MathDB

1

Triangle inequality and symmetry

ABCD

is a

4\times 4

square.

E

is the midpoint of

\left[AB\right]

.

M

is an arbitrary point on

\left[AC\right]

. How many different points

M

are there such that \left|EM\right|\plus{}\left|MB\right| is an integer?

<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ 6

35

1

max no of subsets, having none is a subset of another

What is the maximum number of subsets, having property that none of them is a subset of another, can a set with 10 elements have?

<span class='latex-bold'>(A)</span>\ 126 \qquad<span class='latex-bold'>(B)</span>\ 210 \qquad<span class='latex-bold'>(C)</span>\ 252 \qquad<span class='latex-bold'>(D)</span>\ 420 \qquad<span class='latex-bold'>(E)</span>\ 1024

34

1

an^3 + bn^2 + cn + d is integer, a is rational

Let

a,b,c,d

be rational numbers with

a>0

. If for every integer

n\ge 0

, the number an^{3} \plus{}bn^{2} \plus{}cn\plus{}d is also integer, then the minimal value of

a

will be

<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ \frac{1}{2} \qquad<span class='latex-bold'>(C)</span>\ \frac{1}{6} \qquad<span class='latex-bold'>(D)</span>\ \text{Cannot be found} \qquad<span class='latex-bold'>(E)</span>\ \text{None}

33

1

Circle and symmetry

Chord

\left[AD\right]

is perpendicular to the diameter

\left[BC\right]

of a circle. Let

E

and

F

be the midpoints of the arcs

AC

and

CD

, respectively. If AD\bigcap BE\equal{}\left\{G\right\}, AF\bigcap BC\equal{}\left\{H\right\} and m(AC)\equal{}\alpha, find the measure of angle

BHC

in terms of

\alpha

.
(A)\ 90{}^\circ \minus{}\frac{\alpha }{2} \qquad(B)\ 60{}^\circ \minus{}\frac{\alpha }{3} \qquad(C)\ \alpha \minus{}30{}^\circ \\ \qquad(D)\ 15{}^\circ \plus{}\frac{\alpha }{2} \qquad(E)\ \frac{180{}^\circ \minus{}2\alpha }{3}

32

1

f(x) + f(y) = f(x)f(y) + 1 - 1/xy

For every x,y\in \Re ^{\plus{}}, the function f: \Re ^{\plus{}} \to \Re satisfies the condition f\left(x\right)\plus{}f\left(y\right)\equal{}f\left(x\right)f\left(y\right)\plus{}1\minus{}\frac{1}{xy}. If

f\left(2\right)<1

, then

f\left(3\right)

will be

<span class='latex-bold'>(A)</span>\ 2/3 \\ <span class='latex-bold'>(B)</span>\ 4/3 \\ <span class='latex-bold'>(C)</span>\ 1 \\ <span class='latex-bold'>(D)</span>\ \text{More information needed} \\ <span class='latex-bold'>(E)</span>\ \text{There is no } f \text{ satisfying the condition above.}

31

1

Winning game strategy on chessboard

A two-player game is played on a chessboard with

m

columns and

n

rows. Each player has only one piece. At the beginning of the game, the piece of the first player is on the upper left corner, and the piece of the second player is on the lower right corner. If two squares have a common edge, we call them adjacent squares. The player having the turn moves his piece to one of the adjacent squares. The player wins if the opponent's piece is on that square, or if he manages to move his piece to the opponent's initial row. If the first move is made by the first player, for which of the below pairs of

\left(m,n\right)

there is a strategy that guarantees the second player win.

<span class='latex-bold'>(A)</span>\ (1998, 1997) \qquad<span class='latex-bold'>(B)</span>\ (1998, 1998) \qquad<span class='latex-bold'>(C)</span>\ (997, 1998) \qquad<span class='latex-bold'>(D)</span>\ (998, 1998) \qquad<span class='latex-bold'>(E)</span>\ \text{None }

30

1

(abab) + (cdcd) is a perfect square

Let m\equal{}\left(abab\right) and n\equal{}\left(cdcd\right) be four-digit numbers in decimal system. If m\plus{}n is a perfect square, what is the largest value of

a\cdot b\cdot c\cdot d

?

