### Problem 1J

If we increase the length of each edge of a cube by

Show / hide solution

### Problem 2J

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

Show / hide answerShow / hide solution

### Problem 3J

A five digit number is divisible by

Show / hide solution

### 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.

Show / hide answerShow / hide solution

### Problem 5J

The whole surface of a rectangular prism-shaped vanilla cake with edges of lengths

Show / hide solution

### Problem 6J

We are given a square with side length and a point in its plane (outside the square) so that

Show / hide solution

### 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?

Show / hide answerShow / hide solution

### Problem 8J

The sum of integers and is at most and their difference is smaller than

Show / hide solution

### Problem 9J

Let

Show / hide solution

### 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

Show / hide solution

### 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.

Show / hide answerShow / hide solution

### Problem 12J / 2S

A rectangle is given with and

Show / hide solution

### Problem 13J / 3S

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

Show / hide answerShow / hide solution

### 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?

Show / hide answerShow / hide solution

### Problem 15J / 5S

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

Show / hide solution

### 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

Show / hide solution

### 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

Show / hide solution

### Problem 18J / 8S

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

Show / hide solution

### 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

Show / hide solution

### Problem 20J / 10S

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

Show / hide answerShow / hide solution

### Problem 21J / 11S

If we write the numbers in some order, we will get an

Show / hide solution

### Problem 22J / 12S

Find all triples of positive real numbers for which

Show / hide solution

### 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?

Show / hide answerShow / hide solution

### 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

Show / hide solution

### Problem 25J / 15S

A trapezoid has bases of lengths and and legs of lengths and

Show / hide solution

### 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?

Show / hide answerShow / hide solution

### Problem 27J / 17S

Parsley lives in Vegetable State where one can pay only by coins with values and

Show / hide solution

### 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.

Show / hide answerShow / hide solution

### Problem 29J / 19S

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

Show / hide answerShow / hide solution

### Problem 30J / 20S

How many

Show / hide solution

### Problem 31J / 21S

Consider a right triangle with sides of integer length. One of the sides has length

Show / hide solution

### 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

Show / hide solution

### Problem 33J / 23S

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

Show / hide solution

### 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

Show / hide solution

### Problem 35J / 25S

If we extend the sides and of a convex quadrilateral, they intersect at a point

Show / hide solution

### 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?

Show / hide answerShow / hide solution

### Problem 37J / 27S

We are given a circle with radius

Show / hide solution

### Problem 38J / 28S

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

Show / hide solution

### 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?

Show / hide answerShow / hide solution

### 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

Show / hide solution

### 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.

Show / hide answerShow / hide solution

### 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

Show / hide solution

### Problem 43J / 33S

A hitchhiker is walking along the road. The probability that a car picks him up in the next minutes is

Show / hide solution

### 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?

Show / hide answerShow / hide solution

### 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.

Show / hide solution

### Problem 46J / 36S

Matthew has one

Show / hide solution

### 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

Show / hide solution

### Problem 48J / 38S

We have a triangle with altitudes of sizes

Show / hide solution

### Problem 49J / 39S

Find the greatest integer for which is rational.

Show / hide answerShow / hide solution

### 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

Show / hide solution

### 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
.

Show / hide solution

### Problem 52J / 42S

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

Show / hide solution

### Problem 53J / 43S

Find the sum of all five-digit palindromes.

Show / hide answerShow / hide solution

### Problem 54J / 44S

How many ordered quadruples of odd positive integers satisfy

Show / hide solution

### 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.

Show / hide solution

### Problem 56J / 46S

Two different triangles with side lengths

Show / hide solution

### 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?

Show / hide answerShow / hide solution

### Problem 58J / 48S

Two distinct points and lie inside a square with side length

Show / hide solution

### Problem 59J / 49S

The parking lot consists of parking places regularly distributed in a row with numbers to

- 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).

Show / hide solution

### Problem 60J / 50S

Find all real numbers that satisfy

Show / hide answerShow / hide solution