MathDB

III Soros Olympiad 1996 - 97 (Russia)

Part of Soros Olympiad in Mathematics

Subcontests

(30)
11.10
1
11.9
1
10.9
1
10.10
1
10.8
1
9.10
1
9.9
1
11.3
3

y=x^4+a , x=\1/y^4+b (III Soros Olympiad 1996-97 R1 11.3)

Find the greatest aa for which there is bb such that the system {y=x4+ax=1y4+b\begin{cases} y=x^4+a \\ x=\dfrac{1}{y^4}+b \end{cases} has exactly two solutions.

root of x^3- x- 3 = 0 vs \sqrt[5]{13} (III Soros Olympiad 1996-97 R2 11.3)

Prove that the equation x^3- x- 3 = 0 has a unique root. Which is greater, the root of this equation or 135\sqrt[5]{13}? (Use of a calculator is prohibited.)

2 circles tangent to an arc and a line

A chord ABAB is drawn in a certain circle. The smaller of the two arcs ABAB corresponds to a central angle of 120o120^o. A tangent pp to this arc is drawn. Two circles with radii RR and rr are constructed, touching this smaller arc ABAB and straight lines ABAB and pp. Find the radius of the original circle.
11.8
2
11.7
2
11.6
3

max no of obtuse triangles by 16 segments (III Soros Olympiad 1996-97 R1 11.6)

What is the largest number of obtuse triangles that can be composed of 1616 different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?

199 corrupted ministers

In one criminal kingdom, an underdeveloped state, the King decided to start a fight against corruption and, as an example, punish one of his 199199 ministers. The ministers were summoned to the palace and seated at a large round table. At first they wanted to find the one who had the most money in his bank account and declare him the main corrupt official. It takes 2020 minutes to determine the amount of money in the bank account of one minister. But the King ordered that the accused be found within four hours while he underwent medical procedures. According to the Noble Court Administrator, any minister can be accused, you just need to find a legal justification.The Chief Lawyer proposed that the first minister discovered, who has more money in his bank account than each of his two neighbors (one on the right and one on the left), be declared corrupt. How can one be sure to find a minister who meets this condition within the allotted 44 hours? (During this time, it is possible to consistently determine the size of the bank accounts of no more than 1212 ministers. It is assumed that the amount of money in bank accounts is different.)

log_{x+y} (x^2+y^2)<= 1 (III Soros Olympiad 1996-97 R3 11.6)

On the coordinate plane, draw a set of points M(x,y)M(x,y), the coordinates of which satisfy the inequality log⁡x+y(x2+y2)≤1.\log_{x+y} (x^2+y^2) \le 1.
11.5
3

parallelogram by 4 similar isosceles triangles

The area of a convex quadrilateral is SS, and the angle between the diagonals is aa. On the sides of this quadrilateral, as on the bases, isosceles triangles with vertex angle equal to ϕ\phi, wherein two opposite triangles are located on the other side of the corresponding side of the quadrilateral than the quadrilateral itself, and the other two are located on the other side. Prove that the vertices of the constructed triangles, different from the vertices of the quadrilateral, serve as the vertices of a parallelogram. Find the area of this parallelogram.

folding a paper triangle to get surface of a regular unit tetradragon

Prove that this triangle cut out of paper can be folded so that the surface of a regular unit tetradragon (i.e., a triangular pyramid, all edges of which are equal to 11) is obtained if: a) this triangle is isosceles, the lateral sides are equal to 22 , the angle between them is 120o120^o, b) two sides of this triangle are equal to 22 and 232\sqrt3, the angle between them is 150o150^o.

