MathDB

V Soros Olympiad 1998 - 99 (Russia)

Part of Soros Olympiad in Mathematics

Subcontests

(34)
11.10
2

concurrent wanted, mixtilinear incircle from '99

Consider a circle tangent to sides ABAB and ACAC (these sides are not equal) of triangle ABCABC and the circumscribed circle around it. Let KK, MM and PP be the touchpoints of this circle with the sides of the triangle and with the circle circumscribed around it, respectively, and let LL be the midpoint of the arc BCBC (not containing AA). Prove that the lines KMKM, PLPL and BCBC intersect at one point.

<ADP wanted , pyramid , 2 spheres

The plane angles at vertex DD of the pyramid ABCDABCD are equal to α\alpha,β\beta and γ\gamma (∠CDB=a\angle CDB = a). An arbitrary point MM is taken on edge CBCB. A ball is inscribed in each of the pyramids ABDMABDM and ACDMACDM. Let us draw through DD a plane distinct from BCDBCD, tangent to both balls and not intersecting the segment connecting the centers of the balls. Let this plane intersect the segment AMAM at point PP. What is ∠ADP\angle ADP equal to?
11.9
2
11.8
2

angle chasing, 3 angles 120^o (V Soros Olympiad 1998-99 Round 1 11.8)

Inside triangle ABCABC, point PP is taken so that angles ∠ARB=∠BPC=∠CPA=120o\angle ARB= \angle BPC = \angle CPA= 120^o. Lines BPBP and CPCP intersect lines ACAC and ABAB at points MM and KK. It is known that the quadrilateral AMPKAMPK has same areq with the triangle BCPBCP. What is the angle ∠BAC\angle BAC?

area of non convex quad wanted

Side BCBC of triangle ABCABC is equal to aa and the opposite angle is equal to θ\theta. A straight line passing through the midpoint of BCBC and the center of the inscribed circle intersects lines ABAB and ACAC at points MM and PP, respectively. Find the area of the quadrilateral (non-convex) BMPCBMPC.
11.7
2
11.6
3

cut 10x20 rectangle in 2 pieces III (V Soros Olympiad 1998-99 Round 1 11.6)

Cut the 1010 cm x20x 20 cm rectangle into two pieces with one straight cut so that they can be placed inside the 19.419.4 cm diameter circle without intersecting.

angle chasing, <ABK = <CBL = 2a - 180^o, <ABC=a, angle bisector

In triangle ABCABC, angle BB is obtuse and equal to aa. The bisectors of angles AA and CC intersect opposite sides at points PP and MM, respectively. On the side ACAC, points KK and LL are taken so that ∠ABK=∠CBL=2a−180o\angle ABK = \angle CBL = 2a - 180^o. What is the angle between straight lines KPKP and LMLM?

x^x=\frac{1}{\sqrt2}(V Soros Olympiad 1998-99 Round 3 11.6)

Solve the equation (for positive xx) xx=12x^x=\frac{1}{\sqrt2}
11.5
3

min (x -y)^2 + (z - u)^2 (V Soros Olympiad 1998-99 Round 1 11.5)

Find the smallest value of the expression (x−y)2+(z−u)2,(x -y)^2 + (z - u)^2, if (x−1)2+(y−4)2+(z−3)2+(u−2)2=1.(x -1)^2 + (y -4)^2 + (z-3)^2 + (u-2)^2 = 1.

edge of a cube, distances from a plane (V Soros Olympiad 1998-99 Round 2 11.5)

It is known that the distances from all the vertices of a cube and the centers of its faces to a certain plane (1414 values in total) take two different values. The smallest is 11. What can the edge of a cube be equal to?

x^4 - 5x + a = 0 (V Soros Olympiad 1998-99 Round 3 11.5)

Find all values of the parameter aa for which the sum of all solutions (meaning real solutions) of the equation x4−5x+a=0x^4 - 5x + a = 0 is equal to aa
11.4
3

circumsphere of pyramid (V Soros Olympiad 1998-99 Round 1 11.4)

Given a triangular pyramid in which all the plane angles at one of the vertices are right. It is known that there is a point in space located at a distance of 33 from the indicated vertex and at distances 5,6,7\sqrt5, \sqrt6, \sqrt7 from three other vertices. Find the radius of the sphere circumscribed around this pyramid. (The circumscribed sphere for a pyramid is the sphere containing all its vertices.)

