The traditional mathematics and recreation camp for the winners of the Kangaroo Mathematics competition took place from July 29 to August 7, 2025, at the Dafne II resort in Zakopane. This year, participants came from seven countries: China, Lithuania, Germany, Poland, the United States, Turkey, and Ukraine. Following past years, we welcomed representatives of the Polish communities from Lithuania and Ukraine. The official language of the camp was English, although participants also communicated in other languages.

Participants took part in workshops and mathematics competitions, hiking trips in the mountains, tourist trips to Kraków and Wieliczka, sports competitions, and team-building games. This camp preceded the 35th edition of the Kangaroo competition in Poland, and on this occasion, we refreshed the camp format and added a few historical elements.

Mathematics

Most of the mathematics activities were organized in groups: Bobiński, Jarek, and Uscki. The group names reminded us of the founders of the competition in Poland and the Zakopane camps. Participants in each group took part in lectures, workshops, and mathematics competitions.

Opening lecture

The mathematical part of the camp was opened by Witold Kraśkiewicz with a lecture titled How Much Does It Take to Calculate the Result? On Addition Chains. He addressed a very important aspect of applied mathematics — the time required to perform calculations, which, even for standard operations, can sometimes be surprisingly long…

Workshops

• Witold Kraśkiewicz, Selected topics in graph theory

  Despite their simple definition, graphs turn out to be surprisingly complex and interesting objects. We will discuss selected topics in graph theory (including trees, planar graphs, coloring).

• Agnieszka Krause, Inequalities in geometry

  In mathematics, we often deal with inequalities, but usually from an algebraic perspective. This time, we want to consider some of the basic inequalities in geometry — namely, the triangle inequality, the shortest path problem, and Ptolemy’s inequality. We also present some interesting applications of these inequalities.

• Mieczysław K. Mentzen, Polynomials and complex numbers

  The workshop concerns the application of complex numbers to the decomposition of polynomials with real coefficients into the product of polynomials irreducible over reals.

• Andrzej Sendlewski, Simson lines and curves for cyclic polygons

  We will explore how the concept of the Simson line can be extended from triangles to arbitrary polygons inscribed in a circle. For each such polygon, we consider the family of Simson lines corresponding to all points on the circumcircle. We investigate the relative positions of these lines, identifying when they are parallel or orthogonal. Moreover, we construct curves that are the loci of intersection points of pairs of Simson lines forming a fixed angle. These surprising and often intricate closed curves are introduced as Simson multi-circles.

• Robert Skiba, From Zero to Tangent: The Magic of Calculus Theorems

  Join us for an exciting international workshop where we explore the beauty and logic behind Bolzano's, Rolle's, and Lagrange's Theorems – the backbone of differential calculus!

• Mateusz Topolewski, Good Deal -- Math in Business

  To make a good deal, you need to earn as much as possible while spending as little as possible. Every company has its own business strategy for achieving this. Discover a mathematical method of production planning called linear programming. Find out if you have a knack for business too!

• Magdalena Wysokińska-Pliszka, Divisibility

  During the workshop we will bring up a topic of divisibility. In particular we will show how to determine if a number is divisible by another one or it is not. We are going to use various methods, including non typical rules of divisibility. Moreover, we intend to apply the concept in interesting problems.

Funny Mathematics

Mathematics is not only about formulas, theorems, and proofs. Camp participants had the chance to take part in slightly different workshop. Piotr Nodzyński — the last of the founders of Kangaroo in Poland who still actively co-creates the Zakopane camp — demonstrated that mathematics also has its entertaining side, known to us as logic games, puzzles, and brainteasers. During the workshop, he shared with the participants his impressive collection of games.

Competitions

The camp participants took part in two mathematics competitions.

• Problem of the Day was a team competition. For five consecutive days, in between other scheduled activities, participants worked on a given mathematics problem. The solution had to be submitted the next morning. The competition tasks required not only mathematical knowledge and skills, but also discussion and teamwork. The main prize was a dinner for the winning team with Piotr Nodzyński at Bubuja Bistro.

• Main Competition was the traditional contest in which participants, working within a limited time, tackled mathematics problems that could be solved more effectively using the skills acquired during the workshops.