plane tangent to insphere of parallelepiped

All faces of the parallelepiped ABCDA1B1C1D1ABCDA_1B_1C_1D_1 are equal rhombuses. Plane angles at vertex AA are equal. Points KK and MM are taken on the edges A1B1A_1B_1 and A1D1A_1D_1. It is known that A1K=aA_1K = a, A1M=bA_1M = b, anda+b a + b is an edge of the parallelepiped. Prove that the plane AKMAKM touches the sphere inscribed in the parallelepiped. Let us denote by QQ the touchpoint of this sphere with the plane AKMAKM . In what ratio does the straight line AQAQ divide the segment KMKM?
11.4
3

no of parts 3=4 half planes divide space (III Soros Olympiad 1996-97 R1 11.4)

How many parts can space be divided into by :
a) three half-plane?
b) four half-planes?

four circles , tangencies

There are four circles. The chordAB AB is drawn in the first one, and the distance from the midpoint of the smaller of the two formed arcs to ABAB is equal to 11. The second, third and fourth circles are located inside the larger segment and touch the chord ABAB. The second and fourth circles touch internally the first and externally the third. The sum of the radii of the last three circles is equal to the radius of the first circle. Find the radius of the third circle if it is known that the line passing through the centers of the first and third circles is not parallel to the line passing through the centers of the other two circles.

min y = cos8x + 3cos4x +3cos2x + 2cosx (III Soros Olympiad 1996-97 R3 11.4)

Find the smallest value of a function y=cos⁡8x+3cos⁡4x+3cos⁡2x+2cos⁡x.y = \cos 8x + 3\cos 4x +3\cos2x + 2\cos x.
11.2
3
11.1
3
10.7
2

equilateral ABC, B_1C_1 = AM, C_1A_1 = BM, A_1B_1 = CM,

An arbitrary point MM is taken inside a regular triangle ABCABC. Prove, that on sides ABAB, BCBC and CACA one can choose points C1C_1, A1A_1 and B1B_1, respectively, so that B1C1=AMB_1C_1 = AM, C1A1=BMC_1A_1 = BM, A1B1=CMA_1B_1 = CM. Find BABA if AB1=aAB_1= a, AC1=bAC_1 = b, a>ba>b.

never too late for another locus

Let AA be a fixed point on a circle, BB andC C be arbitrary points on the circle different from AA and at different distances. The bisector of the angle ∠BAC\angle BAC intersects the chord BCBC and the circle at points KK and PP, DD is the projection of AA onto the straight line BCBC. A circle passing through points KK, PP and DD intersects the straight line ADAD for the second time at point MM. Find the locus of points MM.
10.6
3

2 parallel lines and a rhombus

There is a rhombus with acute angle bb and side aa. Two parallel lines, the distance between which is equal to the height of the rhombus, intersect all four sides of the rhombus. What can be the sum of the perimeters of two triangles cut off from a rhombus by straight lines? (These two triangles lie outside the strip between parallel lines.)

76 numbere cards in a circle

There are 7676 cards with different numbers written on them. These cards are laid out on the table in a circle, number down. Try to find some three cards in a row such that the number written on the middle of these three cards is greater than on each of the two neighboring ones. You can turn over no more than 1010 cards in succession. How should one proceed to be sure to find three cardboard boxes for which the specified condition is met?

|y-2x|=x <=> |mx + ny| = y (III Soros Olympiad 1996-97 R3 10.6)

Find mm and nn such that the set of points whose coordinates xx and yy satisfy the equation ∣y−2x∣=x|y-2x|=x, coincides with the set of points specified by the equation ∣mx+ny∣=y|mx + ny| = y.
10.4
3

MR bisects <AMD, convex ABCD with equal diagonals

The diagonals of a convex quadrilateral ABCDABCD are equal to each other and intersect at point MM. Points KK and LL are taken on ABAB and CDCD, respectively, so that AKKB=DLLC\frac{AK}{KB}=\frac{DL}{LC}. Lines ABAB and KDKD intersect at point PP. Prove that MPMP is the bisector of angle AMDAMD.

