1990 IMO Longlists

Part of IMO Longlists

Subcontests

(73)

1

Prove that AB = CD = EF - ILL 1990 USA4

In convex hexagon

ABCDEF

\angle BCA = \angle DEC = \angle AFB = \angle CBD = \angle EDF.

AB = CD = EF.

1

The projection of T on OG - ILL 1990 LUX1

AB, AC

are two chords of the circle centered at

O

. The diameter, which is perpendicular to

BC

AB, AC

F, G

F

is in the circle). The tangent from

G

tangents the circle at

T

F

is the projection of

T

OG.

1

Find all real numbers c - ILL 1990 CZS5

For any permutation

p

\{1, 2, \ldots, n\}

d(p) = |p(1) - 1| + |p(2) - 2| + \ldots + |p(n) - n|

i(p)

the number of integer pairs

(i, j)

p

1 \leqq < j \leq n

p(i) > p(j)

. Find all the real numbers

c

, such that the inequality

i(p) \leq c \cdot d(p)

holds for any positive integer

n

and any permutation

p.

1

n variables inequality - ILL 1990 YUG4

\{ a_1, a_2, \ldots, a_n\} = \{1, 2, \ldots, n\}

\frac 12 +\frac 23 +\cdots+\frac{n-1}{n} \leq \frac{a_1}{a_2} + \frac{a_2}{a_3} +\cdots+\frac{a_{n-1}}{a_n}.

1

Inequality on the side lengths of a triangle - ILL 1990 YUG3

a, b, c

be the side lengths and

P

be area of a triangle, respectively. Prove that

(a^2+b^2+c^2-4\sqrt 3 P) (a^2+b^2+c^2) \geq 2 \left(a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2\right).

1

Find the minimal value of the function - ILL 1990 VIE1

Find the minimal value of the function

\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}

1

Prove that S(ABC) is no larger than 9/2

In coordinate plane, we call a point

(x, y)

"lattice point" if both

x

y

are integers. Knowing that the vertices of triangle

ABC

are all lattice points, and there exists exactly one lattice point interior to triangle

ABC

(there might exist lattice points on the sides of

ABC

). Prove that the area of triangle

ABC

is no larger than

\frac 92.

1

Find the location when Sigma is maximal - ILL 1990 USS4

The side-lengths of two equilaterals

ABC

KLM

1

1/4

, respectively. And triangle

KLM

located inside triangle

ABC

\Sigma

the sum of the distances from

A

KL, LM

MK

. Find the location of triangle

KLM

\Sigma

1

Unit squares with the same color - ILL 1990 USS2

10 \times 10

chessboard colored as black-and-white alternately. Prove that for any

46

unit squares without common edges, there are at least

30

unit squares with the same color.

1

Parallelogram and fibonacci sequence - ILL 1990 USA1

n > 1

and real number

t \geq 1

P

is a parallelogram with four vertices

(0, 0), (0, t), (tF_{2n+1}, tF_{2n}), (tF_{2n+1}, tF_{2n} + t)

{F_n}

n

-th term of Fibonacci sequence defined by

F_0 = 0, F_1 = 1

F_{m+1} = F_m + F_{m-1}

L

be the number of integral points (whose coordinates are integers) interior to

P

M

P

t^2F_{2n+1}.

i) Prove that for any integral point

(a, b)

, there exists a unique pair of integers

(j, k)

j(F_{n+1}, F_n) + k(F_n, F_{n-1}) = (a, b)

jF_{n+1} + kF_n = a

jF_n + kF_{n-1} = b.

ii) Using i) or not, prove that

|\sqrt L-\sqrt M| \leq \sqrt 2.

1

m distinct points - ILL 1990 TUR3

n

be a positive integer and

m = \frac{(n+1)(n+2)}{2}

. In coordinate plane, there are

n

L_1, L_2, \ldots, L_n

m

distinct points

A_1, A_2, \ldots, A_m

, satisfying the following conditions:
i) Any two lines are non-parallel.
ii) Any three lines are non-concurrent.
iii) Only

A_1

does not lies on any line

L_k

, and there are exactly

k + 1

A_j

's that lie on line

L_k

(k = 1, 2, \ldots, n).

Prove that there exist a unique polynomial

p(x, y)

n

p(A_1) = 1

p(A_j) = 0

j = 2, 3, \ldots, m.