Closing Lecture

At the closing ceremony, Mieczysław K. Mentzen gave a talk titled Solving Kangaroo Problems..., in which he showed that choosing the correct answer in a Kangaroo problem does not always require mathematical knowledge. Sometimes, it is enough to make use of the competition’s rules.

Tourism

Both the young mathematician and the older one sometimes need a break from the board, formulas, and solving successive problems. And there is no better way to relax than a walk in nature. Here, Zakopane offered us plenty of opportunities.

The Charms of Zakopane

There were three mountain trips during the camp: to Nosal, Gubałówka, and along the Trail above the Regls. Each offered slightly different experiences. Nosal allows us to feel the mountain atmosphere, giving a bird’s-eye view of Zakopane and its surroundings. Of course after a climb to the summit. Gubałówka is an interesting tourist spot where we can get closer to the Highlander culture. Meanwhile, the Trail above the Regls offers a view of the mountains from below and guarantees a pleasant walk among the trees, with the mountains in the background.

Zakopane Thermal Baths offer an excellent experience for those who want to let their muscles relax. After the mountain trips, the Kangaroos went there to loosen their muscles and rejuvenate their bodies. An additional attraction of the Zakopane Thermal Baths is the outdoor rooftop pool, from which visitors can enjoy a beautiful view of the Tatra mountains.

Integration games

During one of the first evenings, participants had the chance to get to know each other better through team-building games organized at the Dafne resort. Forming international teams, the participants tackled challenges set by the camp Leaders. The activities included a bit of mathematics, a bit of computer science, but above all, plenty of fun.

During the camp, there was also a dance evening. Participants tried their hand at dancing the Macarena and other popular dances.

Cracow and Wieliczka

The Kangaroos took part in two full-day bus trips: to the Wieliczka Salt Mine and to Cracow.

There is probably no need to say much about Cracow. The historic capital of Poland, with Wawel Hill and a picturesque old town, is full of history and legends. The Kangaroos learned about some of them during a guided walk. There were also stops for souvenirs and local delicacies. This time we didn’t manage to visit the Wawel Dragon, but maybe next year...

The Wieliczka Salt Mine is one of the world’s major tourist attractions. Since 1978, it has been on the UNESCO World Cultural and Natural Heritage List. The mine has existed for over 700 years and was operational until June 30, 1996. Since then, it has been a purely tourist site, visited by nearly a million people annually. The tourist route includes 20 historic chambers connected by corridors totaling approximately 2 km. At the end of this route, at a depth of 135 meters, is the Cracow Saltworks Museum. Camp participants admired exhibits showcasing the mine’s history, the tools used for salt extraction and transport, and the history of the city.

Games and Sports

Although chess has been known since the 6th century AD, it only became an official sport in 1999. It is a game with many variations, differing in how the matches are played. During the camp, participants took part in a traditional individual tournament. At the camp, chess also serves as a mental exercise, promotes logical thinking, and helps integrate participants through healthy competition.

Table tennis tournaments also took place during the camp. Singles tournaments for girls and boys were held using the Portuguese system — a player is eliminated only after losing two games.

Young Talents Competition

On Tuesday evening, a gala was held where participants showcased their talents. They demonstrated their skills in magic, dancing, and singing.

Kangaroo That Brings Us Together

The Kangaroo competition, like all of mathematics, entertains, teaches, and leaves memories. Former camp participants return to Zakopane years later as leaders of their national teams. One of them is Arpit Ranasaria from the United States. He first came to the Zakopane camp in 2017 as a young math enthusiast. Even then, he felt right at home and received distinctions in mathematics competitions (evidence can be found in this year’s photo gallery, which we encourage you to check out).

After the camp, Arpit got back home and graduated from school. He remained actively involved in math education and even organizes his own math competition in the USA. This year, we met Arpit again, this time in Zakopane as a Leader of the American team. Thanks to his passion and dedication, he was the kind of leader every team needs. Thank you, Arpit!

We also thank the other leaders and lecturers who contributed to the camp in Zakopane. Thanks to your dedication, the map of valuable mathematics camps has gained another location. See you next year!