sum of digits -= 25, 99 divides them (V Soros Olympiad 1998-99 Round 2 11.4)

Find all natural numbers less than 105105 that are divisible by 19991999 and whose digits sum (in decimal notation) to 2525.

locus of intersections of tangents

A chord ABAB is drawn in a circle. On its extensions beyond points AA and BB, points PP and QQ respectively are taken such that AP=BQAP = BQ. Through PP and QQ two tangents to the circle are drawn, intersecting at point MM. Find the locus of points MM (PP and QQ move along a straight line and for any PP and QQ all possible pairs of tangents are taken, which determine four points from the desired locus of points) .
11.3
3

a cos x + b cos 2x <= 0 (V Soros Olympiad 1998-99 Round 1 11.3)

For what a from the interval [0,π][0,\pi] do there exist aa and bb that are not simultaneously equal to zero, for which the inequality acos⁡x+bcos⁡2x≤0a \cos x + b \cos 2x \le 0 is satisfied for all xx belonging to the segment [a,π][a, \pi]?

x - \sqrt{x^2-a^2} = (x-a)^2}/2(x+a) (V Soros Olympiad 1998-99 Round 2 11.3)

For each value of parameter aa, solve the the equation x−x2−a2=(x−a)22(x+a) x - \sqrt{x^2-a^2} = \frac{(x-a)^2}{2(x+a)}

y =\sqrt{x^3+1},y=-\sqrt[3]{x^2+ 2x} (V Soros Olympiad 1998-99 Round 3 11.3)

Find the area of the figure on the coordinate plane bounded by the straight lines x=0x = 0, x=2x = 2 and the graphs of the functions y=x3+1y =\sqrt{x^3+ 1} and y=−x2+2x3y = - \sqrt[3]{x^2+ 2x}.
11.2
3

max C: (x + y +z + u)^2 >= Cyz (V Soros Olympiad 1998-99 Round 1 11.2)

Find the greatest value of CC for which, for any x,y,z,ux, y, z,u, and such that for 0≤x≤y≤z≤u0\le x\le y \le z\le u, holds the inequality (x+y+z+u)2≥Cyz.(x + y +z + u)^2 \ge Cyz .

2 cyclists and 1 fly (V Soros Olympiad 1998-99 Round 2 11.2)

From the two cities of Dobruisk and Bodruisk, the distance between which is 4040 km, two cyclists Dobi and Bodi simultaneously rode towards each other. Dobie was traveling at 2323 km/h and Bodie was traveling at 1717 km/h. Before departure, a fly landed on Dobie’s nose, which, at the moment of his departure from the city, flew towards Bodruisk at a speed of 4040 km/h. The fly met Bodie, immediately turned back and flew towards Dobruisk at a speed of 3030 km/h. (The fact is that the wind was blowing from Dobruisk to Bodruisk.) Having met Doby, the fly turned back again, etc. Determine the total path that the fly flew until the moment the cyclists met. (The speed of the fly in each direction did not change.)

triangular pyramid (V Soros Olympiad 1998-99 Round 3 11.2)

Five edges of a triangular pyramid are equal to 11. Find the sixth edge if it is known that the radius of the ball circumscribed about this pyramid is equal to 11.
11.1
3
10.10
2

circle tangent to incircle (V Soros Olympiad 1998-99 Round 1 10.10)

A circle inscribed in triangle ABCABC touches BCBC at point KK, MM is the midpoint of the altitude drawn on BCBC. The straight line KMKM intersects the circle inscribed in ABCABC for the second time at point PP. Prove that the circle passing through BB, CC and PP touches the circle inscribed in triangle ABCABC.

fixed point for constant angle

A chord ABAB is drawn in a circle. The line ℓ\ell is parallel to ABAB and does not intersect the circle. Let CC be a certain point on the circle (points CC located on one side of ABAB are considered). Lines CACA and CBCB intersect ℓ\ell at points DD and EE. Prove that there exists a fixed point FF of the plane, not lying on line ℓ\ell , such that ∠DFE\angle DFE is constant.
10.9
2

min network in equiangular hexagon (V Soros Olympiad 1998-99 Round 1 10.9)

Six cities are located at the vertices of a convex hexagon, all angles of which are equal. Three sides of this hexagon have length aa, and the remaining three have length bb (a≤ba \le b). It is necessary to connect these cities with a network of roads so that from each city you can drive to any other (possibly through other cities). Find the shortest length of such a road network.

