To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.

Anett is drawing $X$-es on a $5 \times 5$ grid. For each newly drawn $X$ she gets points in the following way: She checks how many $X$-es there are in the same row (including the new one) that can be reached from the newly drawn $X$ with horizontal steps, moving only on fields that were previously marked with $X$-es. For the vertical $X$-es, she gets points the same way. a) What is the maximum number of points that she can get with drawing $25$ $X$-es? b) What is the minimum number of points that she can get with drawing $25$ $X$-es? For example, if Anett put the $X$ on the field that is marked with the circle, she would get $3$ points for the horizontal fields and $1$ point for the vertical ones. Thus, she would get $4$ points in total.

$n$ students, numbered from $1$ to $n$ are sitting next to each other in a class. In the beginning the $1$st student has $n$ pieces of paper in one pile. The goal of the students is to distribute the $n$ pieces in a way that everyone gets exactly one. The teacher claps once in a minute and for each clap the students can choose one of the following moves (or do nothing): $\bullet$ They divide one of their piles of paper into two smaller piles. $\bullet$ They give one of their piles of paper to the student with the next number. At least how many times does the teacher need to clap in order to make it possible for the students to distribute all the pieces of paper amongst themselves?

Show that the divisors of a number $n \ge 2$ can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of $n$ is a square number.

Annie drew a rectangle and partitioned it into $n$ rows and $k$ columns with horizontal and vertical lines. Annie knows the area of the resulting $n \cdot k$ little rectangles while Benny does not. Annie reveals the area of some of these small rectangles to Benny. Given $n$ and $k$ at least how many of the small rectangle’s areas did Annie have to reveal, if from the given information Benny can determine the areas of all the $n \cdot k$ little rectangles? For example in the case $n = 3$ and $k = 4$ revealing the areas of the $10$ small rectangles if enough information to find the areas of the remaining two little rectangles.

In duck language, only letters $q$, $a$, and $k$ are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there? In duck language, too, the letter $a$ is a vowel, while $q$ and $k$ are consonants.

Csaba stands in the middle of a $15$ m $\times 15$ m room at a workplace where everyone strictly adheres to $1,5$ m social distancing. At least how many people are there other than Csaba in the room if Csaba cannot reach any wall without the others moving? The people are viewed as points.

Three palaces, each rotating on a duck leg, make a full round in $30$, $50$, and $70$ days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards?

At least how many regular triangles are needed to cover the lines of the following diagram? (Only the perimeter of the triangles is involved in the covering, and the entire perimeter need not be incident on the diagram.)

Benedek draws circles with the same center in the following way. The first circle he draws has radius $1$. Next, he draws a second circle such that the ring between the first and second circles has twice the area of the first circle. Next, he draws a third circle such that the ring between the second and third circles is three times the area of the first circle, and so on (see the diagram). What is the smallest $n$ fow which the radius of the $n$-th circle is an integer greater than $1$?

In Kacs Aladár street, houses are only found on one side of the road, so that only odd house numbers are found along the street. There are an odd number of allotments, as well. The middle three allotments belong to Scrooge McDuck, so he only put up the smallest of the three house numbers. The numbering of the other houses is standard, and the numbering begins with $1$. What is the largest number in the street if the sum of house numbers put up is $3133$?

The fragments of a positive integer are the numbers seen when reading one or more of its digits in order. The fragment sum equals the sum of all the fragments, including the number itself. For example, the fragment sum of $2022$ is $2022+202+022+20+02+22+2+0+2+2 = 2296$. There is another four-digit number with the same fragment sum. What is it? As the example shows, if a fragment occurs multiple times, then all its occurrences are added, and the fragments beginning with $0$ also count (for instance, $022$ is worth $22$).

The product of Albrecht’s three favorite numbers is $2022$, and if we add one to each number, their product will be $1514$. What is the sum of their squares, if we know their sum is $0$?

Every side of a right triangle is an integer when measured in cm, and the difference between the hypotenuse and one of the legs is $75$ cm. What is the smallest possible value of its perimeter?

The pair of positive integers $(a, b)$ is such that a does not divide $b$, $b$ does not divide a, both numbers are at most $100$, and they have the maximal possible number of common divisors. What is the largest possible value of $a \cdot· b$?

In rectangle $ABCD$, diagonal $AC$ is met by the angle bisector from $B$ at $B'$ and the angle bisector from $D$ at $D'$. Diagonal $BD$ is met by the angle bisector from $A$ at $A'$ and the angle bisector from $C$ at $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ the area of rectangle $ABCD$. What is the ratio of the longer side and shorter side of rectangle $ABCD$?

