1. A group of 16 numbers contains one 1, two 2's, three 3, four 4's, five 5's and another digit. Split the numbers into four groups of four numbers each so that the numbers in all four groups have the same sum.

2. There are 12 equal squares whose sides can be joined to form other shapes. Small combinations of squares can be joined to form bigger shapes. (For example, two 1×2 rectangles can be joined to form a 1×4 rectangle, a 2×2 square, an L shape and a two-stairs shape.) Describe the fastest way to form a 3×4 rectangle?

3. John wanted to buy 39 toy soldiers but found to be 14 cents short. After he bought 35 soldiers, he was left with 18 cents. How much money did John have, to start with?

4. Five integers were chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8}. Prove that, out of these five, at least one divides at least one other.

5. One year, the month of February had five Fridays. How many Tuesdays there were in July?

6. Four fluorescent markers are more expensive than five pens, while four pens are more expensive than three markers. Finally, two pencils cost as much as a marker and a pen. Anton bought eight pencils; Boris bought seven pens. Which boy has spent more money?

Grade 6

1. A group of 36 numbers contains one 1, two 2's, three 3, four 4's, five 5's, six 6's, seven 7's, and eight 8's. Split the numbers into six groups of six numbers each so that the numbers in all six groups have the same sum.

2. There are 15 equal squares whose sides can be joined to form other shapes. Small combinations of squares can be joined to form bigger shapes. (For example, two 1×2 rectangles can be joined to form a 1×4 rectangle, a 2×2 square, an L shape and a two-stairs shape.) Describe the fastest way to form a 3×5 rectangle?

3. Six integers were chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Prove that, out of these six, at least one divides at least one other.

4. The first term of a sequence is 3 and the second is 5. All successive terms are obtained by dividing the preceding two. For example, the third term is 5/3, the fourth term is (5/3)/5 = 1/3, and so on. Find the term number 2009.

5. After the price of a product was reduced by 50%, a store sold 100% more of that product than before. How much have the proceeds changed?

6. Four fluorescent markers are more expensive than five pens, while four pens are more expensive than three markers. Finally, two pencils cost as much as a marker and a pen. Anton bought eight pencils; Boris bought nine pens. Which boy has spent more money?

Grade 7

1. A group of 36 numbers contains one 1, two 2's, three 3, four 4's, five 5's, six 6's, seven 7's, and eight 8's. Split the numbers into six groups of six numbers each so that the numbers in all six groups have the same sum.

2. There are 20 equal squares whose sides can be joined to form other shapes. Small combinations of squares can be joined to form bigger shapes. (For example, two 1×2 rectangles can be joined to form a 1×4 rectangle, a 2×2 square, an L shape and a two-stairs shape.) Describe the fastest way to form a 4×5 rectangle?

3. Find all pairs of natural numbers A and B for which (А

^{2}─В

^{2})

^{2}─(2

^{В})

^{2}=2009 .

4. None of the group of seven distinct natural numbers exceeds 12. Prove that, out of these seven, at least one divides at least one other.

5. As a consequence of a 20% reduction in price of a product, a store sold 20% more of that product. Express the change in the proceeds in percents.

6. Eight Flo-Master markers are more expansive than four pencils and five pens. Four pens are more expansive than three markers. Two pencils cost the same as a marker and a pen. John bought six markers; Paul bought seven pens. Which boy has spent more money?

Grade 8

1. A group of 36 numbers contains one 1, two 2's, three 3, four 4's, five 5's, six 6's, seven 7's, and eight 8's. Split the numbers into six groups of six numbers each so that the numbers in all six groups have the same sum.

2. There are 24 equal squares whose sides can be joined to form other shapes. Small combinations of squares can be joined to form bigger shapes. (For example, two 1×2 rectangles can be joined to form a 1×4 rectangle, a 2×2 square, an L shape and a two-stairs shape.) Describe the fastest way to form a 4×6 rectangle?

3. Find all pairs of natural numbers A and B for which (А

^{В}─В

^{В})

^{2}─(2

^{В})

^{2}=2009 .

4. None of the group of eight distinct natural numbers exceeds 14. Prove that, out of these eight, at least one divides at least one other.

5. One year, the month of February had five Fridays. Which months of that year had five Sundays?

6. Two pencils cost as much as a pencil and a fluorescent marker. Sixteen markers are more expensive than nine pencils and ten pens. Kyle bought twenty three markers; Kim bought twenty nine pens. Which boy has spent more money?

Grade 9

1. A group of 49 numbers contains four 4's, five 5's, six 6's, seven 7's, eight 8's, nine 9's and ten 3's. Split the numbers into seven groups of seven numbers each so that the numbers in all seven groups have the same sum.

2. Find the area of the ring between two concentric circles, if the chord of the bigger circle that is tangent to the smaller one has the length of 2009.

3. Find all pairs of natural numbers A and B for which (А

^{В}─В

^{2})

^{2}─(2

^{В})

^{2}=2009 .

4. None of the group of nine distinct natural numbers exceeds 16. Prove that, out of these nine, at least one divides at least one other.

5. Find on all natural numbers that equal the sum of squares of their digits.

6. Through the vertices of a 9×9 grid square draw a broken line with nodes at the grid points that encloses the least possible area.

Grade 10

1. A group of 49 numbers contains four 4's, five 5's, six 6's, seven 7's, eight 8's, nine 9's and ten 3's. Split the numbers into seven groups of seven numbers each so that the numbers in all seven groups have the same sum.

2. Find the area of the ring between two concentric circles, if the chord of the bigger circle that is tangent to the smaller one has the length of 2009.

3. Find all natural numbers M for which (7√М─М)

^{√М}─(√М)

^{М}=2009 .

4. By definition, the shortest pass between two points on the surface of a cube is the shortest broken line on the surface of the cube joining the two points. (Which is a straight line segment whenever the given points lie on the same face.) Three shortest passes joining three points split the surface of a cube into two regions. The region with the smallest area is called a triangle. Find the largest area of a triangle on the surface of a unit cube.

5. Find all natural numbers that equal the sum of squares of their digits.

6. In a Cartesian system of coordinates, a circle of radius r with center at (p, q) meets the parabola with the equation y = ax² + bx + c at four distinct points. Prove that there is another parabola passing through the same four points and find its equation.

Grade 11-12

1. A group of 49 numbers contains four 4's, five 5's, six 6's, seven 7's, eight 8's, nine 9's and ten 3's. Split the numbers into seven groups of seven numbers each so that the numbers in all seven groups have the same sum.

2. Find the area of the ring between two concentric circles, if the chord of the bigger circle that is tangent to the smaller one has the length of 2009.

3. Find all natural numbers M for which (7√М─М)

^{√М}─(М)

^{√М}=2009 .

4. By definition, the shortest pass between two points on the surface of a cube is the shortest broken line on the surface of the cube joining the two points. (Which is a straight line segment whenever the given points lie on the same face.) Three shortest passes joining three points split the surface of a cube into two regions. The region with the smallest area is called a triangle. Find the largest area of a triangle on the surface of a unit cube.

5. Give examples of two functions f(x) and g(x) of which one is monotone increasing and the other monotone decreasing that satisfy f(sin(g(x))) = g(sin(f(x))), for all real x.

6. The derivatives of two polynomials P(x) and Q(x) are divisible by x

^{2009}. Prove that the same is true of the derivative of their product.

matholimpnovember 26, 17:30 55## Journal information

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