cut a paper triangle of area 1 (V Soros Olympiad 1998-99 Round 3 10.9)

A triangle of area 11 is cut out of paper. Prove that it can be bent along a straight segment so that the area of the resulting figure is less than s0s_0, where s0=5−12s_0=\frac{\sqrt5-1}{2}.
Note. The value s0s_0 specified in the condition can be reduced (the smallest value ofs0s_0 is unknown to the authors of the problem). If you manage to do this (and justify it), write.
10.8
2
10.7
2

cut 10x25 rectangle in 2 pieces II (V Soros Olympiad 1998-99 Round 1 10.7)

Cut the 1010 cm ×25\times 25 cm rectangle into two pieces with one straight cut so that they can fit inside the 22.122.1 cm circle without crossing.

error in multplication of 2 naturals (V Soros Olympiad 1998-99 Round 3 10.7)

High school graduate Igor Petrov, who dreamed of becoming a diplomat, took the entrance exam in mathematics to Moscow University. Igor remembered all the problems offered during the exam, but forgot some numerical data in one. This is the task:
“When multiplying two natural numbers, the difference of which is 1010, an error was made: the hundreds digit in the product was increased by 22. When dividing the resulting (incorrect) product by the smaller of the factors, the result was quotient kk and remainder rr.. Find the numbers that needed to be multiplied.” .
The values of kk and rr were given in the condition, but Igor forgot them. However, he remembered that the problem had two answers. What could the numbers k k and rr be equal to (they are both integers and positive)?
Note. The problem in question was proposed at one of the humanities faculties of Moscow State University in 1991.
10.6
3

angle chasing, angle bisectors (V Soros Olympiad 1998-99 Round 1 10.6)

In triangle ABCABC, the bisectors of the internal angles AA1AA_1 , BB1BB_1 and CC1CC_1 are drawn (A1,B1A_1, B_1, C1C_1 - on the sides of the triangle). It is known that ∠AA1C=∠AC1B1\angle AA_1C = \angle AC_1B_1. Find ∠BCA\angle BCA.

angle chasing, OI line (V Soros Olympiad 1998-99 Round 2 10.6)

The straight line containing the centers of the circumscribed and inscribed circles of triangle ABCABC intersects rays BABA and BCBC and forms an angle with the altitude to side BCBC equal to half the angle ∠BAC\angle BAC. What is angle ∠ABC\angle ABC?

a_{n+1} = 3a_n-2a_{n-1} (V Soros Olympiad 1998-99 Round 3 10.6)

Find the formula for the general term of the sequence an, for which a1=1a_1 = 1, a2=3a_2 = 3, an+1=3an−2an−1a_{n+1} = 3a_n-2a_{n-1} (you need to express an in terms of nn).
10.4
3

length of projection, isosceles, circle (V Soros Olympiad 1998-99 Round 1 10.4)

A straight line tangent to a circle circumscribed about an isosceles triangle ABCABC (AB=ACAB = AC) at point BB intersects straight line ACAC at point PP, EE is the midpoint of ABAB (fig.). What is the projection of DEDE onto ABAB if PA=aPA = a? https://cdn.artofproblemsolving.com/attachments/e/3/59c67e8f5eb3d399656d86613bc699c8baf1c1.png

x+\sqrt{x^2-9} = 2(x+3)/(x-3)^2 (V Soros Olympiad 1998-99 Round 2 10.4)

Solve the equation x+x2−9=2(x+3)(x−3)2 x + \sqrt{x^2-9} = \frac{2(x+3)}{(x-3)^2}

min <MQA, MQ=QA, Q incenter (V Soros Olympiad 1998-99 Round 3 10.4)

Let MM be the midpoint of side BCBC of triangle ABCABC, QQ the point of intersection of its angle bisectors. It is known that MQ=QAMQ=QA. Find the smallest possible value of angle ∠MQA\angle MQA.
10.5
3

\sqrt{2+\sqrt{2-\sqrt{2+x}}}=x (V Soros Olympiad 1998-99 Round 1 10.5)

Solve the equation 2+2−2+x=x.\sqrt{2+\sqrt{2-\sqrt{2+x}}}=x.

locus, <MAC + <CMB = 90^o (V Soros Olympiad 1998-99 Round 2 10.5)

An isosceles triangle ABCABC (AB=BCAB = BC) is given on the plane. Find the locus of points MM of the plane such that ABCMABCM is a convex quadrilateral and ∠MAC+∠CMB=90o\angle MAC + \angle CMB = 90^o.

