MATHEMATICS OLYMPIADE FOR JUNIOR HIGH SCHOOL PART 3
11 Januari 2011 3 Komentar
- Solve for a:
- a + 3 =1 (mod 5)
- a + 2 =4 (mod 7) and a − 2 =4 (mod 9)
- 4a + 4 =3 (mod 6)
- What are the possible choices for the digits A and B if the umber 2A1B6 is divisible by both 9 and 4?
- A certain integer n has 40 positive divisors including 1 and n. What is the largest number of primes that could divide n?
- How many of the positive factors of 720 are also divisible by 12?
- How many natural numbers are divisors of exactly 2 of the 3 integers 240, 600, and 750?
- If 14 times a natural number has a units digit of 2 when xpressed in base 12, what is the Units digit of the natural number when expressed in base 6?
- If a and b are natural numbers and 5a3 = 6b4, what is the mallest possible value of a + b?
- Prove that every perfect square has an odd number of positive factors and that every Nonsquare has an even number of positive factors.
- How many of the first 1000 natural numbers have a remainder of 1 when divided by 5, a remainder of 2 when divided by 6, and a remainder of 5 when divided by 7?
- A point D is placed on side BC of triangle ABC. Circles are inscribed in ABD and ACD. Their common exterior tangent (other than BC) meets AD at K. Prove that the length of AK is independent of D.
- From vertex A of triangle ABC, perpendiculars AM and AN are drawn to the bisectors of the exterior angles of the triangle at B and C. Prove that MN is equal to half the perimeter of ABC.
- Given is cube ABCD A’B’C’D’ (A connected to A’,B,C, and like wise for the others). Points M and N are on AA’ and BC’ so that line MN intersects line B0D. Find BC/ BN− AM / AA’ .
- ABC is an acute-angled triangle. The incircle touches BC at K. The altitude AD has midpoint M. The line KM meets the incircle again at N. Show that the circumcircle of BCN is tangent to the incircle of ABC at N.
- Let I be the incenter of triangle ABC. Prove that (IA)(IB)(IC) = 4Rr2, where R is the circumradius of ABC and r is the inradius of ABC.