<span class='latex-bold'>(A)</span>\ 392 \qquad<span class='latex-bold'>(B)</span>\ 420 \qquad<span class='latex-bold'>(C)</span>\ 588 \qquad<span class='latex-bold'>(D)</span>\ 600 \qquad<span class='latex-bold'>(E)</span>\ 750

29

1

Cyclic quadrilateral with a side which is diameter

Let

ABCD

be convex quadrilateral with \angle C\equal{}\angle D\equal{}90{}^\circ. The circle

K

passing through

A

and

B

is tangent to

CD

at

C

. Let

E

be the intersection of

K

and

\left[AD\right]

. If \left|BC\right|\equal{}20, \left|AD\right|\equal{}16, then

\left|CE\right|

is

<span class='latex-bold'>(A)</span>\ 9 \qquad<span class='latex-bold'>(B)</span>\ 6\sqrt{2} \qquad<span class='latex-bold'>(C)</span>\ 4\sqrt{5} \qquad<span class='latex-bold'>(D)</span>\ 7\sqrt{2} \qquad<span class='latex-bold'>(E)</span>\ 10

28

1

sqrt( x + 4sqrt(x-4)) - sqrt( x + 2sqrt(x-1)) = 1

How many distinct real roots does the equation \sqrt{x\plus{}4\sqrt{x\minus{}4} } \minus{}\sqrt{x\plus{}2\sqrt{x\minus{}1} } \equal{}1 have?

<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 4

27

1

covering board with L-shaped pieces and at most one unit

For which of the following

n

,

n\times n

chessboard cannot be covered using at most one unit square piece and many L-shaped pieces (an L-shaped piece is a 2x2 piece with one square removed)?

<span class='latex-bold'>(A)</span>\ 96 \qquad<span class='latex-bold'>(B)</span>\ 97 \qquad<span class='latex-bold'>(C)</span>\ 98 \qquad<span class='latex-bold'>(D)</span>\ 99 \qquad<span class='latex-bold'>(E)</span>\ 100

26

1

y = sqrt( x +1988 and so on)

How many ordered integer pairs

\left(x,y\right)

are there satisfying following equation: y \equal{} \sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1997\plus{}\sqrt{x\plus{}1997\plus{}\ldots \plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}\sqrt{x} } } } } } } }.

<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 1998 \qquad<span class='latex-bold'>(D)</span>\ 3996 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}

25

1

P inside ABC

In triangle

ABC

with

\left|BC\right|>\left|BA\right|

,

D

is a point inside the triangle such that \angle ABD\equal{}\angle DBC, \angle BDC\equal{}150{}^\circ and \angle DAC\equal{}60{}^\circ. What is the measure of

\angle BAD

?

<span class='latex-bold'>(A)</span>\ 45 \qquad<span class='latex-bold'>(B)</span>\ 50 \qquad<span class='latex-bold'>(C)</span>\ 60 \qquad<span class='latex-bold'>(D)</span>\ 75 \qquad<span class='latex-bold'>(E)</span>\ 80

24

1

x^6 - 2x^4 + x^2 = A

Let

n\left(A\right)

be the number of distinct real solutions of the equation x^{6} \minus{}2x^{4} \plus{}x^{2} \equal{}A. When

A

takes every value on real numbers, the set of values of

n\left(A\right)

is

<span class='latex-bold'>(A)</span>\ \left\{0,1,2,3,4,5,6\right\} \\ <span class='latex-bold'>(B)</span>\ \left\{0,2,4,6\right\} \\ <span class='latex-bold'>(C)</span>\ \left\{0,3,4,6\right\} \\ <span class='latex-bold'>(D)</span>\ \left\{0,2,3,4,6\right\} \\ <span class='latex-bold'>(E)</span>\ \left\{0,2,3,4\right\}