sum of inradii of AMB , AMC (V Soros Olympiad 1998-99 Round 3 10.5)

The radius of the circle inscribed in triangle ABCABC is equal to rr. This circle is tangent to BCBC at point MM and divides the segment AMAM in ratio kk (starting from vertex AA). Find the sum of the radii of the circles inscribed in triangles AMBAMB and AMCAMC.
10.3
3

5x^6 - 16x^4 - 33x^3 - 40x^2 +8 = 0 (V Soros Olympiad 1998-99 Round 1 10.3)

Find two roots of the equation 5x6−16x4−33x3−40x2+8=0,5x^6 - 16x^4 - 33x^3 - 40x^2 +8 = 0, whose product is equal to 11.

P (cos x/ 1999t) = 1/2 (V Soros Olympiad 1998-99 Round 2 10.3)

It is known that sin⁡3x=3sin⁡x−4sin⁡3x\sin 3x = 3 \sin x - 4 \sin^3x. It is also easy to prove that sin⁡nx\sin nx for odd nn can be represented as a polynomial of degree nn of sin⁡x\sin x. Let sin⁡1999x=P(sin⁡x)\sin 1999x = P(\sin x), where P(t)P(t) is a polynomial of the 19991999th degree of tt. Solve the equation P(cos⁡x1999)=12.P \left(\cos \frac{x}{1999}\right) = \frac12 .

29^{200} 2^{151} ? 5^{279} 3^{300} (V Soros Olympiad 1998-99 Round 3 10.3)

Without using a calculator, find out which number is greater: 29200⋅2151   or   5279⋅330029^{200}\cdot 2^{151} \,\,\, or \,\,\, 5^{279} \cdot 3^{300}
10.2
3

draw cos(x + y)^2 <= cos(x - y)^2 (V Soros Olympiad 1998-99 Round 1 10.2)

On the coordinate plane, draw all pointsM(x,y) M(x, y), the coordinates of which satisfy the inequalities cos⁡(x+y)2≤cos⁡(x−y)2,   0≤x3,   0≤y3.\cos(x + y)^2 \le \cos(x - y)^2, \,\,\, 0 \le x^3, \,\,\, 0 \le y^3.

infinite sum 1/ (2p+1)^2 (V Soros Olympiad 1998-99 Round 2 10.2)

In 17481748, the great Russian mathematician Leonhard Euler published one of his most important works, Introduction to the Analysis of Infinites. In this work, in particular, Euler finds the values of two infinite sums 1+14+19+116+...1 +\frac14 +\frac19+ \frac{1}{16}+... and 1+19+116+...1 +\frac19+ \frac{1}{16}+... (the terms in the first sum are the inverses of the squares of the natural numbers, and in the second are the inverses of the squares of the odd numbers of the natural series). The value of the first sum, as Euler proved, equals π26\frac{\pi^2}{6}. Given this result, find the value of the second sum.

|cos3x-tgt+ |cos3x+tgt|=|tg^2t -3|(V Soros Olympiad 1998-99 Round 3 10.2)

Solve the equation ∣cos⁡3x−tgt∣+∣cos⁡3x+tgt∣=∣tg2t−3∣. |\cos 3x - tgt| + |\cos 3x + tgt| = |tg^2t -3|.
10.1
3

perfect powers (V Soros Olympiad 1998-99 Round 1 10.1)

Find some natural number aa such that 2a2a is a perfect square, 3a3a is a perfect cube, 5a5a is the fifth power of some natural number.

average speed from 3 different speeds (V Soros Olympiad 1998-99 Round 2 10.1)

A car drove from one city to another. She drove the first third of the journey at a speed of 5050 km/h, the second third at 6060 km/h, and the last third at 7070 km/h. What is the average speed of the car along the entire journey?

y=(a-6x)/{2+x) summteric to y=1/x(V Soros Olympiad 1998-99 Round 3 10.1)

It is known that the graph of the function y=a−6x2+xy =\frac{a-6x}{2+x} is centrally summetric to the graph of the function y=1xy = \frac{1}{x} with respect to some point. Find the value of the parameter aa and the coordinates of the center of symmetry.
9.10
2

tangent circles , given equal angles (V Soros Olympiad 1998-99 Round 1 9.10)

