KöMaL Problems in Mathematics, December 2022
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Problems with sign 'K'Deadline expired on January 10, 2023. |
K. 744. I have 1000 forints (HUF, Hungarian currency) in my pocket. If I buy two sandwiches and one soft drink then I will have as many forints remaining as the amount I should add to my 1000 in order to be able to buy three sandwiches and one soft drink. Two sandwiches and two soft drinks cost 1100 forints.
What is the price of one sandwich, and what is the price of one soft drink?
(5 pont)
solution (in Hungarian), statistics
K. 745. In a game, players are collecting points. The players take turns in playing and scoring points. When a certain player is playing, he or she may get any non-negative integer of points (including 0). The points scored by a player in successive turns add up. The game terminates when the total of the points scored by the players reaches 1000 (that is, the last player may only score as many points as needed to make the total equal to 1000). The player with the largest number of points will win the game. In the case of equality, the player reaching the same score earlier will win. The player with the second largest number of points will finish in the second place, and so on.
At the moment, the scores of the players are as follows: Kate has 314 points, Sam has 207 points, John has 58 points, Gillian has 31 and Joe has 0.
\(\displaystyle a)\) If it is Kate's turn now, what is the minimum number of points she needs to gain in this turn in order to be certain that she will finish in the first or second place?
\(\displaystyle b)\) If it is Sam's turn now, what is the minimum number of points he needs to gain in this turn in order to be certain that he will finish in the first or second place?
\(\displaystyle c)\) If it is Joe's turn now, what is the minimum number of points he needs to gain in this turn in order to be certain that he will finish in the first or second place?
(5 pont)
solution (in Hungarian), statistics
K. 746. In a certain small country, citizens are charged for gas consumption as follows: in the first year of the new regulations, the first 1700 m\(\displaystyle {}^3\) of gas costs 100 pennies/m\(\displaystyle {}^3\), and any further consumption costs 750 pennies/m\(\displaystyle {}^3\). From the following year onwards, the quantity sold for the reduced price is determined each year from the nationwide average consumption of the previous year. John Average lives in this country, and he used gas with these conditions for a year. Then he realized that he was paying too much, and decided to economise with the gas used in his household. He succeeded in reducing his consumption by 10% in the following year. However, since everyone was trying to save money, the mean consumption decreased by 15%. Although John Average used less gas than previously, he still had to pay more (for the whole year) than in the previous year. What may have been John's consumption during the first year?
(5 pont)
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Problems with sign 'K/C'Deadline expired on January 10, 2023. |
K/C. 747. A forty-sided polygon is divided into two polygons by one of its diagonals. The two polygons altogether have 298 fewer diagonals than the forty-gon. How many sides do these polygons have?
(5 pont)
solution (in Hungarian), statistics
K/C. 748. The integers 1 to 100 are written along the circumference of a circle. First every even number is connected to each odd number that is smaller than the even number. Then every odd number is connected to each even number that is smaller than the odd number. How many connecting lines are drawn?
(5 pont)
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Problems with sign 'C'Deadline expired on January 10, 2023. |
C. 1743. Seven natural numbers form an arithmetic sequence with a common difference of \(\displaystyle 30\). Show that exactly one of the numbers is divisible by \(\displaystyle 7\).
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1744. In a triangle \(\displaystyle ABC\), \(\displaystyle \angle CAB =45^{\circ}\) and \(\displaystyle \angle ABC =60^{\circ}\). \(\displaystyle D\) is a point of the line segment \(\displaystyle AB\). The circumscribed circle of triangle \(\displaystyle CAD\) passes through the orthocentre of triangle \(\displaystyle ABC\). Without the use of trigonometry, find the exact value of the ratio \(\displaystyle \frac{AD}{BD}\).
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1745. Solve the equation \(\displaystyle x^2+8x-y=\frac{y-5}{y+6}\), where \(\displaystyle x\), \(\displaystyle y\) are integers.
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1746. Side \(\displaystyle AB\) of a square \(\displaystyle ABCD\) is extended beyond point \(\displaystyle A\) by the line segment \(\displaystyle AE=2\), and beyond point \(\displaystyle B\) by the line segment \(\displaystyle BF=3\). Lines \(\displaystyle ED\) and \(\displaystyle FC\) enclose an angle of \(\displaystyle 45^{\circ}\). Determine the possible values of the length of the sides of the square.
Proposed by L. Németh, Fonyód
(5 pont)
solution (in Hungarian), statistics
C. 1747. Let \(\displaystyle n \ge 3\) be a positive integer, and let \(\displaystyle k\) be the sum of the digits in the number \(\displaystyle 10^n-4!\). What is the sum of the digits in the number \(\displaystyle \frac{10^{n+1}-7}{3}\)?
Proposed by M. Szalai, Szeged
(5 pont)
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Problems with sign 'B'Deadline expired on January 10, 2023. |
B. 5278. In Noname School of Nowhere, the graduating class consists of four groups, each specializing in a different subject: math, physics, chem and bio. On day, they are all sitting around a large round table in the school cafeteria: everyone has someone sitting directly opposite them, and everyone has neighbours sitting right next to them on either side. However a student is selected, it is observed that this student, the two neighbours, and the one sitting straight across the table all belong to different groups. How many students may there be in the graduating class if there are less than \(\displaystyle 20\) of them?