23

1

A game on nxn board

Ahmet and Betül play a game on

n\times n

\left(n\ge 7\right)

board. Ahmet places his only piece on one of the

n^{2}

squares. Then Betül places her two pieces on two of the squares at the border of the board. If two squares have a common edge, we call them adjacent squares. When it is Ahmet's turn, Ahmet moves his piece either to one of the empty adjacent squares or to the out of the board if it is on one of the squares at the border of the board. When it is Betül's turn, she moves all her two pieces to the adjacent squares. If Ahmet's piece is already on one of the two squares that Betül has just moved to, Betül attacks to his piece and wins the game. If Ahmet manages to go out of the board, he wins the game. If Ahmet begins to move, he guarantees to win the game putting his piece on one of the

\dots

squares at the beginning of the game.
(A)\ 0 \qquad(B)\ n^{2} \qquad(C)\ \left(n\minus{}2\right)^{2} \qquad(D)\ 4\left(n\minus{}1\right) \qquad(E)\ 2n\minus{}1

22

1

(x1 x2 .. x1998) = 7x10^1996 (x1 + x2 + .. + x1998)

\left(x_{1} x_{2} \ldots x_{1998} \right)

shows a number with 1998 digits in decimal system. How many numbers

\left(x_{1} x_{2} \ldots x_{1998} \right)

are there such that \left(x_{1} x_{2} \ldots x_{1998} \right) \equal{} 7\cdot 10^{1996} \left(x_{1} \plus{} x_{2} \plus{} \ldots \plus{} x_{1998} \right) ?

<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 4

21

1

Orthocenter and Circumcircles

In an acute triangle

ABC

, let

D

be a point on

\left[AC\right]

and

E

be a point on

\left[AB\right]

such that \angle ADB\equal{}\angle AEC\equal{}90{}^\circ. If perimeter of triangle

AED

is 9, circumradius of

AED

is

\frac{9}{5}

and perimeter of triangle

ABC

is 15, then

\left|BC\right|

is

<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ \frac{24}{5} \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 8 \qquad<span class='latex-bold'>(E)</span>\ \frac{48}{5}

20

1

x^3.3^(1/x^3) + 1/x^3.3^(x^3) = 6

How many real solutions does the equation x^{3} 3^{1/x^{3} } \plus{}\frac{1}{x^{3} } 3^{x^{3} } \equal{}6 have?

<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ \text{Infinitely many} \qquad<span class='latex-bold'>(E)</span>\ \text{None}

19

1

22 black & 3 blue balls

There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer

n

in between 1 and 25. Betül draws

n

balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the

n

balls are blue and the second blue ball is drawn at

n^{th}

order, Ahmet wins, otherwise Betül wins. To increase the possibility to win, Ahmet must choose

<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 11 \qquad<span class='latex-bold'>(C)</span>\ 12 \qquad<span class='latex-bold'>(D)</span>\ 13 \qquad<span class='latex-bold'>(E)</span>\ 23

18

1

sum of 99! th power of first 24 odd prime numbers

Let

p_{1} <p_{2} <\ldots <p_{24}

be the prime numbers on the interval

\left[3,100\right]

. Find the smallest value of

a\ge 0

such that \sum _{i\equal{}1}^{24}p_{i}^{99!} \equiv a\, \, \left(mod\, 100\right).

