a?f(x) + f(2x^2 - 1) = 2x + a?

m \ne k_1^n+ k_2^n+... k_n^n, m not equal to sum of perfect powers

Prove that for any natural n>1n>1 there are infinitely many natural numbers mm such that for any nonnegative integers k1k_1,k2k_2, ......,kmk_m, m≠k1n+k2n+...knn,m \ne k_1^n+ k_2^n+... k_n^n,
11.4
3

infinite queens on an infinite chessboard (I Soros Olympiad 1994-95 R1 11.4)

Given a chessboard that is infinite in all directions. Is it possible to place an infinite number of queens on it so that on each horizontally, on each vertical and on each diagonal of both directions (i.e. on a set of cells located at an angle of 45o45^o or 135o135^o to the horizontal) was exactly one queen?

all points at distance &lt;=1 from a square (I Soros Olympiad 1994-99 Round 2 11.4)

The wire is bent in the form of a square with side 22. Find the volume of the body consisting of all points in space located at a distance not exceeding 11 from at least one point of the wire.

A_1X / A_1O +B_1X/B_1O +C_1X/C_1O +D_1X/D_1O =4, insphere of tetrahedron

A tetrahedron ABCDABCD is given, in which each pair of adjacent edges are equal segments. Let OO be the center of the sphere inscribed in this tetrahedron . XX is an arbitrary point inside the tetrahedron, X≠OX \ne O. The line OXOX intersects the planes of the faces of the tetrahedron at the points marked by A1A_1, B1B_1, C1C_1, D1D_1. Prove that A1XA1O+B1XB1O+C1XC1O+D1XD1O=4\frac{A_1X}{A_1O} +\frac{B_1X}{B_1O} +\frac{C_1X}{C_1O}+\frac{D_1X}{D_1O}=4
11.2
3

&lt;ACB=? BL _|_AC, AL _|_DK , rectangle (I Soros Olympiad 1994-95 R1 11.2)

Given a rectangle ABCDABCD with AB>BCAB> BC. On the side CDCD, take a point LL such that BLBL and ACAC are perpendicular. Let KK be the intersection point of segments BLBL and ACAC. It is known that segments ALAL. and DKDK are perpendicular. Find ∠ACB.\angle ACB.

sin x &lt;= sin (x+1)&lt;=. ..&lt;= sin (x+4) (I Soros Olympiad 1994-99 Round 2 11.2)

Find the smallest positive xx for which holds the inequality sin⁡x≤sin⁡(x+1)≤sin⁡(x+2)≤sin(x+3)≤sin⁡(x+4).\sin x \le \sin (x+1)\le \sin (x+2)\le sin (x+3)\le \sin (x+4) .

partition of set of all finite ordered sets of 0 and 1

The set of all finite ordered sets of 00 and 1 1 is somehow partitioned into two disjoint classes. Prove that any infinite sequence of 00 and 11 can be cut into non-intersecting finite parts such that all of these parts (except perhaps the first) belong to the same class.
11.1
3
10.9
1
10.6
3

f(x+2^y)=f(2^x)+f(y) (I Soros Olympiad 1994-95 R1 10.6 11.6)

Find all functions f:R→Rf:R\to R such that for any real x,yx, y , f(x+2y)=f(2x)+f(y)f(x+2^y)=f(2^x)+f(y)

max of inradii and exradii in different circles

The radius of the circle inscribed in triangle ABCABC is equal to rr, and the radius of the circle tangent to the segment BCBC and the extensions of sides ABAB and ACAC (the exscribed circle corresponding to angle AA) is equal to RR. A circle with radius x<rx < r is inscribed in angle ∠BAC\angle BAC. Tangents to this circles passing through points BB and CC and different from BABA and ACAC intersect at point A′A'. Let yy be the radius of the circle inscribed in triangle BCKBCK. Find the greatest value of the sum x+yx + y as x changes from 00 to rr. (In this case, it is necessary to prove that this largest value is the same in any triangle with given rr and RR).

all the turtles will be at the vertices of some convex polygon.