2x2 radical system (III Soros Olympiad 1996-97 R2 10.4)

Solve the system of equations {y2+x4x+yy2+x=y244xy2+xx+x−y−1=(y+1)(x−x−y−1)\begin{cases} \sqrt{\dfrac{y^2+x}{4x}}+\dfrac{y}{\sqrt{y^2+x}}=\dfrac{y^2}{4}\sqrt{\dfrac{4x}{y^2+x}} \\ \sqrt{x}+ \sqrt{x-y-1}=(y+1)(\sqrt{x}- \sqrt{x-y-1}) \end{cases}

2x2 diophantine: ab+cd=34, ac-bd=19 (III Soros Olympiad 1996-97 R3 10.4)

Find natural a,b,c,da, b, c, d that satisfy the system {ab+cd=34ac−bd=19\begin{cases} ab+cd=34 \\ ac-bd=19 \end{cases}
10.5
3

\frac{x+y}{1+xy}=\frac{1-2y}{2-y} (III Soros Olympiad 1996-97 R1 10.5)

Solve the system of equations {x+y1+xy=1−2y2−yx−y1−xy=1−3x3−x\begin{cases} \dfrac{x+y}{1+xy}=\dfrac{1-2y}{2-y} \\ \dfrac{x-y}{1-xy}=\dfrac{1-3x}{3-x} \end{cases}

circumcenter constrcuction using at most 5 circles, given more circle

A circle is drawn on a plane, the center of which is not indicated. On this circle, point AA is marked and a second circle with center at AA is constructed. The second circle has a radius greater than the radius of the first and intersects the first at two points. Construct the center of the first circle using only a compass, drawing no more than five more circles.

area of quad. wanted, starting with intersecting circles

Two circles intersect at two points AA and BB. The radii of these circles are equal to RR and rr, respectively; the angle between the radii going to the points of intersection is equal to aa. A chord KMKM of length bb is taken in a circle of radius rr. Straight lines KAKA, KBKB, MAMA and MBMB intersect the other circle for second time at four points. Find the area of the quadrilateral with vertices at these points.
10.3
3

2 geometric progressions (III Soros Olympiad 1996-97 R1 10.3)

An infinite sequence of numbers a,b,c,d,...a, b, c, d,... is obtained by term-by-term addition of two geometric progressions. Can this sequence begin with the following numbers:. a) 1,1,3,51,1,3,5 ? b) 1,2,3,51,2,3,5 ? c) 1,2,3,41,2,3, 4 ?

cos<AOB by two tangents in trinomial (III Soros Olympiad 1996-97 R2 10.3)

Let's consider the graph of a square trinomial having roots 11 and 44. Let's draw two tangents to it from point OO (the origin of coordinates), touching it at points AA and BB. What values can the cosine of angle ∠AOB\angle AOB take?

\sqrt{x(x+7)}+\sqrt{(x+7)(x+17)}+... (III Soros Olympiad 1996-97 R3 10.3)

Solve the equation x(x+7)+(x+7)(x+17)+(x+17)(x+24)=12+172\sqrt{x(x+7)}+\sqrt{(x+7)(x+17)}+\sqrt{(x+17)(x+24)}=12+17\sqrt2
10.2
3

x^3 + px^2 + q = 0, 2 of 3 are primes (III Soros Olympiad 1996-97 R1 10.2)

It is known that the equation x3+px2+q=0x^3 + px^2 + q = 0 where qq is non-zero, has three different integer roots, the absolute values of two of which are prime numbers. Find the roots of this equation.

computational geo with 4 points on the same triangle side

On a side of the triangle, take four points KK, PP, HH and MM, which are respectively the midpoint of this side, the foot of the bisector with the opposite angle of the triangle, the touchpoint of this side of the circle inscribed in the triangle and the foot of the corresponding altitude. Find KHKH if KP=aKP = a, KM=bKM =b.