1

Prove that OM = OK - ILL 1990 TUR2

ABCD

has an inscribed circle with center

O

AB = CD

M, K

are the midpoints of

BC, AD

respectively. Prove that

OM = OK.

1

Find the locus of the incenter - ILL 1990 TUR1

P

be a variable point on the circumference of a quarter-circle with radii

OA, OB

\angle AOB = 90^\circ

. H is the projection of

P

OA

. Find the locus of the incenter of the right-angled triangle

HPO.

1

Find number of functions - ILL 1990 THA3

n

be a positive integer.

S_1, S_2, \ldots, S_n

are pairwise non-intersecting sets, and

S_k

k

(k = 1, 2, \ldots, n)

S = S_1\cup S_2\cup\cdots \cup S_n

f: S \to S

maps all elements in

S_k

to a fixed element of

S_k

k = 1, 2, \ldots, n

. Find the number of functions

g: S \to S

f(g(f(x))) = f(x).

1

The number is divisiblr by 112 for every m - ILL 1990 THA1

m

be an positive odd integer not divisible by

3

\left[4^m -(2+\sqrt 2)^m\right]

is divisible by

112.

1

Prove that there exists real number p - ILL 1990 SWE4

f(x) = \sin x + \sin \pi x

and positive number

d

. Prove that there exists real number

p

|f(x + p) - f(x)| < d

holds for all real numbers

x

, and the value of

p

can be arbitrarily large.

1

there exists a point belonging to all the sets-ILL1990 SWE3

A_1, A_2, \ldots, A_n (n \geq 4)

n

convex sets in plane. Knowing that every three convex sets have a common point. Prove that there exists a point belonging to all the sets.

1

Exist numbers their sum is divisible by 2n - ILL 1990 SWE2

n \geq 4

a_1, a_2, \ldots, a_n \in (0, 2n)

n

distinct integers. Prove that there exists a subset of the set

\{a_1, a_2, \ldots, a_n \}

such that the sum of its elements is divisible by

2n.

1

Prove that S(ABC)=AD^2 - ILL 1990 SWE1

D

is on the hypotenuse

BC

of right-angled triangle

ABC

. The inradii of triangles

ADB

ADC

are equal. Prove that

S_{ABC} = AD^2

S

is the area function.

1

Find f(1990,31) - ILL 1990 SPA1

f(x, y): \mathbb N \times \mathbb N \to \mathbb Q

satisfies the conditions:
(i)

f(1, 1) =1

f(p + 1, q) + f(p, q + 1) = f(p, q)

p, q \in \mathbb N

qf(p + 1, q) = pf(p, q + 1)

p, q \in \mathbb N

f(1990, 31).

1

At least two non-congruence quadrilaterals - ILL 1990 ROM2

Prove that there exist at least two non-congruent quadrilaterals, both having a circumcircle, such that they have equal perimeters and areas.

1

Subset in the coordinate plane - ILL 1990 KOR3

L

be a subset in the coordinate plane defined by

L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}

\mathbb Z

is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area

1990

, there exist at least two points of

L

1

Three variables inequality - ILL 1990 ROM3

a, b, c \in \mathbb R

(a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2) \geq (ab + bc + ca)^3.

When does the equality hold?

1

Nice functional equation - ILL 1990 KOR2

\mathbb Q

be the set of all rational numbers and

\mathbb R

be the set of real numbers. Function

f: \mathbb Q \to \mathbb R

satisfies the following conditions:
(i)

f(0) = 0

, and for any nonzero

a \in Q, f(a) > 0.

f(x + y) = f(x)f(y) \qquad \forall x,y \in \mathbb Q.

f(x + y) \leq \max\{f(x), f(y)\} \qquad \forall x,y \in \mathbb Q , x,y \neq 0.

x

be an integer and

f(x) \neq 1

f(1 + x + x^2+ \cdots + x^n) = 1

for any positive integer

n.

1

There exists a pair of indexes - ILL 1990 KOR1

n \geq 5

be a positive integer.

a_1, b_1, a_2, b_2, \ldots, a_n, b_n

( a_i, b_i)

are pairwisely distinct for

i = 1, 2, \ldots, n

|a_1b_2 - a_2b_1| = |a_2b_3 -a_3b_2| = \cdots = |a_{n-1}b_n -a_nb_{n-1}| = 1

. Prove that there exists a pair of indexes

i, j

2 \leq |i - j| \leq n - 2

|a_ib_j -a_jb_i| = 1.

