Wettbewerbsaufgaben 2012

### Problem 1J

If we increase the length of each edge of a cube by , by what percent does the volume increase?

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### Problem 2J

What is the largest number of pieces a ring can be divided into using three straight lines?

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### Problem 3J

A five digit number is divisible by . Find the value of .

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### Problem 4J

A math teacher decided to organize two rounds of a math competition. Each team consisted of five members. In the first round, the students divided themselves into teams on their own. In the second round the teacher divided them so that nobody was in the same team with anyone he or she has played with in the first round. Determine the minimum number of students in which such division is possible.

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### Problem 5J

The whole surface of a rectangular prism-shaped vanilla cake with edges of lengths , and is covered by a thin layer of chocolate. Let us cut the cake into cubes of volume . What percentage of cubes have no chocolate on them?

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### Problem 6J

We are given a square with side length and a point in its plane (outside the square) so that . What is the length of the largest diagonal in the pentagon ?

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### Problem 7J

Jacob had written on a blackboard. Now he is wondering how to increase or decrease each of the digits by so that the result is correct. What will be the right-hand side after the change?

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### Problem 8J

The sum of integers and is at most and their difference is smaller than . Find the maximum value of the expression .

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### Problem 9J

Let , , be real numbers such that the arithmetic mean of and is equal to , and arithmetic mean of and is equal to . What is the arithmetic mean of , , and ?

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### Problem 10J

trees grow on a square grid. The gardener cut out one of the corner trees and is now standing at that corner facing the rest of the grid. However, he does not see some of the trees because they are aligned with the other trees. (A tree is aligned with another tree if there exists a tree on the line segment between the gardener and .) How many trees does the gardener see?

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### Problem 11J / 1S

How many ways are there to color the faces of a cube with two colors? Two colorings are considered identical if we can get one from the other by rotating the cube.

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### Problem 12J / 2S

A rectangle is given with and . Let be the point on the ray such that and the point inside such that the distance from to both and is . Let line intersect and at points and , respectively. Find the area of .

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### Problem 13J / 3S

For how many positive integers () is a square of a positive integer?

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### Problem 14J / 4S

An equilateral triangle with side length is lying on the floor, with one altitude perpendicular to the floor. We color one of its vertices red and we ”roll” the triangle on the floor (in the plane of the triangle) through one full rotation. What is the length of the red vertex's trajectory?

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### Problem 15J / 5S

What is the smallest positive integer consisting only of the digits and that is divisible by ?

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### Problem 16J / 6S

Bill is old enough to vote but not old enough to use the senior discount. (His age is between 18 and 70.) It is known that years ago, the square of his age was the same as his current age increased by . Moreover, Bill's age is a square of an integer. Find .

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### Problem 17J / 7S

Let us fold the bottom left corner of a rectangular paper to its top right corner. The resulting figure consists of three triangles created by the edges of the paper and the fold. For what ratio of side lengths of the paper is the ratio of the areas of the triangles ?

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### Problem 18J / 8S

How many three digit numbers are divisible by if each digit is larger than ?

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### Problem 19J / 9S

Three circles with radius are given, such that each two are externally tangent. We put all the circles into a greater circle . Each small circle is internally tangent to the greater circle. Find the radius of the circle .

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### Problem 20J / 10S

Let be a positive integer. If has in the tens place, which digits can be in the units place?

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### Problem 21J / 11S

If we write the numbers in some order, we will get an -chain. For example, one possible -chain is

What is the smallest with such that there exists an -chain that is a palindrome? (A number is a palindrome if it may be read the same way in either direction.)

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### Problem 22J / 12S

Find all triples of positive real numbers for which , , .

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### Problem 23J / 13S

A circle is given with a radius and two perpendicular chords inside it, dividing the circle into parts. We color the part with the greatest and the part with the smallest area in black and leave the rest white. We know that the area of the white parts is the same as the area of the black parts. What is the maximum possible distance of the longer chord to the centre?

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### Problem 24J / 14S

An officer's route consists of three circles (shown in the picture). He must start at and travel the entire route, without visiting any portion twice except perhaps the intersection points, and return to . If his direction matters, how many such routes exist?

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### Problem 25J / 15S

A trapezoid has bases of lengths and and legs of lengths and . What is its area?

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### Problem 26J / 16S

There are people in a row. They want to order themselves according to their height, so that the tallest one will stand in front. In one step, two people who are next to each other can switch position. At most, how many steps are necessary for them to order as they want?

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### Problem 27J / 17S

Parsley lives in Vegetable State where one can pay only by coins with values and . If Parsley had an unlimited supply of both kinds of coins what would be the highest integer price he could not pay with them?

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### Problem 28J / 18S

Let us divide a circle with radius into parts. What is the smallest possible perimeter of the part with the greatest area? If there is more than one part with the greatest area, we take into account the one with the smallest perimeter.

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### Problem 29J / 19S

Find the sum of all real numbers for which the equations and have at least one common real root.

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### Problem 30J / 20S

How many -digit numbers are there such that after crossing out its first digit (from the left), one gets a number times smaller than the original one?

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### Problem 31J / 21S

Consider a right triangle with sides of integer length. One of the sides has length . What is the maximum area it can have?

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### Problem 32J / 22S

Let be a triangle with circumcentre and orthocentre in the Cartesian plane. No two of these five points coincide and all of them have integer coordinates. What is the second smallest possible radius of a circle circumscribed around triangle ?

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### Problem 33J / 23S

Find the largest positive integer such that the number is divisible by .

