KöMaL Problems in Mathematics, March 2026
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Problems with sign 'K'Deadline expired on April 10, 2026. |
K. 894. Knowing that \(\displaystyle \frac{x}{y}=\frac{4}{7}\) and \(\displaystyle \frac{y}{z}=\frac{14}{3}\) find \(\displaystyle \frac{x+y}{z}\).
(5 pont)
solution (in Hungarian), statistics
K. 895. The sum of a few distinct positive prime numbers is 40. What can be these prime numbers?
(5 pont)
solution (in Hungarian), statistics
K. 896. A circle with radius \(\displaystyle R\) is given. We draw the tangent lines to the circle from an external point \(\displaystyle P\). These two lines form a \(\displaystyle 60^\circ\) angle, and the circle is in one of the \(\displaystyle 60^\circ\) angles formed by the two lines. How many circles of unit radius can be drawn that is tangent to exactly two of the circle and the two lines, if
a) \(\displaystyle R=\)3; b) \(\displaystyle R=4\)?
(5 pont)
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Problems with sign 'K/C'Deadline expired on April 10, 2026. |
K/C. 897. We have two dice with six faces, both containing the numbers from 1 to 6. On one of the dice the sum of the opposite numbers is always the same, on the other one the difference of the opposite numbers is always the same. We rolled the dice, and the same number came out on top for both. What can be the second smallest and the second biggest value of the product of the eight numbers on the side faces of the two dice?
(5 pont)
solution (in Hungarian), statistics
K/C. 898. In parallelogram \(\displaystyle ABCD\) the angle at \(\displaystyle A\) is \(\displaystyle 60\) degrees, \(\displaystyle AD=20\) cm, \(\displaystyle AB=30\) cm. Let \(\displaystyle E\) be the trisection point of \(\displaystyle AB\) closer to \(\displaystyle B\). Let \(\displaystyle F\) and \(\displaystyle G\) be the midpoints of line segments \(\displaystyle ED\) and \(\displaystyle BC\), respectively. The circle with diameter \(\displaystyle ED\) intersects \(\displaystyle FG\) at \(\displaystyle H\). Find the length of line segment \(\displaystyle DH\).
(5 pont)
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Problems with sign 'C'Deadline expired on April 10, 2026. |
C. 1893. How many rectangular prisms are there with a volume of \(\displaystyle 2560~\mathrm{cm}^3\) where every edge has an integer length measured in centimeters.
Proposed by Márton Ujházy, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1894. Prove that if \(\displaystyle n\) is an integer, then the value of \(\displaystyle \frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}\) is also an integer.
Scottish competition problem
(5 pont)
solution (in Hungarian), statistics
C. 1895. The surface of a cube shaped small planet is covered in grass, its edges have a length of 1 km. A goat is tied to one of the vertives with a rope of length \(\displaystyle \sqrt{2}\) km. What ratio of the surface of the planet can be consumed by the goat?
Proposed by Márton Ujházy, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1896. Let \(\displaystyle E\) be the center of square \(\displaystyle ABCD\). A quarter circle with center \(\displaystyle D\) and radius \(\displaystyle DA\) inside the square intersects the semicircle with diameter \(\displaystyle AB\) inside the square at point \(\displaystyle F\). Line \(\displaystyle EF\) intersects line \(\displaystyle AB\) at \(\displaystyle M\), and the semicircle with diameter \(\displaystyle DA\) outside the circle at point \(\displaystyle N\). Prove that \(\displaystyle F\) is the midpoint of line segment \(\displaystyle MN\).
Proposed by Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1897. We choose three vertices of a regular \(\displaystyle n\)-gon such that the chance of picking any of the triples is the same. Prove that if n is even, then the probability of the triangle formed by the three vertices being obtuse is three times as big as the probability of it being acute.
Proposed by Márton Ujházy, Budapest
(5 pont)
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Problems with sign 'B'Deadline expired on April 10, 2026. |
B. 5518. Let \(\displaystyle n\) and \(\displaystyle k\) be positive integers. How many ways are there to partition the lattice trapezoid with vertices \(\displaystyle (0,0)\), \(\displaystyle (n,0)\), \(\displaystyle (k,1)\) and \(\displaystyle (0,1)\) into lattice triangles of area \(\displaystyle 1/2\)? (A polygon is called a lattice polygon if both coordinates of all its vertices are integers.)