Proposed by K. A. Kozma, Győr
(3 pont)
solution (in Hungarian), statistics
B. 5279. Two circles are drawn inside a right angle. One circle touches one of the arms at point \(\displaystyle A\), and the other circle touches the other arm at point \(\displaystyle B\). The circles also touch each other at point \(\displaystyle C\). What is the measure of \(\displaystyle \angle ACB\)?
Proposed by B. Bíró, Eger
(3 pont)
solution (in Hungarian), statistics
B. 5280. Let \(\displaystyle a>2\), \(\displaystyle b\) and \(\displaystyle c\) be real numbers. Consider the three statements below.
(1) The equation \(\displaystyle ax^2 + bx + c = 0\) has no real solution.
(2) The equation \(\displaystyle (a-1)x^2 + (b-1)x + (c-1) = 0\) has 1 real solution.
(3) The equation \(\displaystyle (a-2) x^2 + (b-2)x + (c-2) = 0\) has 2 real solutions.
\(\displaystyle a)\) Given that statements (1) and (2) are true, can we conclude that statement (3) is also true?
\(\displaystyle b)\) Given that statements (2) and (3) are true, can we conclude that statement (1) is also true?
Proposed by B. Hujter, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5281. Let \(\displaystyle d > 1\) denote a positive integer. Prove that there exists a positive integer that has the same number of factors divisible by \(\displaystyle d\) as factors not divisible by \(\displaystyle d\).
Proposed by B. Hujter, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5282. In an acute-angled triangle \(\displaystyle ABC\), the foot of the altitude drawn from vertex \(\displaystyle A\) is \(\displaystyle T\). The projection of \(\displaystyle T\) on side \(\displaystyle AB\) is \(\displaystyle D\), and its projection on side \(\displaystyle AC\) is \(\displaystyle E\). Let \(\displaystyle F\) be the intersection of side \(\displaystyle BC\) and the circle \(\displaystyle ABE\) that is different from \(\displaystyle B\). Analogously, let \(\displaystyle G\) be the intersection of side \(\displaystyle BC\) and circle \(\displaystyle ACD\), different from \(\displaystyle C\). Show that \(\displaystyle TF=TG\).
Proposed by G. Kós, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5283. A convex quadrilateral \(\displaystyle N\) contains a circular disc of radius \(\displaystyle r\). Show that the perimeter of \(\displaystyle N\) is at least \(\displaystyle 8r\).
Proposed by V. Vígh, Sándorfalva
(4 pont)
solution (in Hungarian), statistics
B. 5284. Let \(\displaystyle n>2\). Alan has selected an edge of the complete graph of \(\displaystyle 2n\) vertices. Paula wants to find out which. If she pays 1 forint (Hungarian currency, HUF) she may name any pairing of all vertices and ask whether the selected edge is contained in it. What is the minimum number of forints that Paula needs to have in her pockets in order to be certain that she can find out the selected edge by asking the appropriate questions?
Proposed by P. P. Pach, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5285. In an acute-angled triangle \(\displaystyle ABC\), \(\displaystyle AB=AC\). Points \(\displaystyle A'\), \(\displaystyle B'\) and \(\displaystyle C'\) are moving along the circumscribed circle of the triangle, so that triangle \(\displaystyle A'B'C'\) always remain congruent to triangle \(\displaystyle ABC\), and have the same orientation. Let \(\displaystyle P\) be the intersection of lines \(\displaystyle BB'\) and \(\displaystyle CC'\). Show that the lines \(\displaystyle A'P\) all pass through a certain point.
Proposed by G. Kós, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on January 10, 2023. |
A. 839. We are given a finite, simple, non-directed graph. Ann writes positive real numbers on each edge of the graph such that for all vertices the following is true: the sum of the numbers written on the edges incident to a given vertex is less than one. Bob wants to write non-negative real numbers on the vertices in the following way: if the number written at vertex \(\displaystyle v\) is \(\displaystyle v_0\), and Ann's numbers on the edges incident to \(\displaystyle v\) are \(\displaystyle e_1,e_2, \dots, e_k\), and the numbers on the other endpoints of these edges are \(\displaystyle v_1, v_2, \dots, v_k\), then \(\displaystyle v_0=\sum\limits_{i=1}^{k}e_iv_i+2022\). Prove that Bob can always number the vertices in this way regardless of the graph and the numbers chosen by Ann.
Proposed by Boldizsár Varga, Verőce
(7 pont)
A. 840. The incircle of triangle \(\displaystyle ABC\) touches the sides in \(\displaystyle X\), \(\displaystyle Y\) and \(\displaystyle Z\). In triangle \(\displaystyle XYZ\) the feet of the altitude from \(\displaystyle X\) and \(\displaystyle Y\) are \(\displaystyle X'\) and \(\displaystyle Y'\), respectively. Let line \(\displaystyle X'Y'\) intersect the circumcircle of triangle \(\displaystyle ABC\) at \(\displaystyle P\) and \(\displaystyle Q\). Prove that points \(\displaystyle X\), \(\displaystyle Y\), \(\displaystyle P\) and \(\displaystyle Q\) are concyclic.
Proposed by László Simon, Budapest
(7 pont)
A. 841. Find all non-negative integer solutions of the equation \(\displaystyle 2^a+p^b=n^{p-1}\), where \(\displaystyle p\) is a prime number.
Proposed by Máté Weisz, Cambridge
(7 pont)
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