<span class='latex-bold'>(A)</span>\ 24 \qquad<span class='latex-bold'>(B)</span>\ 25 \qquad<span class='latex-bold'>(C)</span>\ 48 \qquad<span class='latex-bold'>(D)</span>\ 50 \qquad<span class='latex-bold'>(E)</span>\ 99

17

1

Internal Bisector

In triangle

ABC

, internal bisector of angle

A

intersects with

BC

at

D

. Let

E

be a point on

\left[CB\right.

such that \left|DE\right|\equal{}\left|DB\right|\plus{}\left|BE\right|. The circle through

A

,

D

,

E

intersects

AB

at

F

, again. If \left|BE\right|\equal{}\left|AC\right|\equal{}7, \left|AD\right|\equal{}2\sqrt{7} and \left|AB\right|\equal{}5, then

\left|BF\right|

is

<span class='latex-bold'>(A)</span>\ \frac {7\sqrt {5} }{5} \qquad<span class='latex-bold'>(B)</span>\ \sqrt {7} \qquad<span class='latex-bold'>(C)</span>\ 2\sqrt {2} \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ \sqrt {10}

16

1

Quadratic equation system with three variables

If x^{2} \plus{}y^{2} \plus{}z\equal{}15, x\plus{}y\plus{}z^{2} \equal{}27 and xy\plus{}yz\plus{}zx\equal{}7, then
(A)\ 3\le \left|x\plus{}y\plus{}z\right|\le 4 \\ (B)\ 5\le \left|x\plus{}y\plus{}z\right|\le 6 \\ (C)\ 7\le \left|x\plus{}y\plus{}z\right|\le 8 \\ (D)\ 9\le \left|x\plus{}y\plus{}z\right|\le 10 \\ (E)\ \text{None}

15

1

12 couples on a circle, swapping

Twelve couples are seated around a circular table such that all of men are seated side by side, and every women are seated to opposite of her husband. In every step, a woman and a man next to her are swapping. What is the least possible number of swapping until all couples are seated side by side?

<span class='latex-bold'>(A)</span>\ 36 \qquad<span class='latex-bold'>(B)</span>\ 55 \qquad<span class='latex-bold'>(C)</span>\ 60 \qquad<span class='latex-bold'>(D)</span>\ 66 \qquad<span class='latex-bold'>(E)</span>\ \text{None}

14

1

x^4 + 2x^3 + 3x^2 - x + 1 = 0 (mod 30)

Find the number of distinct integral solutions of x^{4} \plus{}2x^{3} \plus{}3x^{2} \minus{}x\plus{}1\equiv 0\, \, \left(mod\, 30\right) where

0\le x<30

.

<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 4

13

1

Orthocenter

Let

H

be the intersection of altitudes in triangle

ABC

. If \angle B\equal{}\angle C\equal{}\alpha and

O

is the center of circle passing through

A

,

H

and

C

, then find

\angle HOC

in terms of

\alpha

.
(A)\ 90{}^\circ \minus{}\alpha \qquad(B)\ 90{}^\circ \plus{}\frac{\alpha }{2} \qquad(C)\ 180{}^\circ \minus{}\alpha \\ \qquad(D)\ 180{}^\circ \minus{}\frac{\alpha }{2} \qquad(E)\ 180{}^\circ \minus{}2\alpha

12

1

ratio of the hypotenuse over perimeter

In a right triangle, ratio of the hypotenuse over perimeter of the triangle determines an interval on real numbers. Find the midpoint of this interval?
(A)\ \frac{2\sqrt{2} \plus{}1}{4} \qquad(B)\ \frac{\sqrt{2} \plus{}1}{2} \qquad(C)\ \frac{2\sqrt{2} \minus{}1}{4} \\ \qquad(D)\ \sqrt{2} \minus{}1 \qquad(E)\ \frac{\sqrt{2} \minus{}1}{2}

11

1

Number of ways to create a dice

If two faces of a dice have a common edge, the two faces are called adjacent faces. In how many ways can we construct a dice with six faces such that any two consecutive numbers lie on two adjacent faces?

<span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 14 \qquad<span class='latex-bold'>(C)</span>\ 18 \qquad<span class='latex-bold'>(D)</span>\ 56 \qquad<span class='latex-bold'>(E)</span>\ \text{None}

10

1

number of divisor of p+q

Let

p

and

q

be two consecutive terms of the sequence of odd primes. The number of positive divisor of p \plus{} q, at least

<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ 6

9

1

Externally tangent circles

C_{1}

and

C_{2}

be two externally tangent circles with diameter

\left[AB\right]

and

\left[BC\right]

, with center

D

and

E

, respectively. Let

F

be the intersection point of tangent line from A to

C_{2}

and tangent line from

C

to

C_{1}

(both tangents line on the same side of

AC

). If \left|DB\right|\equal{}\left|BE\right|\equal{}\sqrt{2}, find the area of triangle

AFC

.