cutting surface of regular triangular pyramid

Let ABCDABCD be a regular triangular pyramid with base ABCABC (this means that ABCABC is a regular triangle, and edges ADAD, BDBD and CDCD are equal) and plane angles at the opposite vertex equal to aa. A plane parallel to ABCABC intersects ADAD, BDBD and CDCD, respectively, at points A1A_1, B1B_1 and C1C_1. The surface of the polyhedron ABCA1B1C1ABCA_1B_1C_1 is cut along five edges: A1B1A_1B_1, B1C1B_1C_1, C1CC_1C, CACA and ABAB, after which this surface is turned onto a plane. At what values of aa will the resulting scan necessarily cover itself?
10.1
3
9.4
3

2x2 x^4-2x^3+x=y^2-y , y^4-2y^3+y=x^2-x (III Soros Olympiad 1996-97 R1 9.4)

Solve the system of equations {x4−2x3+x=y2−yy4−2y3+y=x2−x\begin{cases} x^4-2x^3+x=y^2-y \\ y^4-2y^3+y=x^2-x \end{cases}

x^2+ax+1996=0, x^2+1996x+a=0 (III Soros Olympiad 1996-97 R2 9.4)

At what values of aa does each of the equations x2+ax+1996=0x^2 + ax + 1996 = 0 or x2+1996x+a=0x^2 + 1996x + a = 0 have two integer roots?

computational geo with 2 circles

A chord AB=aAB = a is drawn in a circle of radius BB. A circle with center on line ABAB passes through AA and intersects this circle a second time at point CC. Let MM be an arbitrary point of the second circle. Straight lines MAMA and MCMC intersect the first circle a second time at points PP and QQ. Find PQPQ.
9.6
3

orthocenter of A_1BC_1 is the incenter of A_2BC_2

In triangle ABCABC, angle BB is not right. The circle inscribed in ABCABC touches ABAB and BCBC at points C1C_1 and A1A_1, and the feet of the altitudes drawn to the sides ABAB and BCBC are points C2C_2 and A2A_2. Prove that the intersection point of the altitudes of triangle A1BC1A_1BC_1 is the center of the circle inscribed in triangle A2BC2A_2BC_2.

collinear wanted, starting with right isosceles triangle

Let ABCABC be an isosceles right triangle with hypotenuse ABAB, DD be some point in the plane such that 2CD=AB2CD = AB and point CC inside the triangle ABDABD. We construct two rays with a start in CC, intersecting ADAD and BDBD and perpendicular to them. On the first one, intersecting ADAD, we will plot the segment CK=ADCK = AD, and on the second one - CM=BDCM = BD. Prove that points MM, DD and KK lie on the same line.

96/36 ... ? ... 97/36 (III Soros Olympiad 1996-97 R3 9.6)

Find the common fraction with the smallest positive denominator lying between the fractions 9635\frac{96}{35} and 9736\frac{97}{36} .
9.8
2

card with 10 numbered cells in a lottery (III Soros Olympiad 1996-97 R1 9.8)

Some lottery is played as follows. A lottery participant buys a card with 1010 numbered cells. He has the right to cross out any 44 of these 1010 cells. Then a drawing occurs, during which some 77 out of 1010 cells become winning. The player wins when all 44 squares he crosses out are winning. The question arises, what is the smallest number of cards that can be used so that, if filled out correctly, at least one of these cards will win in any case? We do not suggest that you answer this question (we ourselves do not know the answer), although, of course, we will be very glad if you do and will evaluate this achievement accordingly. The task is; to indicate a certain number nn and a method of filling n cards that guarantees at least one win. The smaller nn, the higher the rating of the work.

min max cos in triangle with sides 1-x (III Soros Olympiad 1996-97 R3 9.8)

The two sides of the triangle are equal to 11 and xx, and x≥1 x \ge 1. The values aa and bb are the largest and smallest angles of this triangle, respectively. Find the greatest value of cos⁡a\cos a and the smallest value of cos⁡b\cos b.
9.7
2
9.5
3