Several (at least three) turtles are crawling along the plane, the velocities of which are constant in magnitude and direction (all are equal in magnitude, but pairwise different in direction). Prove that regardless of the initial location, after some time all the turtles will be at the vertices of some convex polygon.
10.5
3

| a_1+a_2+...+a_n - nS | &lt;= 997 (I Soros Olympiad 1994-95 R1 10.5)

Let a1,a2,...,a1994a_1,a_2,...,a_{1994} be real numbers in the interval [−1,1][-1,1], S=a1+a2+...+a19941994.S=\frac{a_1+a_2+...+a_{1994}}{1994}. Prove that for an arbitrary natural , 1≤n≤19941\le n \le 1994, holds the inequality ∣a1+a2+...+an−nS∣≤997.| a_1+a_2+...+a_n - nS | \le 997.

LP bisects BD

A circle can be drawn around the quadrilateral ABCDABCD. Let straight lines ABAB and CDCD intersect at point MM, and straight lines BCBC and ADAD intersect at point KK. (Points BB and PP lie on segments AMAM and AKAK, respectively.) Let PP be the projection of point MM onto straight line AKAK, LL be the projection of point KK on the straight line AMAM. Prove that the straight line LPLP divides the diagonal BDBD in half.

product sin \frac{k&#039;\pi}{2n} (I Soros Olympiad 1994-95 Ukraine R2 10.5)

For an arbitrary natural nn, prove the equality sin⁡π2nsin⁡3π2nsin⁡5π2n...sin⁡n′π2n=21−n2\sin \frac{\pi}{2n}\sin \frac{3\pi}{2n}\sin \frac{5\pi}{2n}...\sin \frac{n'\pi}{2n}=2^{\dfrac{1-n}{2}} where n′n' is the largest odd number not exceeding nn.
9.10
1
9.9
1
9.7
2
9.6
3

max no of circles in regular hexagon (I Soros Olympiad 1994-95 R1 9.6 10.4 11.3)

Given a regular hexagon, whose sidelength is 1 1 . What is the largest number of circles of radius 34\frac{\sqrt3}{4} can be placed without overlapping inside such a hexagon? (Circles can touch each other and the sides of the hexagon.)

tangent circles

A circle can be drawn around the quadrilateral ABCDABCD. KK is a point on the diagonal BDBD . The straight line CKCK intersects the side ADAD at the point MM. Prove that the circles circumscribed around the triangles BCKBCK and ACMACM are tangent.

orthocenter lies on incircle (I Soros Olympiad 1994-95 Ukraine R2 9.6)

In the triangle ABCABC, the orthocenter HH lies on the inscribed circle. Is this triangle necessarily isosceles?
9.5
3

1/p +q/(q^r -1)=1 (I Soros Olympiad 1994-95 R1 9.5 10.3)

Find the triplets of natural numbers (p,q,r)(p,q,r) that satisfy the equality 1p+qqr−1=1.\frac{1}{p}+\frac{q}{q^r -1}=1.

2 player game wit 8 numbered cards (I Soros Olympiad 1994-95 R2 9.5)

Kolya and Vasya each have 88 cards with numbers from 11 to 88 (each has all the numbers from 11 to 88). Kolya put 44 cards on the table, and Vasya put a card with a larger number on each of them. Now Vasya puts his remaining 44 cards on the table.
a) Can Kolya always put his own card with a larger number on each of Vasya’s cards?
b) Can Kolya always put on each of Vasya’s cards his own card with a number no less than on Vasya’s card?

1,995 soldiers lined up in a column

On the square, 1,9951,995 soldiers lined up in a column, and some of them stood correctly, and some turned backwards. Sergeant Smith remembers only the command "as". With this command, each soldier who sees an even number of faces facing him turns 180o180^o, while the rest remain stationary. All movements on command are performed simultaneously. Prove that the sergeant can orient all the soldiers in one direction.
9.4
3

triangle construction, given G,H,A (I Soros Olympiad 1994-95 R1 9.4)

Use a compass and a ruler to construct a triangle, given the intersection point of its median, the orthocenter, and one from the vertices.