On the bisector of angle AA of triangle ABCABC, points DD and FF are taken inside the triangle so that ∠DBC=∠FBA\angle DBC = \angle FBA. Prove that: a) ∠DCB=∠FCA\angle DCB = \angle FCA b) a circle passing through BB and FF and tangent to the segment BCBC is tangle to the circumscribed circle of the triangle ABCABC.

perimeter of triangle wanted (V Soros Olympiad 1998-99 Round 3 9.10)

The bisector of angle ∠BAC\angle BAC of triangle ABCABC intersects arc BCBC (not containing point AA) of the circle circumscribed around this triangle at point PP. Segment APAP is divided by side BCBC in ratio kk (counting from vertex AA). Find the perimeter of triangle ABCABC if BC=aBC = a.
9.9
2

max area of right triangle (V Soros Olympiad 1998-99 Round 1 9.9)

What is the largest area of a right triangle, the vertices of which are located at distances aa, bb and cc from a certain point (where aa is the distance to the vertex of the right angle)?

4 receive a prize out of 9 (V Soros Olympiad 1998-99 Round 3 9.9)

Of the 99 people who reached the final stage of the competition, only 44 should receive a prize. The candidates were renumbered and lined up in a circle. Then a certain number mm (possibly greater than 99) and the direction of reference were determined. People began to be counted, starting from the first. Each one became a winner and was eliminated from the drawing, and counting, starting from the next, continued until four winners were identified. The first three prizes were awarded to three people who had numbers 22, 77 and 55 in the original lineup (they were eliminated in that order). What number did the fourth winner of the competition have in the initial lineup?
9.8
2
9.7
2
9.6
3

{x [x] } = 0.5 (V Soros Olympiad 1998-99 Round 1 9.6)

How many solutions satisfying the condition 1<x<51 < x < 5 does the equation {x[x]}=0.5\{x[x]\} = 0.5 have?
(Here [x][x] is the integer part of the number xx, {x}=x−[x]\{x\} = x - [x] is the fractional part of the number xx.)

ratio that (BMF) cuts segment BE (V Soros Olympiad 1998-99 Round 2 9.6)

On side ABAB of triangle ABCABC, points MM and KK are taken (MM on segment AKAK). It is known that AM:MK:MB=a:b:cAM: MK: MB = a: b: c. Straight lines CMCM and CKCK intersect for the second time the circumscribed circle of the triangle ABCABC at points EE and FF, respectively. In what ratio does the circumscribed circle of the triangle BMFBMF divide the segment BEBE?

|x- y| + |1 -x| + |у| = 1 (V Soros Olympiad 1998-99 Round 3 9.6)

On the coordinate plane, draw all points M(x,y)M(x, y), whose coordinates satisfy the equation: ∣x−y∣+∣1−x∣+∣y∣=1 |x-y| + |1-x| + |y|=1
9.5
3

area wanted, 3 angles 120^o (V Soros Olympiad 1998-99 Round 1 9.5)

In triangle ABCABC, ∠BAC=60o\angle BAC= 60^o. Point PP is taken inside the triangle so that angles ∠APB=∠BPC=∠CPA=120o\angle APB=\angle BPC= \angle CP A=120^o. It is known that AP=aAP = a. Find the area of triangle BPCBPC.

locus of vertice of isosceles triangle (V Soros Olympiad 1998-99 Round 2 9.5)

An angle with vertex AA is given on the plane. Points KK and PP are selected on its sides so that AK+AP=aAK + AP = a, where aa is a given segment. Let MM be a point on the plane such that the triangle KPMKPM is isosceles with the base KPKP and the angle at the vertex MM equal to the given angle. Find the locus of points MM (as KK and PP move).

computational geo with trapezoid (V Soros Olympiad 1998-99 Round 3 9.5)

In the trapezoid ABCDABCD with bases BC=aBC = a, AD=bAD = b, the equality holds: ∠BAC+∠ACD=180o\angle BAC + \angle ACD = 180^o. The straight line ACAC intersects the common tangents to the circumcircles of the triangles ABCABC and ACDACD at the points Find PQPQ.
9.4
3

n points marked on a circle (V Soros Olympiad 1998-99 Round 1 9.4)

There are n points marked on the circle. It is known that among all possible distances between two marked points there are no more than 100100 different ones. What is the largest possible value for nn?

simplify fraction (V Soros Olympiad 1998-99 Round 2 9.4)