1

n-dimensional space - ILL 1990 PRK5

P = (p_1, p_2, \ldots, p_n)

n

-dimensional space . Find point

X = (x_1, x_2, \ldots, x_n)

x_1 \leq x_2 \leq\cdots \leq x_n

\sqrt{(x_1-p_1)^2 + (x_2-p_2)^2+\cdots+(x_n-p_n)^2}

1

Set of Cuboids - ILL 1990 PRK3

Consider the set of cuboids: the three edges

a, b, c

from a common vertex satisfy the condition

\frac ab = \frac{a^2}{c^5}

(i) Prove that there are

100

pairs of cuboids in this set with equal volumes in each pair.
(ii) For each pair of the above cuboids, find the ratio of the sum of their edges.

1

Apolonius Theorem - ILL 1990 PRK4

BC

M

BC

A

is a point such that

A, B, C

are non-collinear.
(i) Prove that if

M

is the midpoint of

BC

AB^2 + AC^2 = 2(AM^2 + BM^2).

(ii) If there exists another point (except

M

BC

satisfying (i), find the region of point

A

1

Variable point M - ILL 1990 PRK2

In coordinate plane, a variable point

M

, starting from the origin

O(0, 0)

, moves on the line

l

k

k

is an irrational number.
(i) Prove that point

O(0, 0)

is the only rational point (namely, the coordinates of which are both rationals) on the line

l.

(ii) Prove that for any number

\varepsilon > 0

, there exist integers

m, n

such that the distance between

l

(m, n)

\varepsilon.

1

Find the range of r - ILL 1990 PRK1

a + bi

c + di

be two roots of the equation

x^n = 1990

n \geq 3

is an integer and

a,b,c,d \in \mathbb R

. Under the linear transformation

f =\left(\begin{array}{cc}a&c\\b &d\end{array}\right)

(2, 1) \to (1, 2)

r

to be the distance from the image of

(2, 2)

to the origin. Find the range of

r.

1

Find all continuous functions - ILL 1990 POL4

Find all the continuous bounded functions

f: \mathbb R \to \mathbb R

(f(x))^2 -(f(y))^2 = f(x + y)f(x - y) \text{ for all } x, y \in \mathbb R.

1

Number of functions that have path - ILL 1990 POL2

m

M

k

-element subset

K \subset M

. We call a function

f: K \to M

has "path", if there exists an element

x_0 \in K

f(x_0) = x_0

, or there exists a chain

x_0, x_1, \ldots, x_j = x_0 \in K

_xi = f(x_{i-1})

i = 1, 2, \ldots, j

. Find the number of functions

f: K \to M

which have path.

1

We can fill in the three dimensional space - ILL 1990 NET4

Prove that we can fill in the three dimensional space with regular tetrahedrons and regular octahedrons, all of which have the same edge-lengths. Also find the ratio of the number of the regular tetrahedrons used and the number of the regular octahedrons used.

1

Inequality on eight variables - ILL 1990 NET2

Given eight real numbers

a_1 \leq a_2 \leq \cdots \leq a_7 \leq a_8

x = \frac{ a_1 + a_2 + \cdots + a_7 + a_8}{8}

y = \frac{ a_1^2 + a_2^2 + \cdots + a_7^2 + a_8^2}{8}

2 \sqrt{y-x^2} \leq a_8 - a_1 \leq 4 \sqrt{y-x^2}.

1

Find the greatest common divisor - ILL 1990 MOR2

\{u_n\}

u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3

. Prove that for any positive integers

n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}

. Also find the greatest common divisor of

u_n

u_{n+3}.

1

Find the sums - ILL 1990 MOR1

For positive integers

n, p

n \geq p

, define real number

K_{n, p}

K_{n, 0} = \frac{1}{n+1}

K_{n, p} = K_{n-1, p-1} -K_{n, p-1}

1 \leq p \leq n.

S_n = \sum_{p=0}^n K_{n,p} , \ n = 0, 1, 2, \ldots

\lim_{n \to \infty} S_n.