<span class='latex-bold'>(A)</span>\ \frac{7\sqrt{3} }{2} \qquad<span class='latex-bold'>(B)</span>\ \frac{9\sqrt{2} }{2} \qquad<span class='latex-bold'>(C)</span>\ 4\sqrt{2} \qquad<span class='latex-bold'>(D)</span>\ 2\sqrt{3} \qquad<span class='latex-bold'>(E)</span>\ 2\sqrt{2}

8

1

quadratic sequence

a_{1} \equal{}1, a_{n\plus{}1} \equal{}\frac{a_{n} }{\sqrt{1\plus{}4a_{n}^{2} } } for

n\ge 1

. What is the least

k

such that a_{k} <10^{\minus{}2} ?

<span class='latex-bold'>(A)</span>\ 2501 \qquad<span class='latex-bold'>(B)</span>\ 251 \qquad<span class='latex-bold'>(C)</span>\ 2499 \qquad<span class='latex-bold'>(D)</span>\ 249 \qquad<span class='latex-bold'>(E)</span>\ \text{None}

7

1

combination of drawing of n sets

Find the minimal value of integer

n

that guarantees: Among

n

sets, there exits at least three sets such that any of them does not include any other; or there exits at least three sets such that any two of them includes the other.

<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ 8

6

1

Cubic equation on mod p

Find the number of primes

p

, such that x^{3} \minus{}5x^{2} \minus{}22x\plus{}56\equiv 0\, \, \left(mod\, p\right) has no three distinct integer roots in

\left[0,\left. p\right)\right.

.

<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ \text{None}

5

1

Cyclic Quadrilateral with angle 90,90,30,150

ABCD

is a cyclic quadrilateral. If \angle B \equal{} \angle D, AC\bigcap BD \equal{} \left\{E\right\}, \angle BCD \equal{} 150{}^\circ, \left|BE\right| \equal{} x, \left|AC\right| \equal{} z, then find

\left|ED\right|

in terms of

x

and

z

.
(A)\ \frac {z \minus{} x}{\sqrt {3} } \qquad(B)\ \frac {z \minus{} 2x}{3} \qquad(C)\ \frac {z \plus{} x}{\sqrt {3} } \qquad(D)\ \frac {z \minus{} 2x}{2} \qquad(E)\ \frac {2z \minus{} 3x}{2}

4

1

Minimal value of an 3 variable quadratic function

x,y,z\in \mathbb R

, find the minimal value of

f\left(x,y,z\right) = 2x^{2} + 5y^{2} + 10z^{2} - 2xy - 4yz - 6zx + 3

.

<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ -3 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None}

3

1

Divide set into 3 subsets

How many ways are there to divide a set with 6 elements into 3 disjoint subsets?

<span class='latex-bold'>(A)</span>\ 90 \qquad<span class='latex-bold'>(B)</span>\ 105 \qquad<span class='latex-bold'>(C)</span>\ 120 \qquad<span class='latex-bold'>(D)</span>\ 180 \qquad<span class='latex-bold'>(E)</span>\ 243

2

1

(2^1998)(5^1998)

Let

A

,

B

be the number of digits of

2^{1998}

and

5^{1998}

in decimal system. A \plus{} B \equal{} ?

