User:Vbatz

From Wikipedia, the free encyclopedia

[edit] IMO 1959

Prove that $21n+4\over14n+3$ is irreducible for every natural number $n$

For what real values of

x

$\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A$

Given $A=\sqrt2$

$A=\sqrt1$

$A=\sqrt4$

where only non-negative real numbers are allowed in square roots and the root always denotes the non-negative root?

Let

a, b, c

be real numbers. Given the equation for

cos x

a cos 2 x + b cos x + c = 0

, form a quadratic equation in

cos2 x

whose roots are the same values of

x

. Compare the equations in

cos x

and

cos2 x

for

a = 4, b = 2, c = - 1

Given the length

| A C |

, construct a triangle

A B C

with $\angle ABC=90^o$ , and the median

B M

satisfying $BM^2=AB\cdot BC$ .

An arbitrary point

M

is taken in the interior of the segment

A B

. Squares

A M C D

and

M B E F

are constructed on the same side of

A B

. The circles circumscribed about these squares, with centers

P

and

Q

, intersect at

M

and

N

(a) prove that $A F$ and $B C$ intersect at $N$ ;

(b) prove that the lines $M N$ pass through a fixed point $S$ (independent of $M$ );

The planes $P$ and $Q$ are not parallel. The point $A$ lies in $P$ but not $Q$ , and the point $C$ lies in $Q$ but not $P$ . Construct points $B$ in $P$ and $D$ in $Q$ such that the quadrilateral $A B C D$ satisfies the following conditions: (1) it lies in a plane, (2) the vertices are in the order $A, B, C, D$ , (3) it is an isosceles trapezoid with $A B$ parallel to $C D$ (meaning that $A D = B C$ , but $A D$ is not parallel to $B C$ unless it is a square), and (4) a circle can be inscribed in $A B C D$ touching the sides.

Retrieved from "http://en.wikipedia.org../../../v/b/a/User%7EVbatz_cd14.html"

User:Vbatz

From Wikipedia, the free encyclopedia

[edit] IMO 1959

Views

Navigation

interaction

Search