T_n = \sum_{p=0}^n (-1)^p K_{n,p} , \ n = 0, 1, 2, \ldots

1

Bijection and mapping - ILL 1990 MON3

M = \{1, 2, \ldots, n\}

\phi : M \to M

be a bijection.
(i) Prove that there exist bijections

\phi_1, \phi_2 : M \to M

\phi_1 \cdot \phi_2 = \phi , \phi_1^2 =\phi_2^2=E

E

is the identity mapping.
(ii) Prove that the conclusion in (i) is also true if

M

is the set of all positive integers.

1

Construct triangle ABC - ILL 1990 MON4

A, M, M_1

and rational number

\lambda \neq -1

. Construct the triangle

ABC

M

BC

M_1

B_1C_1

B_1, C_1

are the projections of

B, C

AC, AB

respectively), and

\frac{BM}{MC}=\frac{B_1M_1}{M_1C_1}=\lambda.

1

Real solutions of the system of equations - ILL 1990 MON2

Find the real solution(s) for the system of equations

\begin{cases}x^3+y^3 &=1\\x^5+y^5 &=1\end{cases}

1

Inequality over the sums - ILL 1990 MON1

Let real numbers

a_1, a_2, \ldots, a_n

0 < a_i \leq a, \ i = 1, 2, \ldots, n

. Prove that
(i) If

n = 4

\frac 1a \sum_{i=1}^4 a_i - \frac{a_1a_2 + a_2a_3 + a_3 a_4 + a_4 a_1}{a^2} \leq 2.

n = 6

\frac 1a \sum_{i=1}^6 a_i - \frac{a_1a_2 + a_2a_3 + \cdots + a_5 a_6 + a_6 a_1}{a^2} \leq 3.$$

1

Find the value of n - ILL 1990 MEX1

During the class interval,

n

children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...
Find the value of

n

for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).

1

Prove that the number is irrational - ILL 1990 JAP3

\sqrt 2 +\sqrt 3 +\sqrt{1990}

1

Find the least number of doors - ILL 1990 ITA4

The tourist on an island can play the "getting treasure" game. He has to open a series of doors, each door is colored with one of n colors, according to the following rules:
(i) The tourist has n keys, each key with a different color.
(ii) Once a key is used, it is not permitted to change until it is destroyed.
(iii) Each key can open any door, and keeps intact when it opens the door having different color with it, but is destroyed when it opens the door having the same color with it.
Find the least number of doors to ensure that no tourist, no matter how he choose the order of the keys to use, can get the treasure.

1

Number of odd integers among binomial coefficients

Prove that for any positive integer

n

, the number of odd integers among the binomial coefficients

\binom nh \ ( 0 \leq h \leq n)

is a power of 2.

1

Geometric Inequality - ILL 1990 ITA2

V

be a finite set of points in three-dimensional space. Let

S_1, S_2, S_3

be the sets consisting of the orthogonal projections of the points of

V

yz

zx

xy

-plane, respectively. Prove that

| V|^2 \leq | S1|\cdot|S2|\cdot |S3|

| A|

denotes the number of elements in the finite set

A.

1

Find n points such that the sum is maximal - ILL 1990 ITA1

n

p_1, p_2, \ldots, p_n

on the circumference of a unit circle, such that

\sum_{1\leq i< j \leq n} p_i p_j

1

Calculate the sum for every natural n - ILL 1990 ISR2

n

be an arbitrary positive integer. Calculate

S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n}.

1

Letter Keys - ILL 1990 ISR1

Given three letters

X, Y, Z

, we can construct letter sequences arbitrarily, such as

XZ, ZZYXYY, XXYZX

, etc. For any given sequence, we can perform following operations:

T_1

: If the right-most letter is

Y

, then we can add

YZ

after it, for example,

T_1(XYZXXY) = (XYZXXYYZ).

T_2

: If The sequence contains

YYY

, we can replace them by

Z

T_2(XXYYZYYYX) = (XXYYZZX).

T_3

: We can replace

Xp

p

is any sub-sequence) by

XpX

T_3(XXYZ) = (XXYZX).

T_4

: In a sequence containing one or more

Z

, we can replace the first

Z

XY

T_4(XXYYZZX) = (XXYYXYZX).

T_5

: We can replace any of

XX, YY, ZZ

X

T_5(ZZYXYY) = (XYXX)

(XYXYY)

(ZZYXX).

Using above operations, can we get

XYZZ

XYZ \ ?

1

Prove that there exist p_i, q_i, r_i - ILL 1990 IRE4

a, b, c

be integers. Prove that there exist integers

p_1, q_1, r_1, p_2, q_2

r_2

a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1

c = p_1q_2 - p_2q_1.