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### Problem 34J / 24S

If we calculate the product of the digits of some number, the product of the digits of the product, and so on, we arrive after a finite number of steps at a one digit number. The number of required steps is called the \textit*{persistence} of the number. For example, the number has persistence because then . Find the greatest even number with mutually distinct digits that has persistence .

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### Problem 35J / 25S

If we extend the sides and of a convex quadrilateral, they intersect at a point . Let us denote and the midpoints of and , respectively. Find the ratio between the area of a triangle and the area of a quadrilateral . (We let you know that this ratio is the same for all convex quadrilaterals with non-parallel sides.)

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### Problem 36J / 26S

Cube termites bore four straight square tunnels of sidelength in each direction inside a cube (as you can see on the picture) and now they have left the cube. How many cm of paint do we need to cover the surface of what is left of the cube if the original cube had side length cm?

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### Problem 37J / 27S

We are given a circle with radius . We are standing on the leftmost point of the circle. It is possible to move only up and right. What is the length of the longest trajectory that we can travel inside the circle?

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### Problem 38J / 28S

What is the greatest divisor of that gives a remainder of when divided by ?

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### Problem 39J / 29S

A cube and of its points are given: the vertices, midpoints of the edges, centers of the faces, and the center of the cube. How many lines are passing through exactly three of the given points?

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### Problem 40J / 30S

participants took part in a competition that lasted for days. Each day each participant received an integer amount of points between and , inclusive. No two participants had the same score in a given day. At the end of the competition (the -th evening) each participant had a score of points (when all the points for the whole competition were added). Find the sum of all for which this is possible (regardless of ).

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### Problem 41J / 31S

We want to cut a cylindrical cake with straight cuts. What is the maximum number of resulting pieces? For example, by cuts we can divide the cake into pieces.

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### Problem 42J / 32S

An eight-branched star (Stella octangula) is a solid, which results from sticking eight regular tetrahedrons to the faces of an octahedron. All the edges of each tetrahedron and the octahedron have length . What is the volume of the eight-branched star?

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### Problem 43J / 33S

A hitchhiker is walking along the road. The probability that a car picks him up in the next minutes is . What is the probability of a car picking him up in the next five minutes, if the probability that he gets picked up by a car is the same in each moment?

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### Problem 44J / 34S

A vandal and a moderator are editing a Wikipedia article. At the beginning, the article was without a mistake and each day the vandal adds one mistake. At the end of each day the moderator has chance of having found each single mistake that is in the article. What is the probability that after three days the article will be without a mistake?

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### Problem 45J / 35S

We have a large enough heap of red, blue, and yellow cards. We can receive the following number of points:

• for each red card one point,
• for each blue card twice the number of red cards as points,
• for each yellow card three times the number of blue cards as points.
What is the maximum number of points we can receive when we have cards?

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### Problem 46J / 36S

Matthew has one -sided die and his friend CD has three -sided dice. What is the chance that after rolling all the dice, the value on Matthew's die will be greater than the sum of the values on CD's dice?

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### Problem 47J / 37S

Let us have a by board. The rows and columns are numbered from left to right and top to bottom, respectively, by integers from to . In each cell we write the product of the row number and column number. A traveler stands on the top left cell and wants to arrive to the bottom right cell. However, she can only travel right and down (not diagonally). A \textit*{traveler's number} is the product of the numbers on the cells which she had stepped onto (including the first and the last one). What is the greatest common divisor of all possible traveler's numbers?

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### Problem 48J / 38S

We have a triangle with altitudes of sizes , , and . What is its perimeter?

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### Problem 49J / 39S

Find the greatest integer for which is rational.

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### Problem 50J / 40S

A Maxi-square is a square divided into nine square tiles. Each tile is divided into four little squares in which the numbers , , , and are written (each of them exactly once). Two tiles can touch only if their adjacent numbers match (as in dominoes). How many Maxi-squares exist?

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### Problem 51J / 41S

Andrew calls his favourite number \textit*{balloon}. It holds for balloon that

• the sum of its digits is twice the number of its digits
• it does not have more than digits
• its digits are in turn even and odd (it doesn't have to start with an even digit)
• the number greater by one is divisible by .
Find the value of Andrew's balloon.

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### Problem 52J / 42S

Find all four-digit positive integers such that the last four digits of the number is the number .

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### Problem 53J / 43S

Find the sum of all five-digit palindromes.

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### Problem 54J / 44S

How many ordered quadruples of odd positive integers satisfy ?

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### Problem 55J / 45S

Find the only eleven-digit number such that

• it starts with a one
• when it it is written twice in a row, it is a perfect square.

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### Problem 56J / 46S

Two different triangles with side lengths , , and are given such that their incircles coincide and their circumcircles coincide. What is the area of the polygon that the triangles have in common?

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### Problem 57J / 47S

In how many ways can we color the cells of a grid with black and white so that in each column and row there will be exactly two black squares?

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### Problem 58J / 48S

Two distinct points and lie inside a square with side length . By of a vertex of the square we mean its distance to the closer of the points and . What is the smallest possible sum of the remotenesses of the vertices of the square?

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### Problem 59J / 49S

The parking lot consists of parking places regularly distributed in a row with numbers to . cars park there one after each other, in the following way:

• The first car chooses randomly one of the places.
• The following cars choose the place from which the distance to the closest car is the greatest (each of such places with the same probability).
Determine the probability that the last car parks in the parking place number .

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### Problem 60J / 50S

Find all real numbers that satisfy

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