KöMaL Problems in Mathematics, November 2023
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Problems with sign 'K'Deadline expired on December 11, 2023. |
K. 784. Let us replace the letters below with digits \(\displaystyle 0\)–\(\displaystyle 9\) (with the exception of one digit) such that the result of the subtraction of the two 3-digit numbers is the closest possible to \(\displaystyle 300\):
\(\displaystyle ABC - DEF = GHJ. \)
Prove that the result that has the smallest difference from \(\displaystyle 300\) can only be obtained in an unique way. (Different letters has to be replaced with different digits.)
(5 pont)
solution (in Hungarian), statistics
K. 785. In the shops of three merchants Ali, Selim and Khafim a barrel of olives had the same price at the beginning of June. Ali increased the price by 10%, then by 10% again, and then at the beginning of September decreased the price by 20%. Selim increased the price by 20%, then decreased it by 10%, and then at the beginning of September decreased it by 10% again. Khafim increased the price by 20%, and then at the beginning of September decreased it by 20%. We know that at the beginning of September Ali sold a barrel of Olives 4 dinars cheaper than Selim. How much did a barrel of olives cost at the beginning of September in Khafim's shop?
(5 pont)
solution (in Hungarian), statistics
K. 786. Let \(\displaystyle X\) denote the sum of the squares of the first \(\displaystyle 50\) positive integers. Express the sum of the squares of the first \(\displaystyle 50\) positive even integers in terms of \(\displaystyle X\).
(5 pont)
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Problems with sign 'K/C'Deadline expired on December 11, 2023. |
K/C. 787. How many points of intersection can the diagonals of a convex 16-gon have, if the points of intersections are all distinct?
(5 pont)
solution (in Hungarian), statistics
K/C. 788. Sequence \(\displaystyle a_n\) satisfies \(\displaystyle a_1=2\) and \(\displaystyle a_{n+1} = a_n + 2n\). Find the value of \(\displaystyle a_{100}\).
(5 pont)
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Problems with sign 'C'Deadline expired on December 11, 2023. |
C. 1783. Let \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) be positive integers such that the open interval \(\displaystyle \left(\frac{a}{b},\frac{c}{d}\right)\) includes \(\displaystyle 1\). Prove that
\(\displaystyle \frac{a}{b}<\frac{a+c+1}{b+d+1}<\frac{c}{d}. \)
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1784. The shorter leg of right triangle \(\displaystyle ABC\) has length \(\displaystyle 1\). The ratio of the angles \(\displaystyle \varphi\) and \(\displaystyle \varepsilon\) between the height from the right angle (perpendicular to hypotenuse \(\displaystyle AB\)) and the angle bisectors of the acute angles is
\(\displaystyle \frac{\varphi}{\varepsilon}=\frac{4}{5}. \)
Find the angles of the triangle and the length of the height corresponding to the hypotenuse.
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1785. Find all pairs \(\displaystyle (x;y)\) of real numbers that satisfy the system of equations
\(\displaystyle \frac{10}{1+|x|}+y=4;\quad \frac{10}{1+|y|}+x=4 \)
German competition problem
(5 pont)
solution (in Hungarian), statistics
C. 1786. We roll five four-sided tetrahedral dies. Find the probability that the resulting five numbers can be the degrees of a tree on five vertices.
Proposed by B. Kovács, Szombathely
(5 pont)
solution (in Hungarian), statistics
C. 1787. In acute triangle \(\displaystyle ABC\) we draw a parallel line through orthocenter \(\displaystyle M\) with side \(\displaystyle AB\), which intersects sides \(\displaystyle AC\) and \(\displaystyle BC\) in points \(\displaystyle D\) and \(\displaystyle E\), respectively. Let \(\displaystyle CC_1\) be the diameter of the circumcircle of triangle \(\displaystyle ABC\). Find the perimeter of triangle \(\displaystyle DEC_1\), given that \(\displaystyle AB=14\).
Inspired by Kvant
(5 pont)
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Problems with sign 'B'Deadline expired on December 11, 2023. |
B. 5342. Let us pick four consecutive integers of the same parity, and take the products of all possible pairs. Prove that the sum of these products cannot be a perfect square.