Simplify the fraction 123456788...876543211234567899...987654321\frac{123456788...87654321}{1234567899...987654321}’ if the digit 88 in the numerator occurs 20002000 times, and the digit 99 in the denominator 19991999 occurs times (as a result you need to get an irreducible fraction).

max &lt;AKM, 2 midpoints (V Soros Olympiad 1998-99 Round 3 9.4)

Let ABCABC be a triangle without obtuse angles, MM the midpoint of BCBC, KK the midpoint of BMBM. What is the largest value of the angle ∠KAM\angle KAM?
9.3
3

R(x - 1)^2+ y^2} +R{x^2 + (y -1)^2} = R2 (V Soros Olympiad 1998-99 Round 1 9.3)

On the coordinate plane, draw a set of points M(x;y)M(x;y), whose coordinates satisfy the equation (x−1)2+y2+x2+(y−1)2=2.\sqrt{(x - 1)^2+ y^2} +\sqrt{x^2 + (y -1)^2} = \sqrt2.

floor + fractional parts 3x3 system (V Soros Olympiad 1998-99 Round 2 9.3)

Solve the system of equations: {x+[y]+{z}=3.9y+[z]+{x}=3.5z+[x]+{y}=2.\begin{cases} x + [y] + \{z\}=3.9 \\ y + [z] + \{x\}= 3.5 \\ z + [x] + \{y\}= 2. \end{cases}

3x2 non linear system (V Soros Olympiad 1998-99 Round 3 9.3)

Solve the system of equations: x−1xy−3=y−2xy−4=3−x−y7−x2−y2\frac{x-1}{xy-3}=\frac{y-2}{xy-4}=\frac{3-x-y}{7-x^2-y^2}
9.2
3

x^4 + 4x^3 - 8x + 4 = 0 (V Soros Olympiad 1998-99 Round 1 9.2)

Solve the equation x4+4x3−8x+4=0x^4 + 4x^3 - 8x + 4 = 0.

3 equal non intersecting circles (V Soros Olympiad 1998-99 Round 2 9.2)

There are two equal non-intersecting circles on a plane. Two lines were drawn. Each of the lines intersects the circles at four points so that the three segments formed on each of the lines are equal (segments with ends at adjacent points of intersection are considered). For one line these segments have length aa, for the other they have length bb (a<ba < b). Find the radius of the circles.

simplify be reducing gives correct (V Soros Olympiad 1998-99 Round 3 9.2)

As evidence that the correct answer does not mean the correctness of the proof, the teacher cited next example. Let's take the fraction 1995\frac{19}{95}. After crossing out 99 in the numerator and denominator (“reduction” by 99), we get 15\frac{1}{5} which is the correct answer. In the same way, a fraction 19999995\frac{1999}{9995} can be “reduced” by three nines (cross out 999999 in the numerator and denominator). Is it possible that as a result of such a “reduction” we also get the correct answer, equal to 13\frac13 ? (We consider fractions of the form 1aa3\frac{1a}{a3}. Here, with the letter aa we denote several numbers that follow in the same order in the numerator after 11, and in the denominator before 33. “Reduce” by aa.)
9.1
3

parentheses in 2:2 -3:3 - 4: 4-5:5 (V Soros Olympiad 1998-99 Round 1 9.1)

Place parentheses in the expression 2:2−3:3−4:4−5:52:2 -3:3 - 4: 4-5:5 so that the result is a number greater than 3939.

number of letters in this phrase is (V Soros Olympiad 1998-99 Round 2 9.1)

In the phrase given at the end of the condition of the problem, it is necessary to put a number (numeral) in place of the ellipsis, written in verbal form and in the required case, so that the statement formulated in it is true. Here is this phrase: “The number of letters in this phrase is...”

x^2 + ax + b = x^2 + bx + a = 0 (V Soros Olympiad 1998-99 Round 3 9.1)