2 body time and speed problem (I Soros Olympiad 1994-95 R2 9.4)

Two bodies simultaneously emerge from points AA and BB towards each other. Every time each body finds itself at point AA or point BB, it changes the direction of its movement to the opposite (the absolute values of the velocities of both bodies are constant). A body that initially leaves point AA travels the path ABAB in 101101 seconds. A body that initially left point BB travels the same path ABAB in 201201 seconds. After 2⋅101⋅2012 \cdot 101 \cdot 201 seconds, each body will be at its starting point. How many times during this time will the first body overtake the second?

permuting 2 digits (I Soros Olympiad 1994-95 Ukraine R2 9.4)

The natural numbers XX and YY are obtained from each other by permuting the digits. Prove that the sums of the digits of the numbers 5X5X and 5Y5Y coincide.
9.3
3

min (a+b)(b+c)/(a+2b+c), 1&lt;=a,b,c&lt;=2 (I Soros Olympiad 1994-95 R1 9.3 10.2)

Find the smallest possible value of the expression (a+b)(b+c)a+2b+c\frac{(a+b) (b + c)}{a + 2b+c} where a,b,ca, b, c are arbitrary numbers from the interval [1,2][1,2].

equal quads (I Soros Olympiad 1994-95 R2 9.3)

Is there a quadrilateral in which the position of any vertex can be changed, leaving the other three in place, so that the resulting four points serve as the vertices of a quadrilateral equal to the original one?

a fly in a 1995 x 1995 chessboard

Given is a square board measuring 1995×19951 995 \times 1 995. These cells are painted with black and white paints in a checkerboard pattern, so that the corner cells are black. A spider sitting on one of the black cells can crawl to the cell on the same side as the one it occupies in one step. Prove that a spider can always reach a fly sitting motionless in another black cell by visiting all the cells of the board once.
9.2
3

equilateral 2-player game in 72-gon (I Soros Olympiad 1994-95 R1 9.2)

Given a regular 7272-gon. Lenya and Kostya play the game "Make an equilateral triangle." They take turns marking with a pencil on one still unmarked angle of the 7272-gon: Lenya uses red. Kostya uses blue. Lenya starts the game, and the one who marks first wins if its color is three vertices that are the vertices of some equilateral triangle, if all the vertices are marked and no such a triangle exists, the game ends in a draw. Prove that Kostya can play like this so as not to lose.

hour + minute, angle (I Soros Olympiad 1994-95 R2 9.2)

What can be the angle between the hour and minute hands of a clock if it is known that its value has not changed after 3030 minutes?

2 circumcircles of similar triangles and 2 lines concurrent

Triangles MA2B2MA_2B_2 and MA1B1MA_1B_1 are similar to each other and have the same orientation. Prove that the circles circumcribed around these triangles and the straight lines A1A2A_1A_2 , B1B2B_1B_2 have a common point.
9.1
3

2 sets from {3, 4, 5, ...,13, 14} (I Soros Olympiad 1994-95 R1 9.1 10.1)

Divide the set of twelve numbers A={3,4,5,...,13,14}A = \{3, 4, 5, ...,13, 14\} into two sets B B and CC 'of six numbers each according to this condition: for any two different numbers with B B their sum does not belong to B B and for any two different numbers from CC, the sum does not belong to CC.

ab and cd divides abcd (I Soros Olympiad 1994-95 R2 9.1)

The number 19951995 is divisible by both 1919 and 9595. How many four-digit numbers are there that are divisible by two-digit numbers formed by both its first two digits and its last two digits?

wonderful , hour and minutes, clock (I Soros Olympiad 1994-95 Ukraine R2 9.1)

Let's call this position of the hour and minute hands on the analog clock wonderful, during which the hands change places after some time. Count the total number of wonderful clockwise positions.