Proposed by G. Kiss, Csömör
(3 pont)
solution (in Hungarian), statistics
B. 5343. Decide which one of numbers
$$\begin{align*} A & = 1! - 2! + 3! - 4! + \ldots + 2021! - 2022! + 2023! \quad\text{and}\\ B & =(1-2+3-4+\ldots+2021-2022+2023)! \end{align*}$$is the bigger.
Proposed by B. Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5344. Anti and Bandi would like to get from Balatonmáriafürdő to Balatonlelle, which is 30 kilometers away, partly on foot and partly by bicycle. They start at the same time, and they only have a single bicycle. Anti's speed by bicycle is 30 km/h, and he can run with a speed of 15 km/h. Bandi's speed by bicycle is 20 km/h, and he can run with a speed of 12 km/h. Find the least amount of time they need to get to their destination in minutes. (During their trip they can swap the bicycle with each other as many times as they want, and the bicycle can be safely left at the side of the road.)
Proposed by P.\(\displaystyle \,\)P. Pach, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5345. Prove that if the incircle and the nine-point circle of a triangle are concentric, then the triangle is equilateral.
Proposed by G. Kiss, Csömör
(4 pont)
solution (in Hungarian), statistics
B. 5346. For which \(\displaystyle n\) is it possible to find an equilateral (all sides have the same length) \(\displaystyle n\)-gon such that each of its sides is parallel to exactly two other sides?
Proposed by B. Hujter, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5347. Prove that if \(\displaystyle r\) is a positive rational number, and \(\displaystyle r^r\) is also rational, then \(\displaystyle r\) is an integer.
Proposed by Cs. Sándor, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5348. Let \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle n\) be non-negative integers satisfying \(\displaystyle 0\le c\le b-2n\). Prove that
\(\displaystyle \sum_{a=0}^n \binom{2a}{a}\binom{b-2a}{n-a} = \sum_{a=0}^n \binom{2a+c}{a}\binom{b-c-2a}{n-a}. \)
Proposed by L. Tóthmérész, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5349. The edges of parallelepiped \(\displaystyle P\) all have lengths that are at most \(\displaystyle 1\). Let \(\displaystyle X\) be an arbitrary point of \(\displaystyle P\). Show that we can find a vertex of \(\displaystyle P\) such that its distance from \(\displaystyle X\) is at most \(\displaystyle \sqrt 3 /2\).
Proposed by V. Vígh, Sándorfalva
(6 pont)
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Problems with sign 'A'Deadline expired on December 11, 2023. |
A. 863. Let \(\displaystyle n\ge 2\) be a given integer. Find the greatest value of \(\displaystyle N\), for which the following is true: there are infinitely many ways to find \(\displaystyle N\) consecutive integers such that none of them has a divisor greater than 1 that is a perfect \(\displaystyle n^{\mathrm{th}}\) power.
Proposed by Péter Pál Pach, Budapest
(7 pont)
A. 864. Let \(\displaystyle ABC\) be a triangle and \(\displaystyle O\) be its circumcenter. Let \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\) be the respective tangent points of the incircle of \(\displaystyle \triangle ABC\), and sides \(\displaystyle BC\), \(\displaystyle CA\) and \(\displaystyle AB\). Let \(\displaystyle M\) and \(\displaystyle N\) be the respective midpoints of sides \(\displaystyle AB\) and \(\displaystyle AC\). Let \(\displaystyle M'\) and \(\displaystyle N'\) be the respective reflections of points \(\displaystyle M\) and \(\displaystyle N\) across lines \(\displaystyle DE\) and \(\displaystyle DF\). Let lines \(\displaystyle CM'\) and \(\displaystyle BN'\) intersect lines \(\displaystyle DE\) and \(\displaystyle DF\) at points \(\displaystyle H\) and \(\displaystyle J\), respectively.
Prove that the points \(\displaystyle H\), \(\displaystyle J\) and \(\displaystyle O\) are collinear.
Proposed by Luu Dong, Vietnam
(7 pont)
A. 865. A crossword is a grid of black and white cells such that every white cell belongs to some \(\displaystyle 2 \times 2\) square of white cells. A word in the crossword is a contiguous sequence of two or more white cells in the same row or column, delimited on each side by either a black cell or the boundary of the grid.
Show that the total number of words in an \(\displaystyle n\times n\) crossword cannot exceed \(\displaystyle \frac{{(n+1)}^2}{2}\).
Proposed by Nikolai Beluhov, Bulgaria
(7 pont)
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