It is known that each of the equations x2+ax+b=0x^2 + ax + b = 0 and x2+bx+a=0x^2 + bx + a = 0 has two different roots and these four roots form an arithmetic progression in some order. Find aa and bb.
8.5
1
8.1 - 8.4
1

grade 8 problems 8.1-8.4 (V Soros Olympiad 1998-99 Round 3)

p1. Is it possible to write 55 different fractions that add up to 11, such that their numerators are equal to one and their denominators are natural numbers?
p2. The following is known about two numbers xx and yy: if x≥0x\ge 0, then y=1−xy = 1 -x; if y≤1y\le 1, then x=1+yx = 1 + y; if x≤1x\le 1, then x=∣1+y∣x = |1 + y|. Find xx and yy.
p3. Five people living in different cities received a salary, some more, others less (143143, 233233, 313313, 410410 and 413413 rubles). Each of them can send money to the other by mail. In this case, the post office takes 10%10\% of the amount of money sent for the transfer (in order to receive 100100 rubles, you need to send 10%10\% more, that is, 110110 rubles). They want to send money so that everyone has the same amount of money, and the post office receives as little money as possible. How much money will each person have using the most economical shipping method?
p4. a) List three different natural numbers mm, nn and kk for which m!=n!⋅k!m! = n! \cdot k! . b) Is it possible to come up with 19991999 such triplets?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
grade8
2

grade 8 problems (V Soros Olympiad 1998-99 Round 1)

p1. Two proper ordinary fractions are given. The first has a numerator that is 55 less than the denominator, and the second has a numerator that is 19981998 less than the denominator. Can their sum have a numerator greater than its denominator?
p2. On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in 365365 days on the next New Year's Eve?
p3. The number xx is such that 15%15\% of it and 33%33\% of it are positive integers. What is the smallest number xx (not necessarily an integer!) with this property?
p4. In the quadrilateral ABCDABCD, the extensions of opposite sides ABAB and CDCD intersect at an angle of 20o20^o; the extensions of opposite sides BCBC and ADAD also intersect at an angle of 20o20^o. Prove that two angles in this quadrilateral are equal and the other two differ by 40o40^o.
p5. Given two positive integers aa and bb. Prove that aabb≥aaba.a^ab^b\ge a^ab^a.
p6. The square is divided by straight lines into 2525 rectangles (fig.). The areas of some of They are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png
p7. A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly 17o 17^o (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again?
p8. In expression (a−b+c)(d+e+f)(g−h−k)(ℓ+m−n)(p+q)(a-b+c)(d+e+f)(g-h-k)(\ell +m- n)(p + q) opened the brackets. How many members will there be? How many of them will be preceded by a minus sign?
p9. In some countries they decided to hold popular elections of the government. Two-thirds of voters in this country are urban and one-third are rural. The President must propose for approval a draft government of 100100 people. It is known that the same percentage of urban (rural) residents will vote for the project as there are people from the city (rural) in the proposed project. What is the smallest number of city residents that must be included in the draft government so that more than half of the voters vote for it?
p10. Vasya and Petya play such a game on a 10×10board10 \times 10 board. Vasya has many squares the size of one cell, Petya has many corners of three cells (fig.). They are walking one by one - first Vasya puts his square on the board, then Petya puts his corner, then Vasya puts another square, etc. (You cannot place pieces on top of others.) The one who cannot make the next move loses. Vasya claims that he can always win, no matter how hard Petya tries. Is Vasya right? https://cdn.artofproblemsolving.com/attachments/f/1/3ddec7826ff6eb92471855322e3b9f01357116.png
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grade 8 problems (V Soros Olympiad 1998-99 Round 2)

p1. Given two irreducible fractions. The denominator of the first fraction is 44, the denominator of the second fraction is 66. What can the denominator of the product of these fractions be equal to if the product is represented as an irreducible fraction?
p2. Three horses compete in the race. The player can bet a certain amount of money on each horse. Bets on the first horse are accepted in the ratio 1:41: 4. This means that if the first horse wins, then the player gets back the money bet on this horse, and four more times the same amount. Bets on the second horse are accepted in the ratio 1:31:3, on the third -1:1 1:1. Money bet on a losing horse is not returned. Is it possible to bet in such a way as to win whatever the outcome of the race?
p3. A quadrilateral is inscribed in a circle, such that the center of the circle, point OO, is lies inside it. Let KK, LL, MM, NN be the midpoints of the sides of the quadrilateral, following in this order. Prove that the bisectors of angles ∠KOM\angle KOM and ∠LOC\angle LOC are perpendicular (Fig.). https://cdn.artofproblemsolving.com/attachments/b/8/ea4380698eba7f4cc2639ce20e3057e0294a7c.png
p4. Prove that the number33...33⏟1999   3s1\underbrace{33...33}_{1999 \,\,\,3s}1 is not divisible by 77.
p5. In triangle ABCABC, the median drawn from vertex AA to side BCBC is four times smaller than side ABAB and forms an angle of 60o60^o with it. Find the greatest angle of this triangle.
p6. Given a 7×87\times 8 rectangle made up of 1x1 cells. Cut it into figures consisting of 1×11\times 1 cells, so that each figure consists of no more than 55 cells and the total length of the cuts is minimal (give an example and prove that this cannot be done with a smaller total length of the cuts). You can only cut along the boundaries of the cells.
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grade7
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grade 7 problems (V Soros Olympiad 1998-99 Round 1)

