Problem Kangur_2003_0708_1 (3 pts) http://www.mathkangaroo.org

A segment of a length equal to 4 was divided with 4 points into segments of equal length. How long is each segment?

Problem Kangur_2003_0708_2 (3 pts) http://www.mathkangaroo.org

A square built of 16 small squares is cut with a line. What is the greatest number of the little squares that the line can go through?

Problem Kangur_2003_0708_3 (3 pts) http://www.mathkangaroo.org

When 29 is subtracted from the greatest 2-digit number and the difference is divided by the smallest 2-digit number what is the result?

Problem Kangur_2003_0708_4 (3 pts) http://www.mathkangaroo.org

If , then is equal to?

A) 9
B) 2
C) 10
D) 3
E) 15

Problem Kangur_2003_0708_5 (3 pts) http://www.mathkangaroo.org

Tomek saved 120 zl. One day he bought a present for his brother and he spent 1/3 of all his money. The next day he bought a book for himself and he spent 1/4 of the remaining money. How much money did he have left after shopping? (zl is a monetary unit in Poland like dollar in USA)

Problem Kangur_2003_0708_6 (3 pts) http://www.mathkangaroo.org

A rectangular prism was made out of three blocks, each consisting of four cubes (see the picture). Which of the blocks below has the same shape as the white block?

Problem Kangur_2003_0708_7 (3 pts) http://www.mathkangaroo.org

There are 17 trees on one side of the street on Tomek's way from his house to school. One day Tomek marked these trees with white chalk in the following way: on the way from his house to the school he marked every other tree, starting with the first one. On his way back home he marked every third tree, starting with the first one. How many trees were not marked?

Problem Kangur_2003_0708_8 (3 pts) http://www.mathkangaroo.org

In triangle ABC : DA = DB = DC (see the figure). Then:

A) Angle ACB is obtuse.
B) Angle ACB is acute.
C) Angle ACB is right.
D) Angle ACB changes, depending on the lengths of the sides AC and BC.
E) This kind of triangle ABC does not exist.

Problem Kangur_2003_0708_9 (3 pts) http://www.mathkangaroo.org

There were 5 parrots in a pet store. The average price of each of them was 600 zl. One day, the most beautiful parrot was sold. The average price of each of the remaining 4 birds was 500 zl.What was the price of the sold parrot? (zl is a monetary unit in Poland like dollar in USA)

Problem Kangur_2003_0708_10 (3 pts) http://www.mathkangaroo.org

What is the greatest number of inner right angles that a hexagon can have (not necessary a convex hexagon)?

Problem Kangur_2003_0708_11 (4 pts) http://www.mathkangaroo.org

How many integers are there, the squares of which are found between the numbers -100 and 100, inclusive?

Problem Kangur_2003_0708_12 (4 pts) http://www.mathkangaroo.org

Five girls Ania, Beata, Celina, Dorota, and Ela, received the following assignment: they were supposed to draw four segments and to count all the points of intersection of these segments. After they finished, Ania said she counted 2 points, Beata 3 points, Celina 5 points, Dorota 6 points, Ela 7 points. Who made a mistake?

Problem Kangur_2003_0708_13 (4 pts) http://www.mathkangaroo.org

A cube was made from the given configuration (see the figure). Which wall will be opposite to the wall with the letter x?

Problem Kangur_2003_0708_14 (4 pts) http://www.mathkangaroo.org

A square piece of paper is folded twice and cut in the way you can see in the picture. How will the piece of paper look after unfolding?

Problem Kangur_2003_0708_15 (4 pts) http://www.mathkangaroo.org

In how many ways can you express the number 2003 as a sum of two prime numbers?

A) 1
B) 2
C) 3
D) 4
E) This expression doesn't exist

Problem Kangur_2003_0708_16 (4 pts) http://www.mathkangaroo.org

Which of the following sums is equal to 2003?

