KöMaL Problems in Mathematics, October 2022
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Problems with sign 'K'Deadline expired on November 10, 2022. |
K. 734. Alex and his friends bought 6 bags of sunflower seeds and 4 bags of pumpkin seeds for 1900 HUF. Next week they bought 4 bags of sunflower seeds and 2 bags of pumpkin seeds for 1100 HUF. How much does a single bag of each type cost (assuming that the prices did not change during the week)?
(5 pont)
solution (in Hungarian), statistics
K. 735. Logic blocks were developed by Zoltán Dienes. Peter takes all the red and green disks and squares out of a set of blocks, altogether 16 pieces. No two pieces are identical, and they can be classified into two groups having the same number of elements according to each of the following properties:
– either small or large,
– either red or green,
– either a disk or a square,
– either hollow or not.
Can Peter place the 16 blocks along the perimeter of a circle such that any two neighbors have exactly one of the above properties in common?
(5 pont)
solution (in Hungarian), statistics
K. 736. A company has 120 employees: plumbers, tilers, bricklayers and painters. All plumbers and bricklayers have a driving license, the others do not. The bricklayers and painters work in Pipacs street, the others in Kankalin street. The number of employees without a driving license is 64, and 84 employees work in Kankalin street. There are twice as many plumbers as painters. How many employees of each kind are there at the company?
(5 pont)
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Problems with sign 'K/C'Deadline expired on November 10, 2022. |
K/C. 737. Given two threads of known length, we can measure and mark off the sum or difference of their lengths, and also the half length of a thread by folding it into two. We are given two threads of length 2240 cm and 1760 cm. Describe a procedure to mark off a length of 10 cm by using a single measurement (that is, measuring the sum or difference of lengths is allowed only once, but halving a length by folding can be performed many times).
(5 pont)
solution (in Hungarian), statistics
K/C. 738. In a certain calendar, the days of a month are arranged in 7 columns. Read from left to right and then from top to bottom, each column contains the same day of the week. For a certain integer \(\displaystyle n\), we select an \(\displaystyle n\times n\) square array of days and find that their sum is 198. What is the smallest number in this square?
(5 pont)
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Problems with sign 'C'Deadline expired on November 10, 2022. |
C. 1733. At most how many different positive prime divisors can a 3-digit number have, if its digits are consecutive positive integers in a certain order?
Based on the idea of Erzsébet Berkó, Szolnok
(5 pont)
solution (in Hungarian), statistics
C. 1734. A circle \(\displaystyle k\) with diameter \(\displaystyle AB\) has center \(\displaystyle O\). We draw the circle \(\displaystyle k_1\) with diameter \(\displaystyle OB\), and the line parallel with \(\displaystyle AB\) that touches the circle \(\displaystyle k_1\) at point \(\displaystyle C\). This line intersects the circle \(\displaystyle k\) at points \(\displaystyle D_1\) and \(\displaystyle D_2\). Determine the angles \(\displaystyle \angle COD_1\) and \(\displaystyle \angle COD_2\) exactly.
Proposed by Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1735. Find the real solutions of the system
$$\begin{align*} \sqrt{x}+\sqrt{y} & =6,\\ \frac{1}{x}+\frac{1}{y} & =\frac{5}{16}. \end{align*}$$(The Mathematical Association of America)
(5 pont)
solution (in Hungarian), statistics
C. 1736. Let \(\displaystyle P\) be an interior point of side \(\displaystyle CD\) of a parallelogram \(\displaystyle ABCD\), and let \(\displaystyle Q\) be an interior point of side \(\displaystyle AB\) (being parallel with \(\displaystyle CD\)). The line segments \(\displaystyle PA\) and \(\displaystyle QD\) intersect each other at \(\displaystyle M\), while the line segments \(\displaystyle PB\) and \(\displaystyle QC\) intersect each other at \(\displaystyle N\). Find a condition to have \(\displaystyle MN\parallel{AB}\).
(Based on a U.S. mathematics competition problem)
(5 pont)
solution (in Hungarian), statistics
C. 1737. Dick got two dice for his \(\displaystyle 32^\text{th}\) birthday. He labelled the faces of one die with the numbers \(\displaystyle 1, 2, \ldots, 6\), and the faces of the other one with \(\displaystyle 0, 1, 2, 7, 8, 9\). By using these dice, he can form all integers from 10 up to his age, 32, but the next number, 33, cannot be formed. Octavia uses two regular octahedra instead. Similarly, she wrote a digit on each face of both octahedra, so she can also form all integers from 10 up to her current age (in years), but not the next one. How old is Octavia now?
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
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Problems with sign 'B'Deadline expired on November 10, 2022. |
B. 5262. Louisa wrote down a natural number, not containing 0 but containing at least two different digits. Then she also listed all the numbers which can be formed by permuting the digits of the original number. What is the maximum of the greatest common divisor of all the numbers (including the original one)?
Proposed by Katalin Abigél Kozma, Győr
(3 pont)
solution (in Hungarian), statistics
B. 5263. Prove that the sum of the squares of the medians of a triangle is not less than the square of the semiperimeter of the triangle.