p1. Ivan Ivanovich came to the store with 2020 rubles. The store sold brooms for 11 ruble. 1717 kopecks and basins for 11 rub. 6666 kopecks (there are no other products left in the store). How many brooms and how many basins does he need to buy in order to spend as much money as possible? (Note: 11 ruble = 100100 kopecks)
p2. On the road from city A to city B there are kilometer posts. On each pillar, on one side, the distance to city A is written, and on the other, to B. In the morning, a tourist passed by a pillar on which one number was twice the size of the other. After walking another 1010 km, the tourist saw a post on which the numbers differed exactly three times. What is the distance from A to B? List all possibilities.
p3. On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in 365 days on the next New Year's Eve?
p4. What is the smallest number of digits that must be written in a row so that by crossing out some digits you can get any three-digit natural number from 100100 to 999999?
p5. An ordinary irreducible fraction was written on the board, the numerator and denominator of which were positive integers. The numerator was added to its denominator and a new fraction was obtained. The denominator was added to the numerator of the new fraction to form a third fraction. When the numerator was added to the denominator of the third fraction, the result was 13/2313/23. What fraction was written on the board?
p6. The number xx is such that 15%15\% of it and 33%33\% of it are positive integers. What is the smallest number xx (not necessarily an integer!) with this property?
p7. A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly 17o17^o (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again?
p8. The square is divided by straight lines into 2525 rectangles (fig. 1). The areas of some of them are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png
p9. Petya multiplied all natural numbers from 11 to his age inclusive. The result is a number 8  841  761993  739  701954  543  616  000  000.8 \,\, 841 \,\,761993 \,\,739 \,\,701954 \,\,543 \,\,616 \,\,000 \,\,000. How old is Petya?
p10. There are 100100 integers written in a line, and the sum of any three in a row is equal to 1010 or 1111. The first number is equal to one. What could the last number be? List all possibilities.
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grade 7 problems (V Soros Olympiad 1998-99 Round 2)

p1. Due to the crisis, the salaries of the company's employees decreased by 1/51/5. By what percentage should it be increased in order for it to reach its previous value?
p2. Can the sum of six different positive numbers equal their product?
p3. PointsA,B,C A, B, C and BB are marked on the straight line. It is known that AC=aAC = a and BP=bBP = b. What is the distance between the midpoints of segments ABAB and CBCB? List all possibilities.
p4. Find the last three digits of 62519+37699625^{19} + 376^{99}.
p5. Citizens of five different countries sit at the round table (there may be several representatives from one country). It is known that for any two countries (out of the given five) there will be citizens of these countries sitting next to each other. What is the smallest number of people that can sit at the table?
p6. Can any rectangle be cut into 19991999 pieces, from which you can form a square?
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grade 7 problems (V Soros Olympiad 1998-99 Round 3)

p1. There are eight different dominoes in the box (fig.), but the boundaries between them are not visible. Draw the boundaries. https://cdn.artofproblemsolving.com/attachments/6/f/6352b18c25478d68a23820e32a7f237c9f2ba9.png
p2. The teacher drew a quadrilateral ABCDABCD on the board. Vanya and Vitya marked points XX and YY inside it, from which all sides of the quadrilateral are visible at equal angles. What is the distance between points XX and YY? (From point XX, side ABAB is visible at angle AXBAXB.)
pЗ. Several identical black squares, perhaps partially overlapping, were placed on a white plane. The result was a black polygonal figure, possibly with holes or from several pieces. Could it be that this figure does not have a single right angle?
p4. The bus ticket number consists of six digits (the first digits may be zeros). A ticket is called lucky if the sum of the first three digits is equal to the sum of the last three. Prove that the sum of the numbers of all lucky tickets is divisible by 1313.
p5. The Meandrovka River, which has many bends, crosses a straight highway under thirteen bridges. Prove that there are two neighboring bridges along both the highway and the river. (Bridges are called river neighbors if there are no other bridges between them on the river section; bridges are called highway neighbors if there are no other bridges between them on the highway section.)
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