1

Binary Operation(star) - ILL 1990 IRE3

\alpha

be the positive root of the quadratic equation

x^2 = 1990x + 1

m, n \in \mathbb N

, define the operation

m*n = mn + [\alpha m][ \alpha n]

[x]

is the largest integer no larger than

x

(p*q)*r = p*(q*r)

p, q, r \in \mathbb N.

1

Sum of series - ILL 1990 Iran 4

|x| < 1

\frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots

1

Circle exactly passes through three points - ILL 1990 Iran 3

n

non-coplanar points in space. Prove that there exists a circle exactly passes through three points of them.

1

Can a parallelepiped be constructed ? - ILL 1990 Iran1

Using following five figures, can a parallelepiped be constructed, whose side lengths are all integers larger than

1

1990

? (In the figure, every square represents a unit cube.)

\text{Squares are the same and all are } \Huge{1 \times 1}

[asy] import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,4)--(0,4),linewidth(2pt)); draw((0,4)--(0,0),linewidth(2pt)); draw((0,0)--(2,0),linewidth(2pt)); draw((2,0)--(2,1),linewidth(2pt)); draw((2,1)--(0,1),linewidth(2pt)); draw((1,0)--(1,4),linewidth(2pt)); draw((2,4)--(2,3),linewidth(2pt)); draw((2,3)--(0,3),linewidth(2pt)); draw((0,2)--(1,2),linewidth(2pt)); label("(1)", (0.56,-1.54), SE*lsf); draw((4,2)--(4,1),linewidth(2pt)); draw((7,2)--(7,1),linewidth(2pt)); draw((4,2)--(7,2),linewidth(2pt)); draw((4,1)--(7,1),linewidth(2pt)); draw((6,0)--(6,3),linewidth(2pt)); draw((5,3)--(5,0),linewidth(2pt)); draw((5,0)--(6,0),linewidth(2pt)); draw((5,3)--(6,3),linewidth(2pt)); label("(2)", (5.13,-1.46), SE*lsf); draw((9,0)--(9,3),linewidth(2pt)); draw((10,3)--(10,0),linewidth(2pt)); draw((12,3)--(12,0),linewidth(2pt)); draw((11,0)--(11,3),linewidth(2pt)); draw((9,2)--(12,2),linewidth(2pt)); draw((12,1)--(9,1),linewidth(2pt)); draw((9,3)--(10,3),linewidth(2pt)); draw((11,3)--(12,3),linewidth(2pt)); draw((12,0)--(11,0),linewidth(2pt)); draw((9,0)--(10,0),linewidth(2pt)); label("(3)", (10.08,-1.48), SE*lsf); draw((14,1)--(17,1),linewidth(2pt)); draw((15,2)--(17,2),linewidth(2pt)); draw((15,2)--(15,0),linewidth(2pt)); draw((15,0)--(14,0)); draw((14,1)--(14,0),linewidth(2pt)); draw((16,2)--(16,0),linewidth(2pt)); label("(4)", (15.22,-1.5), SE*lsf); draw((14,0)--(16,0),linewidth(2pt)); draw((17,2)--(17,1),linewidth(2pt)); draw((19,3)--(19,0),linewidth(2pt)); draw((20,3)--(20,0),linewidth(2pt)); draw((20,3)--(19,3),linewidth(2pt)); draw((19,2)--(20,2),linewidth(2pt)); draw((19,1)--(20,1),linewidth(2pt)); draw((20,0)--(19,0),linewidth(2pt)); label("(5)", (19.11,-1.5), SE*lsf); dot((0,0),ds); dot((0,1),ds); dot((0,2),ds); dot((0,3),ds); dot((0,4),ds); dot((1,4),ds); dot((2,4),ds); dot((2,3),ds); dot((1,3),ds); dot((1,2),ds); dot((1,1),ds); dot((2,1),ds); dot((2,0),ds); dot((1,0),ds); dot((5,0),ds); dot((6,0),ds); dot((5,1),ds); dot((6,1),ds); dot((5,2),ds); dot((6,2),ds); dot((5,3),ds); dot((6,3),ds); dot((7,2),ds); dot((7,1),ds); dot((4,1),ds); dot((4,2),ds); dot((9,0),ds); dot((9,1),ds); dot((9,2),ds); dot((9,3),ds); dot((10,0),ds); dot((11,0),ds); dot((12,0),ds); dot((10,1),ds); dot((10,2),ds); dot((10,3),ds); dot((11,1),ds); dot((11,2),ds); dot((11,3),ds); dot((12,1),ds); dot((12,2),ds); dot((12,3),ds); dot((14,0),ds); dot((15,0),ds); dot((16,0),ds); dot((15,1),ds); dot((14,1),ds); dot((16,1),ds); dot((15,2),ds); dot((16,2),ds); dot((17,2),ds); dot((17,1),ds); dot((19,0),ds); dot((20,0),ds); dot((19,1),ds); dot((20,1),ds); dot((19,2),ds); dot((20,2),ds); dot((19,3),ds); dot((20,3),ds); clip((-0.41,-10.15)--(-0.41,8.08)--(21.25,8.08)--(21.25,-10.15)--cycle); [/asy]