How to hang a picture on 2/3 nails (III Soros Olympiad 1996-97 R1 9.5)

How to hang a picture? What a strange question? It's simple. We take a piece of rope, attach its ends to the picture frame on the back side, then drive it into the wall. nail and throw a rope over the nail. The picture is hanging. If you pull out the nail, then, of course, it will fall. But Professor No wonder acted differently. At first, he attached the rope to the painting in the same way, only he took it a little longer. Then he hammered two nails into the wall nearby and threw a rope over these nails in a special way. The painting hangs on these nails, but if you pull out any nail, the painting will fall. Moreover, the professor claims that he can hang a painting on three nails so that the painting hangs on all three, but if any nail is pulled out, the painting will fall. You have two tasks: indicate how you can hang the picture in the right way on a) two nails; b) three nails.

shortest path, square, right isosceles (III Soros Olympiad 1996-97 R2 9.5)

An ant sits at vertex AA of unit square ABCDABCD. He needs to get to point CC, where the entrance to the anthill is located. Points AA and CC are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse BDBD. Find the length of the shortest path that an ant must overcome in order to get into the anthill.

n 7-digit numbers in geom. progression (III Soros Olympiad 1996-97 R3 9.5)

For what largest nn are there nn seven-digit numbers that are successive members of one geometric progression?
9.3
3

sets of projections of a square on lines (III Soros Olympiad 1996-97 R1 9.3)

Draw the set of projections of a square given on a plane onto all possible lines passing through a given point OO of the plane lying outside the square.

compurational geo with 2 circumcircles and angle bisector

In triangle ABCABC, sides CBCB and CACA are equal to aa and bb, respectively. The bisector of the angle ACBACB intersects the side ABAB at the point KK and the circumscribed circle of the triangle at point MM. The circumscribed circle of the triangle AMKAMK intersects for second time the straight line CACA at the point PP. Find the length of the segment APAP.

computational 3D geo , ABCD (III Soros Olympiad 1996-97 R3 9.3)

Let ABCDABCD be a three-link broken line in space, all links of which are equal and ∠BCD=90o\angle BCD=90^o. Find the distance from AA to the midpoint of BDBD, if AD=aAD = a.



9.2
3

y = x^2 + px + q, y = 2x +p are tangent (III Soros Olympiad 1996-97 R1 9.2)

It is known that the graph of a quadratic trinomial y=x2+px+qy = x^2 + px + q touches the graph of a straight line y=2x+py = 2x + p. Prove that all such quadratic trinomials have the same minimum value. Find this smallest value.

4 bells ring in different times (III Soros Olympiad 1996-97 R2 9.2)

Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, 43\frac43 seconds, 53\frac53 second and 22 seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in 11 minute? (Include first and last.)

\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a (III Soros Olympiad 1996-97 R3 9.2)

How many solutions, depending on the value of the parameter aa, has the equation x2−4+2x2−7x+5=a?\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?
9.1
3

19...96 divisible by 1997 (III Soros Olympiad 1996-97 R1 9.1)

In the middle of the number 1996 (i.e. between 1919 and 9696) you need to insert several digits so that the resulting number is divisible by 19971997. In this case, you need to get by with the least number of inserted digits.

|\sqrt[3]{5}-\sqrt3|-\sqrt3| ? 0.01 (III Soros Olympiad 1996-97 R2 9.1)

Without using a calculator, find out which number is greater: ∣53−3∣−3∣    or    0.01|\sqrt[3]{5}-\sqrt3|-\sqrt3| \,\,\,\, \text{or} \,\,\,\, 0.01

rational? sum of third radicals (III Soros Olympiad 1996-97 R3 9.1)

Is rational or irrational,the number (2253+153+93+193+63+43)×(253+103+43)?\left(\dfrac{2}{\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}}+\dfrac{1}{\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4}}\right) \times \left(\sqrt[3]{25}+\sqrt[3]{10}+\sqrt[3]{4}\right)?