European Girls' Mathematical Olympiad

Using the Angle Bisector Theorem, we have BD/DC = AB/AC. Substituting the given values, we get BD/DC = 10/12. Solving for BD, we get BD = 10/12 * DC. Since DC = BC - BD, we have BD = 10/12 * (12 - BD). Solving for BD, we get BD = 6 cm.

The area of a sector of a circle is given by the formula A = (θ/360) * πr^2, where θ is the central angle in degrees and r is the radius of the circle. Substituting the given values, we get A = (60/360) * π(10)^2 = 25π cm^2.

Let t be the time in hours after 11:00 AM when the second train overtakes the first train. The distance traveled by the first train in t hours is 60t miles, and the distance traveled by the second train in t hours is 75t miles. Since the second train overtakes the first train, we have 75t = 60t + 60. Solving for t, we get t = 2.5 hours. Therefore, the second train will overtake the first train at 11:00 AM + 2.5 hours = 1:30 PM.

The number of women in the company is 60% of 100, which is 60. The number of men in the company is 40% of 100, which is 40. The number of women promoted is 20% of 60, which is 12. The number of men promoted is 30% of 40, which is 12. Therefore, the total number of employees promoted is 12 + 12 = 24.

The total number of balls in the bag is 10 + 15 + 20 = 45. The number of balls that are not red is 45 - 10 = 35. Therefore, the probability of selecting a ball that is not red is 35/45 = 3/5.

In a right triangle, the sine of an angle is equal to the ratio of the opposite side to the hypotenuse. In this case, sin(B) = AB/AC. Substituting the given values, we get sin(30) = AB/10. Solving for AB, we get AB = 10 * sin(30) = 5√3 cm.

The farmer plants corn on 40% of the land, which is 40% of 100 acres = 40 acres. He plants soybeans on 30% of the land, which is 30% of 100 acres = 30 acres. Therefore, he plants wheat on 100 acres - 40 acres - 30 acres = 30 acres.

The number of women in the company is 60% of 500, which is 300. The number of men in the company is 40% of 500, which is 200. The number of women promoted is 20% of 300, which is 60. The number of men promoted is 30% of 200, which is 60. Therefore, the total number of women promoted is 60.

Let t be the time in hours after 11:00 AM when the second train overtakes the first train. The distance traveled by the first train in t hours is 70t miles, and the distance traveled by the second train in t hours is 80t miles. Since the second train overtakes the first train, we have 80t = 70t + 70. Solving for t, we get t = 70 hours. Therefore, the second train will overtake the first train at 11:00 AM + 70 hours = 1:30 PM.

The number of women in the company is 40% of 200, which is 80. The number of men in the company is 60% of 200, which is 120. The number of women promoted is 25% of 80, which is 20. The number of men promoted is 35% of 120, which is 42. Therefore, the total number of employees promoted is 20 + 42 = 62.

The total number of balls in the bag is 15 + 20 + 25 = 60. The number of blue balls in the bag is 20. Therefore, the probability of selecting a blue ball is 20/60 = 2/5.

In a right triangle, the cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse. In this case, cos(C) = AB/AC. Substituting the given values, we get cos(45) = AB/12. Solving for AB, we get AB = 12 * cos(45) = 6√2 cm.

The farmer plants corn on 50% of the land, which is 50% of 120 acres = 60 acres. He plants soybeans on 30% of the land, which is 30% of 120 acres = 36 acres. Therefore, he plants wheat on 120 acres - 60 acres - 36 acres = 24 acres.

The number of women in the company is 40% of 600, which is 240. The number of men in the company is 60% of 600, which is 360. The number of women promoted is 20% of 240, which is 48. The number of men promoted is 30% of 360, which is 108. Therefore, the total number of women promoted is 48.

Let t be the time in hours after 11:00 AM when the second train overtakes the first train. The distance traveled by the first train in t hours is 80t miles, and the distance traveled by the second train in t hours is 90t miles. Since the second train overtakes the first train, we have 90t = 80t + 80. Solving for t, we get t = 80 hours. Therefore, the second train will overtake the first train at 11:00 AM + 80 hours = 1:30 PM.