<span class='latex-bold'>(A)</span>\ 1998 \qquad<span class='latex-bold'>(B)</span>\ 1999 \qquad<span class='latex-bold'>(C)</span>\ 2000 \qquad<span class='latex-bold'>(D)</span>\ 3996 \qquad<span class='latex-bold'>(E)</span>\ 3998

1

a, 2a, 180-3a Triangle

If \left|BC\right| \equal{} a, \left|AC\right| \equal{} b, \left|AB\right| \equal{} c, 3\angle A \plus{} \angle B \equal{} 180{}^\circ and 3a \equal{} 2c, then find

b

in terms of

a

.

<span class='latex-bold'>(A)</span>\ \frac {3a}{2} \qquad<span class='latex-bold'>(B)</span>\ \frac {5a}{4} \qquad<span class='latex-bold'>(C)</span>\ a\sqrt {2} \qquad<span class='latex-bold'>(D)</span>\ a\sqrt {3} \qquad<span class='latex-bold'>(E)</span>\ \frac {2a\sqrt {3} }{3}

1998 National Olympiad First Round

Subcontests

Triangle inequality and symmetry

max no of subsets, having none is a subset of another

an^3 + bn^2 + cn + d is integer, a is rational

Circle and symmetry

f(x) + f(y) = f(x)f(y) + 1 - 1/xy

Winning game strategy on chessboard

(abab) + (cdcd) is a perfect square

Cyclic quadrilateral with a side which is diameter

sqrt( x + 4sqrt(x-4)) - sqrt( x + 2sqrt(x-1)) = 1

covering board with L-shaped pieces and at most one unit

y = sqrt( x +1988 and so on)

P inside ABC

x^6 - 2x^4 + x^2 = A

A game on nxn board

(x1 x2 .. x1998) = 7x10^1996 (x1 + x2 + .. + x1998)

Orthocenter and Circumcircles

x^3.3^(1/x^3) + 1/x^3.3^(x^3) = 6

22 black & 3 blue balls

sum of 99! th power of first 24 odd prime numbers

Internal Bisector

Quadratic equation system with three variables

12 couples on a circle, swapping

x^4 + 2x^3 + 3x^2 - x + 1 = 0 (mod 30)

Orthocenter

ratio of the hypotenuse over perimeter

Number of ways to create a dice

number of divisor of p+q

Externally tangent circles

quadratic sequence

combination of drawing of n sets

Cubic equation on mod p

Cyclic Quadrilateral with angle 90,90,30,150

Minimal value of an 3 variable quadratic function

Divide set into 3 subsets

(2^1998)(5^1998)

a, 2a, 180-3a Triangle

1998 National Olympiad First Round

Subcontests

Triangle inequality and symmetry

max no of subsets, having none is a subset of another

an^3 + bn^2 + cn + d is integer, a is rational

Circle and symmetry

f(x) + f(y) = f(x)f(y) + 1 - 1/xy

Winning game strategy on chessboard

(abab) + (cdcd) is a perfect square

Cyclic quadrilateral with a side which is diameter

sqrt( x + 4sqrt(x-4)) - sqrt( x + 2sqrt(x-1)) = 1

covering board with L-shaped pieces and at most one unit

y = sqrt( x +1988 and so on)

P inside ABC

x^6 - 2x^4 + x^2 = A

A game on nxn board

(x1 x2 .. x1998) = 7x10^1996 (x1 + x2 + .. + x1998)

Orthocenter and Circumcircles

x^3.3^(1/x^3) + 1/x^3.3^(x^3) = 6

22 black &amp; 3 blue balls

sum of 99! th power of first 24 odd prime numbers

Internal Bisector

Quadratic equation system with three variables

12 couples on a circle, swapping

x^4 + 2x^3 + 3x^2 - x + 1 = 0 (mod 30)

Orthocenter

ratio of the hypotenuse over perimeter

Number of ways to create a dice

number of divisor of p+q

Externally tangent circles

quadratic sequence

combination of drawing of n sets

Cubic equation on mod p

Cyclic Quadrilateral with angle 90,90,30,150

Minimal value of an 3 variable quadratic function

Divide set into 3 subsets

(2^1998)(5^1998)

a, 2a, 180-3a Triangle

22 black & 3 blue balls