作者cuttlefish (終於可以上網了^^)
看板IMO_Taiwan
標題q2
時間Mon Oct 20 22:24:54 2003
1. In an acute - angled triangle ABC , A=30 ^ , H is the orthocentre,
and M is the midpoint of BC . On the line HM , take a point T such that
HM=MT. Show that AT=2BC
2. Show that there are infinitely many pairs (a,b) of relatively prime
integers(not necessarily positive) such that both the quadratic equations
x^ 2 + ax + b=0 and x^ 2 + 2ax + b=0 have integer roots.
3. Show that the number of 3 - element subsets {a,b,c} of {1,2,3,... ,63}
with a + b + c<95 is less than the number of those with a + b + c>95.
4. Let ABC be a triangle and a circle ' be drawn lying inside the
triangle, touching its incircle externally and also touching the two sides
AB and AC . Show that the ratio of the radii of the circles ' and is equal
to 2 (( pi - A) / (4) .).
5. Let a 1 ,a 2 ,a 3 ,... ,a n be n real numbers all greater than 1 and
such that |a k - a k + 1|< ;1 for 1 <= k <= n - 1.
Show that (a 1 ) / (a 2 ) . + (a 2 ) / (a 3 ) . + (a 3 ) / (a 4 ) . +
... + (a n - 1) / (a n ) . + (a n ) / (a 1 ) .<2n - 1.
6. Find all primes p for which the quotient (2 p - 1 - 1)/p is a square.
^^
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