Proposed by László Németh, Fonyód
(3 pont)
solution (in Hungarian), statistics
B. 5264. First Player and Second Player play the following game. First Player starts and prescribes arbitrarily many (even infinitely many) terms of a binary sequence (i.e., any term is \(\displaystyle 0\) or \(\displaystyle 1\)) in a way that infinitely many terms can still be determined. Then Second Player sets the value of the first digit which has not been prescribed yet. They then repeat this procedure forever by taking turns. First Player wins if the binary sequence is periodic from a certain term, otherwise, Second Player wins. Is there a winning strategy, and if yes, who has it?
Proposed by Péter Pál Pach, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5265. Enlarge the incircle of a right-angled triangle by a scale factor of 2, where the center of enlargement is the vertex at the right angle. Show that this circle touches the circumcircle of the triangle.
Proposed by Viktor Vígh, Szeged
(4 pont)
solution (in Hungarian), statistics
B. 5266. Some football players are on holiday together. Altogether they are from \(\displaystyle k\) clubs and from \(\displaystyle n\) nations where \(\displaystyle k<n\). Show that there are at least \(\displaystyle n-k+1\) players having more club fellows than compatriots.
(5 pont)
solution (in Hungarian), statistics
B. 5267. We are given two line segments of length \(\displaystyle p\) and \(\displaystyle q\), and a triangle \(\displaystyle ABC\) determined by the lines \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) (where the usual convention is used: points \(\displaystyle B\) and \(\displaystyle C\) lie on line \(\displaystyle a\), and so on). Construct the point \(\displaystyle P\) on the circumcircle of the triangle for which the point \(\displaystyle P_a\) divides the line segment \(\displaystyle P_bP_c\) in a ratio \(\displaystyle p: q\), where \(\displaystyle P_x\) is the orthogonal projection of the point \(\displaystyle P\) onto the line \(\displaystyle x\).
(5 pont)
solution (in Hungarian), statistics
B. 5268. Let \(\displaystyle I\) denote the incenter of the triangle \(\displaystyle ABC\). Let \(\displaystyle P\) denote an arbitrary interior point of the triangle on the circle \(\displaystyle ABI\). The reflection of the line \(\displaystyle AP\) about the line \(\displaystyle AI\) intersects the circle \(\displaystyle ABI\) at a point \(\displaystyle Q\) different from the point \(\displaystyle A\). Prove that \(\displaystyle CP=CQ\).
Proposed by Szilveszter Kocsis, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5269. Let \(\displaystyle p \ge 19\) be an odd integer, and color the numbers \(\displaystyle 0,1,\dots,p-1\) with two colors. For \(\displaystyle 1\le i\le p\), let \(\displaystyle x_i\) denote a random element of the set \(\displaystyle \{0,1,\dots,p-1\}\) (the choices are independent, and have uniform distribution). Prove that the probability of the event that \(\displaystyle x_1,\dots,x_p\) have the same color and \(\displaystyle p\) divides \(\displaystyle x_1+\cdots + x_p\) is at least \(\displaystyle 3/(2^pp)\).
Proposed by Péter Pál Pach, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on November 10, 2022. |
A. 833. Some lattice points in the Cartesian coordinate system are colored red, the rest of the lattice points are colored blue. Such a coloring is called finitely universal, if for any finite, non-empty \(\displaystyle A \subset \mathbb{Z}\) there exists \(\displaystyle k\in \mathbb{Z}\) such that the point \(\displaystyle (x,k)\) is colored red if and only if \(\displaystyle x\in A\).
\(\displaystyle a)\) Does there exist a finitely universal coloring such that each row has finitely many lattice points colored red, each row is colored differently, and the set of lattice points colored red is connected?
\(\displaystyle b)\) Does there exist a finitely universal coloring such that each row has a finite number of lattice points colored red, and both the set of lattice points colored red and the set of lattice points colored blue are connected?
A set \(\displaystyle H\) of lattice points is called connected if, for any \(\displaystyle x,y\in H\), there exists a path along the grid lines that passes only through lattice points in \(\displaystyle H\) and connects \(\displaystyle x\) to \(\displaystyle y\).
Submitted by Anett Kocsis, Budapest
(7 pont)
A. 834. Let \(\displaystyle A_1A_2\ldots A_8\) be a convex cyclic octagon, and for \(\displaystyle i=1,2\ldots,8\) let \(\displaystyle B_i=A_iA_{i+3}\cap A_{i+1}A_{i+4}\) (indices are meant modulo 8). Prove that points \(\displaystyle B_1,\ldots,B_8\) lie on the same conic section.
(7 pont)
A. 835. A. 835. Let \(\displaystyle f^{(n)}(x)\) denote the \(\displaystyle n^\text{th}\) iterate of function \(\displaystyle f\), i.e \(\displaystyle f^{(1)}(x)=f(x)\), \(\displaystyle f^{(n+1)}(x)=f\Big(f^{(n)}(x)\Big)\).
Let \(\displaystyle p(n)\) be a given polynomial with integer coefficients, which maps the positive integers into the positive integers. Is it possible that the functional equation \(\displaystyle f^{(n)}(n)=p(n)\) has exactly one solution \(\displaystyle f\) that maps the positive integers into the positive integers?
Submitted by Dávid Matolcsi andKristóf Szabó, Budapest
(7 pont)
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