1

Find the number of good subsets of S - ILL 1990 IND4

S = \{1, 2, \ldots, 1990\}

31

-element subset of

S

is called "good" if the sum of its elements is divisible by

5

. Find the number of good subsets of

S.

1

Find the maximum value of f(n) - ILL 1990 IND2

f(n), n \in \mathbb N

, is defined as follows: Let

\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}

A(n), B(n)

are coprime positive integers; if

B(n) = 1

f(n) = 1

B(n) \neq 1

f(n)

is the largest prime factor of

B(n)

. Prove that the values of

f(n)

are finite, and find the maximum value of

f(n).

1

Find and construct the locus - ILL 1990 IND1

ABC

be an arbitrary acute triangle. Circle

\Gamma

satisfies the following conditions: (i) Circle

\Gamma

intersects all three sides of triangle

ABC.

(ii) In the convex hexagon formed by above six intersections, the three pairs of opposite sides are parallel respectively. (The hexagon maybe degenerate, that is, two or more vertices are coincide. In this case, "opposite sides are parallel" is defined through limit opinion.) Find the locus of the center of circle

\Gamma

, and explain how to construct the locus.

1

There exist infinitely many positive integers - ILL1990 ICE1

Prove that there exist infinitely many positive integers

n

such that the number

\frac{1^2+2^2+\cdots+n^2}{n}

is a perfect square. Obviously,

1

is the least integer having this property. Find the next two least integers having this property.

1

Find t s.t. system of equations have solution - ILL1990 HUN2

Find the real number

t

, such that the following system of equations has a unique real solution

(x, y, z, v)

:

\left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right.

1

Area inequality - ILL 1990 GRE1

O

is interior to triangle

ABC

O

, draw three lines

DE \parallel BC, FG \parallel CA

HI \parallel AB

D, G

AB

I, F

BC

E, H

CA

S_1

the area of hexagon

DGHEFI

S_2

the area of triangle

ABC

S_1 \geq \frac 23 S_2.

1

Union of disjoint circumferences - ILL 1990 FRG3

Could the three-dimensional space be expressed as the union of disjoint circumferences?

1

Find the least n - ILL 1990 FRG1

Find, with proof, the least positive integer

n

having the following property: in the binary representation of

\frac 1n

, all the binary representations of

1, 2, \ldots, 1990

(each consist of consecutive digits) are appeared after the decimal point.

1

There exist four mathematicians - ILL 1990 FRA3

1990 mathematicians attend a meeting, every mathematician has at least 1327 friends (the relation of friend is reciprocal). Prove that there exist four mathematicians among them such that any two of them are friends.

1

Infinitely numbers with property P - ILL 1990 FRA2

We call an integer

k \geq 1

having property

P

, if there exists at least one integer

m \geq 1

which cannot be expressed in the form

m = \varepsilon_1 z_1^k + \varepsilon_2 z_2^k + \cdots + \varepsilon_{2k} z_{2k}^k

z_i

are nonnegative integer and

\varepsilon _i = 1

-1

i = 1, 2, \ldots, 2k

. Prove that there are infinitely many integers

k

having the property

P.

1

Determine all superinvariant sets - ILL 1990 FIN2

S

on the real line

R

"superinvariant", if for any stretching

A

S

by the transformation taking

x

A(x) = x_0 + a(x - x_0)

a > 0

, there exists a transformation

B, B(x) = x + b

, such that the images of

S

A

B

agree; i.e., for any

x \in S

y \in S

A(x) = B(y)

t \in S

u \in S

B(t) = A(u).