Proposed by Attila Sztranyák, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5519. Viktor has drawn a parallelogram of unit area. Bálint measured one of its sides and one of its diagonals. What is the smallest possible value of the sum of these two measurements?
Proposed by Bálint Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5520. Two circles are given in the plane, lying outside each other. Two line segments with lengths \(\displaystyle d_1\) and \(\displaystyle d_2\) are also given. How many lines are there that intersect the first circle in a chord of length \(\displaystyle d_1\) and the second circle in a chord of length \(\displaystyle d_2\)?
Proposed by Géza Kiss, Csömör
(4 pont)
solution (in Hungarian), statistics
B. 5521. Positive integer \(\displaystyle k\) is given. Find all positive integers \(\displaystyle n\) such that in the prime factorization of \(\displaystyle \frac{(2^kn)!}{(n!)^{2^k}}\) the exponent of 2 is exactly \(\displaystyle 2^k-1\).
Proposed by Zoltán Bertalan, Békéscsaba
(5 pont)
solution (in Hungarian), statistics
B. 5522. Does there exist a sequence of positive integers \(\displaystyle a_1\), \(\displaystyle a_2\), \(\displaystyle \ldots\) and positive integer \(\displaystyle N\) such that for every \(\displaystyle n\geq N\), \(\displaystyle a_n<a_{n+1}<{a_1+a_2+\ldots+a_n}\) and \(\displaystyle a_{n+1}\mid a_1+a_2+\ldots+a_n\)?
Proposed by Sándor Róka, Nyíregyháza
(5 pont)
solution (in Hungarian), statistics
B. 5523. Find the largest possible number of distinct integer roots of polynomial \(\displaystyle p(x)=a_{12}x^{12}+a_{11}x^{11}+\ldots+a_2x^2+a_1x+a_0\) with integer coefficients if \(\displaystyle a_0+a_2+a_4+a_6+a_8+a_{10}+a_{12}=0\) and \(\displaystyle a_1+a_3+a_5+a_7+a_9+a_{11}=12\).
Proposed by Attila Sztranyák, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5524. In triangle \(\displaystyle ABC\), \(\displaystyle F\) is the midpoint of side \(\displaystyle BC\). A line intersects line segments \(\displaystyle AB\), \(\displaystyle AC\), \(\displaystyle AF\), in points \(\displaystyle X\), \(\displaystyle Y\), \(\displaystyle Z\), respectively. Prove that \(\displaystyle AB\cdot AX+AC\cdot AY>2AF\cdot AZ\).
Proposed by Ábel Szakács, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5525. For edge \(\displaystyle e\) in tetrahedron \(\displaystyle T\) let us define quantity \(\displaystyle r(e)\) as the ratio of the distance of the two heights of the tetrahedron from the endpoints of e and the length of edge \(\displaystyle e\). Let \(\displaystyle R(T)\) be defined as the sum of \(\displaystyle r(e)\) for all the edges of tetrahedron \(\displaystyle T\). Show an example for a tetrahedron \(\displaystyle T\) for which \(\displaystyle R(T)>5.999\).
Based on an idea of Zoltán Bertalan, Békéscsaba
(6 pont)
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Problems with sign 'A'Deadline expired on April 10, 2026. |
A. 929. Let \(\displaystyle \mathcal{P}\) be a set of ten points in the plane. A subset \(\displaystyle \mathcal{Q}\subseteq \mathcal{P}\) is said to be isolable if there exists a closed disk of positive integer radius that contains all points of \(\displaystyle \mathcal{Q}\) but none of the points of \(\displaystyle \mathcal{P}\setminus\mathcal{Q}\). Prove that at least one third of the five-element subsets of \(\displaystyle \mathcal{P}\) are not isolable.
Proposed by Áron Bán-Szabó, Palaiseau
(7 pont)
A. 930. Does there exist a tetrahedron such that the incenters of its faces are coplanar?
Proposed by Gábor Moussong, Budapest
(7 pont)
A. 931. Prove that for any integers \(\displaystyle k>10\) and \(\displaystyle n>k^4\), among the integers \(\displaystyle ({n+1^2})\), \(\displaystyle ({n+2^2})\), \(\displaystyle ({n+3^2})\), \(\displaystyle \ldots\), \(\displaystyle ({n+k^2})\), there is at least one that has a prime divisor greater than \(\displaystyle k\).
Proposed by Géza Kós, Budapest
(7 pont)
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