Känguru der Mathematik 2024 - Grades 9+10

• The "Känguru der Mathematik 2024" competition is on March 21, 2024.
• The level is Junior, for grades 9 + 10.
• The time allowed is 75 minutes.
• There are 30 starting points.
• Each correct answer for questions 1-10 is worth 3 points.
• Each correct answer for questions 11-20 is worth 4 points.
• Each correct answer for questions 21-30 is worth 5 points.
• Each unanswered question is worth 0 points.
• Each incorrect answer deducts ¼ of the points for the question.
• Participants should write the letter (A, B, C, D, E) of the correct answer in the square under the question number.

Sample Questions (3 Points)

• The question asks to identify which square is split into two parts that do not have the same shape.
• The question asks to calculate 2×0.24/(20×2.4) which equals 0.01.
• The number of points on opposite faces of a die always totals 7.
• The vertex sum in a vertex is the sum of the points on the faces that meet in that vertex.
• For a die where faces with 1, 2, and 3 points meet in P, the vertex sum in point P is 1+2+3 = 6.
• The question asks to identify the biggest vertex sum in the vertices Q, R and S.
• A hopscotch game involves jumping from one square to the next with the left foot, both feet, right foot, both feet, etc., alternately.
• Maya plays this game and jumps into exactly 48 squares starting with the left foot.
• The question asks how often Maya's left foot is on the floor during the game.
• Tim wants to draw a figure without lifting their pencil and some parts must be passed over more than once.
• The segment lengths are stated in the figure.
• The questions asks to identify the minimum length of the total line Tim will draw, if they can choose their starting point freely.
• A square contains four touching circles of equal size.
• The question asks what the ratio is of the area of the black part to the grey part.
• Two numbers, a and b, are from the set {1,2,3,4,5,6}.
• For each pair (a,b), a straight line with the equation y = ax + b is drawn, forming a triangle with the coordinate axes.
• The question asks how many pairs (a,b) create an isosceles triangle.
• John starts with a dark cube and adds light cubes to all the visible faces.
• The question asks to identify what minimum number of dark cubes he then requires to add, such that, no light surfaces are visible.
• A square is drawn with vertices A, B, C, D, and a regular hexagon with side OC, where O is the center of the square.
• The question asks how big is angle α.
• Ardal fences a rectangular plot of land with a 40 m fence, where all sides are prime numbers.
• The question asks what the biggest possible area of the plot of land is.

Sample Questions (4 Points)

• A palindrome number reads the same from front to back.
• The question asks to identify the sum of the digits of the largest three-digit palindrome number that is also a multiple of 6.
• A rectangle is split into three pieces with equal areas: an equilateral triangle with sides of length 4 cm, and two trapezoids.
• The question asks how long is the shorter of the parallel sides of the trapezoid.
• Jelena fills a 2×4 table with the letters A, B, C, and D, ensuring each letter appears once in each row and in each 2×2 square.
• The question asks to identify the number of ways Jelena can organise this.
• Sanjay has three differently colored circles and places them on top of each other shown in Figure 1 .
• Then he moves them so that they touch each other pairwise shown in Figure 2.
• In Figure 1 the visible black area is seven times as big as the area of the white circle.
• The question asks what is the ratio of the visible black areas in Figure 1 and Figure 2.
• The daughter of Mary's daughter was born today.
• In two years' time the product of Mary's age, her daughter's age and her grand-daughter's age will be exactly 2024.
• Each of the three ages will be an even number.
• The question asks how old Mary is today.
• A point P is chosen inside an equilateral triangle ABC.
• Segments of 2 m, 3 m, and 6 m are drawn parallel to the sides of the triangles.
• The question asks what is the perimeter of the triangle?
• A number is written into each of the twelve circles shown.
• The numbers in the squares state the product of the four numbers in the vertices of the squares.
• The question is to identify what is the product of the numbers in the eight bold circles.
• Jean-Philippe has n³ cubes of equal size and uses them to form one big cube and paints its surface.
• The number of small cubes with exactly one painted face is the same as the number of small cubes with no painted face.
• The question is to identify what is the value of n.
• Otis builds the net of a solid using a combination of squares and triangles.
• All sides of the squares and the triangles have side length 1.
• He proceeds to fold the net to form the solid shown.
• The question is to identify what is the distance from A to B.
• Vlado has participated in 31 cross-country races in the last five years.
• In the first year, he participated in the smallest number of races and he then successively increased the number of races each year.
• In the fifth year, he participated in three times as many races as in the first year.
• The question asks how many races did he participate in in the fourth year?

Sample Questions (5 Points)

• The question asks how many integers k have the property that k+6 is a multiple of k-6.
• There are four bowls with sweets on a table.
• The number of sweets in the first bowl equals the number of bowls with one sweet.
• The number of sweets in the second bowl equals the number of bowls with two sweets.
• The number of sweets in the third bowl equals the number of bowls with three sweets.
• The number of sweets in the fourth bowl equals the number of bowls with no sweets.
• The question asks how many sweets are in the bowls in total.
• Cristina has 12 cards numbered from 1 to 12.
• She places eight of them in a circle so that the sum of any two adjacent numbers is a multiple of 3.
• The question asks which numbers Cristina doesn't use.
• Carl tells the truth on one day, lies the next, tells the truth again the day after, etc.
• On one day he made exactly four of five statements.
• The question asks which statement he cannot have made on that day: "2024 is divisible by 11", "Tomorrow is Saturday", "I lied yesterday and I will lie tomorrow", "Yesterday was Wednesday", "I am telling the truth today, and I will tell the truth tomorrow.".
• The sum of the digits of N is three times the sum of the digits of N+1.
• The question asks to identify what is the smallest possible sum of the digits of N.
• Jill has some black and some white unit cubes and uses 27 of them to build a 3×3×3 cube.
• She wants exactly one third of the surface to be black.
• If A is the smallest possible number of black cubes that she can use and B the biggest possible number.
• The question asks what is the value of B - A.
• Ann rolled an ordinary die 24 times.
• All numbers from 1 to 6 were rolled at least once, but the number 1 was rolled more often than any other number.
• Ann then added all numbers that were rolled.
• The question asks to identify what is the biggest number Ann could have obtained in this way.
• Olga was walking in the park.
• For half the time, she walked with a speed of 2 km/h.
• For half the distance, she walked with a speed of 3 km/h.
• For the remaining time, she walked with a speed of 4 km/h.
• The question asks which fraction of the time she walked with 4 km/h?
• 20 points are distributed equally along a circle.
• The question asks how many of the segments connecting two of those points are longer than the radius of the circle but shorter than the diameter of the circle.
• Consider n different straight lines in a plane, labelled as l₁,..., l„ .
• The straight line l₁ intersects 5 other straight lines.
• The straight line l₂ intersects 9 other straight lines.
• The line l₃ intersects 11 other straight lines.
• The question asks which of the following numbers is a possible value of n?