Determine all superinvariant sets.

1

Determine the probability - ILL 1990 FIN1

A, B, C, D, E

F

are located on the vertices of a regular hexagon in that order.

G

is the center of the hexagon. The sides of the hexagon are the roads connecting these cities. Further more, there are roads connecting cities

B, C, E, F

G

, respectively. Because of raining, one or more roads maybe destroyed. The probability of the road keeping undestroyed between two consecutive cities is

p

. Determine the probability of the road between cities

A

D

is undestroyed.

1

A group of mathematicians - ILL 1990 CZS 4

In a group of mathematicians, every mathematician has some friends (the relation of friend is reciprocal). Prove that there exists a mathematician, such that the average of the number of friends of all his friends is no less than the average of the number of friends of all these mathematicians.

1

k boxes with the same number of balls - ILL 1990 CZS3

p, k

x

be positive integers such that

p \geq k

x < \left[ \frac{p(p-k+1)}{2(k-1)} \right]

[q]

is the largest integer no larger than

q

. Prove that when

x

balls are put into

p

boxes arbitrarily, there exist

k

boxes with the same number of balls.

2

At least sqrt(n) distinct values - ILL 1990 CUB1

A

B

are two points in the plane

\alpha

r

passes through points

A, B

n

distinct points

P_1, P_2, \ldots, P_n

in one of the half-plane divided by line

r

. Prove that there are at least

\sqrt n

distinct values among the distances

AP_1, AP_2, \ldots, AP_n, BP_1, BP_2, \ldots, BP_n.

S is the incenter of triangle ABC

S

be the incenter of triangle

ABC

A_1, B_1, C_1

are the intersections of

AS, BS, CS

with the circumcircle of triangle

ABC

respectively. Prove that

SA_1 + SB_1 + SC_1 \geq SA + SB + SC.

2

Show that the fraction isn't an integer - ILL 1990 COL2

f : \mathbb Z_{\geq 0}^0 \to \mathbb N

satisfy the following conditions:
(i)

f(0, 0, 0) = 1;

f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);

(iii) when applying above relation iteratively, if any of

x', y', z

' is negative, then

f(x', y', z') = 0.

x, y, z

are the side lengths of a triangle, then

\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)}

is not an integer for any integers

k, m > 1.

Geometric Inequality on the lengths - ILL 1990 YUG1

S, T

be the circumcenter and centroid of triangle

ABC

, respectively.

M

is a point in the plane of triangle

ABC

90^\circ \leq \angle SMT < 180^\circ

A_1, B_1, C_1

are the intersections of

AM, BM, CM

with the circumcircle of triangle

ABC

respectively. Prove that

MA_1 + MB_1 + MC_1 \geq MA + MB + MC.

2

Find the minimal value of b - ILL 1990 COL1

Given the condition that there exist exactly

1990

ABC

with integral side-lengths satisfying the following conditions: (i)

\angle ABC =\frac 12 \angle BAC;

AC = b.

Find the minimal value of

b.

Fractional Inequality - ILL 1990 VIE2

x,y,z

be positive reals and

x \geq y \geq z

\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y} \geq x^2+y^2+z^2

1

ABC and A1B1C1 are congruent

ABC, O

is the circumcenter,

H

is the orthocenter. Construct the circumcircles of triangles

CHB, CHA

AHB

, and let their centers be

A_1, B_1, C_1

, respectively. Prove that triangles

ABC

A_1B_1C_1

are congruent, and their nine-point circles coincide.

1

Find P such that area of triangle is as large as possible

P

inside the triangle

ABC

A(P), B(P)

C(P)

be the points of intersection of the lines

AP, BP

CP

with the sides opposite to

A, B

C

, respectively. Determine

P

in such a way that the area of the triangle

A(P)B(P)C(P)

is as large as possible.

1

|p|<=800

Let S be a 1990-element set and P be a set of 100-ary sequences

(a_1,a_2,...,a_{100})

a_i's

are distinct elements of S.An ordered pair (x,y) of elements of S is said to appear in

(a_1,a_2,...,a_{100})

x=a_i

y=a_j

for some i,j with

1\leq i<j\leq 100

.Assume that every ordered pair (x,y) of elements of S appears in at most one member in P.Show that

|P|\leq 800