Csongi taught Benedek how to fold a duck in 8 steps from a $24$ cm $\times 24$ cm piece of paper. The paper is meant to be folded along the dashed line in the direction of the arrow. Once Benedek folded the duck, he undid all the steps, finding crease lines on the square sheet of paper. On one side of the paper, he drew in blue the folds which opened towards Benedek, and in red the folds which opened toward the table. How many cm is the difference between the total length of the blue lines and the red lines?

Write some positive integers in the following table such that $\cdot$ there is at most one number in each field $\cdot$ each number is equal to how many numbers there are in edge-adjacent fields, $\cdot$ edge-adjacent fields cannot have equal numbers. What is the sum of numbers in the resulting table?

In Durer’s duck school, there are six rows of doors, as seen on the diagram; both rows are made up of three doors. Dodo duck wishes to enter the school from the street in a way that she uses all six doors exactly once. (On her path, she may go to the street again, or leave the school, so long as she finishes her path in the school.) How many ways can she perform this? Two paths are considered different if Dodo takes the doors in a different order.

Doofy duck buy tangerines in the store. All tangerines have equal weight and are divided into $9$, $10$, $11$, $12$, or $13$ equal wedges, although this cannot be seen without peeling them. How many tangerines does Doofy duck need to buy if he wishes to eat exactly one tangerine’s worth while eating at most one wedge from every tangerine? Doofy duck only peels the tangerines at home.

The number $60$ is written on a blackboard. In every move, Andris wipes the numbers on the board one by one, and writes all its divisors in its place (including itself). After $10$ such moves, how many times will $1$ appear on the board?

Let $c \ge 2$ be a fixed integer. Let $a_1 = c$ and for all $n \ge 2$ let $a_n = c \cdot \phi (a_{n-1})$. What are the numbers $c$ for which sequence $(a_n)$ will be bounded? $\phi$ denotes Euler’s Phi Function, meaning that $\phi (n)$ gives the number of integers within the set $\{1, 2, . . . , n\}$ that are relative primes to $n$. We call a sequence $(x_n)$ bounded if there exist a constant $D$ such that $|x_n| < D$ for all positive integers $n$.

Let $x, y, z$ denote positive real numbers for which $x+y+z = 1$ and $x > yz$, $y > zx$, $z > xy$. Prove that $$\left(\frac{x - yz}{x + yz}\right)^2+ \left(\frac{y - zx}{y + zx}\right)^2+\left(\frac{z - xy}{z + xy}\right)^2< 1.$$

$ABCD$ is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let $O$ denote the centre of its circumcircle and $E$ the intersection of the diagonals. $J$ and $K$ denote the perpendicular projections of $E$ on the sides $AB$ and $BC$ . Let $F , G$ and $H$ be the midpoint line segments. Show that lines $GJ$ , $FB$ and $HK$ either pass through the same point or are parallel to each other.

$n$ people sitting at a round table. In the beginning, everyone writes down a positive number $n$ on piece of paper in front of them. From now on, in every minute, they write down the number that they get if they subtract the number of their right-hand neighbour from their own number. They write down the new number and erase the original. Give those number $n$ that there exists an integer $k$ in a way that regardless of the starting numbers, after $k$ minutes, everyone will have a number that is divisible by $n$.

How many $10$-digit sequences are there, made up of $1$ four, $2$ threes, $3$ twos, and $4$ ones, in which there is a two in between any two ones, a three in between any two twos, and a four in between any two threes?

On a circle $k$, we marked four points $(A, B, C, D)$ and drew pairwise their connecting segments. We denoted angles as seen on the diagram. We know that $\alpha_1 : \alpha_2 = 2 : 5$, $\beta_1 : \beta_2 = 7 : 11$, and $\gamma_1 : \gamma_2 = 10 : 3$. If $\delta_1 : \delta_2 = p : q$, where $p$ and $q$ are coprime positive integers, then what is $p$?

Circle $k_1$ has radius $10$, externally touching circle $k_2$ with radius $18$. Circle $k_3$ touches both circles, as well as the line $e$ determined by their centres. Let $k_4$ be the circle touching $k_2$ and $k_3$ externally (other than $k_1$) whose center lies on line $e$. What is the radius of $k_4$?

Benedek scripted a program which calculated the following sum: $1^1+2^2+3^3+. . .+2021^{2021}$. What is the remainder when the sum is divided by $35$?

An ant crawls along the grid lines of an infinite quadrille notebook. One grid point is marked red, this is its starting point. Every time the ant reaches a grid point, it continues forward with probability $\frac13$ , left with probability $\frac13$ , and right with probability $\frac13$. What is the chance that it is after its third turn, but not after its fourth turn that it returns to the red point? If the answer is $\frac{p}{q}$ , where $p$ and $q$ are coprime positive integers, then your answer should be $p + q$. The steps of the ant are independent.