A)162 + 262 + 322
B)162 + 262 + 332
C)152 + 272 + 322
D)172 + 252 + 332
E)172 + 242 + 342

Problem Kangur_2003_0708_17 (4 pts) http://www.mathkangaroo.org

An empty truck weighs 2000 kg. After the truck was loaded, freight made up 80% of the weight of the loaded truck. At the first stop one fourth of the freight was unloaded. What percent of the loaded truck's weight did the load make up after that? (Hint: freight = load.)

Problem Kangur_2003_0708_18 (4 pts) http://www.mathkangaroo.org

The combined capacity of a bottle and a glass is equal to the capacity of a pitcher. The capacity of a bottle is equal to the combined capacity of a glass and a mug. The combined capacity of three mugs is equal to the combined capacity of two pitchers. How many glasses altogether have the capacity of one mug?

Problem Kangur_2003_0708_19 (4 pts) http://www.mathkangaroo.org

Michal has 42 identical cubical blocks, each one with an edge of 1 cm. From all of these blocks, he built a rectangular prism with a base perimeter equal to 18 cm. What is the height of the prism which he built?

Problem Kangur_2003_0708_20 (4 pts) http://www.mathkangaroo.org

A fresh mushroom contains 90% water. How many kilograms of fresh mushrooms are needed in order to make 1 kg of dried mushrooms containing 11% water?

Problem Kangur_2003_0708_21 (5 pts) http://www.mathkangaroo.org

There are five men in a room. Each of them is either a liar that always lies or a knight that always tells the truth. Each of them was asked the question: "How many liars are among you?" The answers were: "one," "two," "three," "four," "five." How many liars were in that room?

Problem Kangur_2003_0708_22 (5 pts) http://www.mathkangaroo.org

In rectangle ABCD, points P, Q, R, S are respectively the midpoints of sides AB, BC, CD, and DA. Let T be the midpoint of segment SR. What part of the area of rectangle ABCD is the area of triangle PQT?

Problem Kangur_2003_0708_23 (5 pts) http://www.mathkangaroo.org

There are six segments with lengths: 1, 2, 3, 2001, 2002, 2003. In how many ways can we select three of these segments to build a triangle?

Problem Kangur_2003_0708_24 (5 pts) http://www.mathkangaroo.org

Six consecutive points were indicated on a number line in this order: A, B, C, D, E, and F. Regardless of the location of these points, if only AD = CF and BD = DF, then the following equation is true:

A) AB = BC
B) BC = DE
C) BD = EF
D) AB = CD
E) CD = EF

Problem Kangur_2003_0708_25 (5 pts) http://www.mathkangaroo.org

In the picture four squares are shown and the lengths of their sides are indicated. What is the difference between the combined area of the shaded regions and the combined area of the black regions?

Problem Kangur_2003_0708_26 (5 pts) http://www.mathkangaroo.org

Which of the following numbers, after it is multiplied by 768, yields a product that ends with the largest number of zeros?

A) 7,500
B) 5,000
C) 3,125
D) 2,500
E) 10,000

Problem Kangur_2003_0708_27 (5 pts) http://www.mathkangaroo.org

A square was divided into 25 identical small squares (see the picture). What is the sum of the measures of angles < MAN, < MBN, < MCN, < MDN, < MEN?

Problem Kangur_2003_0708_28 (5 pts) http://www.mathkangaroo.org

How many natural numbers n have such a property that out of all the positive divisors of number n, which are different from both 1 and n, the greatest one is 15 times greater than the smallest one?

Problem Kangur_2003_0708_29 (5 pts) http://www.mathkangaroo.org

In a number with at least two digits, the last digit was deleted. The resulting number was n times smaller than the previous one. What is the greatest possible value of n?

Problem Kangur_2003_0708_30 (5 pts) http://www.mathkangaroo.org

How many natural numbers n have the property that the remainder of dividing 2003 by n is equal to 23?