Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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KöMaL Problems in Mathematics, November 2014

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Problems with sign 'K'

Deadline expired on December 10, 2014.

K. 433. We have four objects that look all alike, but weigh 3, 5, 8 and 11 kg. We also have an equal-arm balance without extra weights.

$\displaystyle a)$ By using the balance only twice, how can we identify the 11-kg object?

$\displaystyle b)$ With two further measurements, how can we determine the masses of the other objects, too?

(6 pont)

solution, statistics

K. 434. Laura noticed that by subtracting 16 times her age 16 years ago from her age 16 years from now, she would get her age 16 years ago. How old is she now?

(6 pont)

solution, statistics

K. 435. A class of 30 students were given a test. Those passing the test scored 21 points on average. The average score of those failing was 15. The mean score of the whole class was 20. How many students passed the test?

(6 pont)

solution, statistics

K. 436. How many perfect squares are there in the sequence $\displaystyle a_{n}= 1! + 2! + \ldots + n!$? ($\displaystyle k!$ denotes the product of the integers 1 to $\displaystyle k$.)

(6 pont)

solution, statistics

K. 437. Every digit in the product below is one of the digits 2, 3, 5, 7. Determine which digit each letter represents. (Different letters may stand for the same digit, but each digit must be one of the four listed.)

(6 pont)

solution, statistics

K. 438. Leslie took two tests. The maximum achievable score on each test was 100. When he checked the results, he saw that he had the second best score on each of the two tests. Considering the combined score of both tests, what may be his rank?

(6 pont)

solution, statistics

Problems with sign 'C'

Deadline expired on December 10, 2014.

C. 1252. The product of four consecutive odd numbers ends in 9. What may be the digit preceding the 9 in the product?

(Matlap, Cluj)

(5 pont)

solution, statistics

C. 1253. Prove that if the measure of each side of a right-angled triangle is an integer, then the area of the triangle formed by the right-angled vertex and the two points dividing the hypotenuse into three equal parts is also an integer.

(5 pont)

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C. 1255. Determine the four-digit number $\displaystyle \overline{abcd}$, given that in the equation $\displaystyle {n^4=\overline{a6b\;c4d\;641}}$ $\displaystyle n$ denotes a positive integer with the sequence of digits increasing left to right.

(5 pont)

solution, statistics

C. 1256. A king wants to give a convict another chance to escape prison. The convict is blindfolded, and instructed to select one ball from each of three urns. The guards place the three balls in a fourth urn. Then the blindfolded man draws a ball from this fourth urn. If the ball is white, he is let go free. What is the probability of escaping prison if the number of balls of various colours in the three urns is as follows:

 white red black urn 1 2 5 3 urn 2 5 2 3 urn 3 3 3 4

Suggested by J. Czinki, Budapest

(5 pont)

solution, statistics

C. 1257. In a triangle, $\displaystyle b=1.5a$ és $\displaystyle \beta=2\alpha$, with conventional notations. What is the ratio of the length of side $\displaystyle c$ to the length of side $\displaystyle a$?

(5 pont)

solution, statistics

C. 1258. The base radius of a cone is 1, and its height is 2. The base radius is increased by $\displaystyle x$ and the height is decreased by the same amount. For what value of $\displaystyle x$ will the volume of the resulting cone a maximum?

(5 pont)

solution, statistics

Problems with sign 'B'

Deadline expired on December 10, 2014.

B. 4660. In a championship, every team plays every other team exactly once. 3 points are awarded for winning, 0 for losing, and 1 for a draw. In the case of equal scores, the order of the teams is determined at random. The championship is in progress at the moment. Team $\displaystyle A$ is leading the points table. If team $\displaystyle A$ scores exactly $\displaystyle x$ points in the remaining rounds then they will win the championship. However, if $\displaystyle A$ scores more than $\displaystyle x$ points, they will not necessarily win. (It is possible for $\displaystyle A$ to score more than $\displaystyle x$.) How many rounds remain to be played in the championship?

Suggested by V. Vígh, Szeged

(5 pont)

solution, statistics

B. 4661. There are discs placed on some fields of a chessboard of $\displaystyle n$ columns and $\displaystyle k$ rows (at most one disc on each). Two discs are said to be adjacent if they lie in the same row or in the same column, and there is no other disc along the line segment connecting them. Each disc is adjacent to at most three others. What is the maximum possible number of discs on the chessboard?

Suggested by K. Williams, Szeged

(6 pont)

solution, statistics

B. 4662. A regular triangle is drawn over each side of a triangle $\displaystyle ABC$ on the outside. The third vertices of these triangles are $\displaystyle D$, $\displaystyle E$ and $\displaystyle F$. Given the points $\displaystyle D$, $\displaystyle E$ and $\displaystyle F$, construct the triangle $\displaystyle ABC$.

Suggested by Sz. Miklós, Herceghalom

(4 pont)

solution, statistics

B. 4663. Find the integer solutions of the equation $\displaystyle 2x^3 - y^3 = 5$.

(4 pont)

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B. 4664. A rectangle $\displaystyle ABDE$ is drawn to side $\displaystyle AB$ of an acute triangle $\displaystyle ABC$ on the inside, such that point $\displaystyle C$ should lie on the side $\displaystyle DE$. The rectangles $\displaystyle BCFG$ and $\displaystyle CAHI$ are defined in a similar way. ($\displaystyle A$ lies on line segment $\displaystyle FG$, and $\displaystyle B$ lies on line segment $\displaystyle HI$.) The midpoints of sides $\displaystyle AB$, $\displaystyle BC$, and $\displaystyle CA$ are $\displaystyle J$, $\displaystyle K$, and $\displaystyle L$, respectively. Prove that the sum of the angles $\displaystyle GJH\sphericalangle$, $\displaystyle IKD\sphericalangle$ and $\displaystyle ELF\sphericalangle$ is $\displaystyle 180^{\circ}$.

Suggested by Sz. Miklós, Herceghalom

(4 pont)

solution, statistics

B. 4665. Determine the number of solutions of the equation $\displaystyle mX+4= \big|X^2-10X+21\big|$ as a function of the parameter $\displaystyle m$.

Suggested by K. Grallert, Balassagyarmat

(3 pont)

solution, statistics

B. 4666. Prove that $\displaystyle \sum_{k=1}^n (2k-1) \left[\frac{n}{k}\right] = \sum_{k=1}^n \left[\frac{n}{k}\right]^2$ for every positive integer $\displaystyle n$.

(5 pont)

solution, statistics

B. 4667. Prove that if $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are the sides of a triangle of unit perimeter, then $\displaystyle a^2+b^2+c^2+4abc<\frac{1}{2}$.

(5 pont)

solution, statistics

B. 4668. A tetrahedron is said to be strange if the line segments connecting its vertices to the incentres of the opposite faces are concurrent. What condition is true for the edges of a strange tetrahedron?

Suggested by J. Ruff, Pécs

(6 pont)

solution, statistics

Problems with sign 'A'

Deadline expired on December 10, 2014.

A. 626. We have $\displaystyle 4n+5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $\displaystyle n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors such that all points occurring as vertices of the $\displaystyle n$ triangles have the same color.

Miklós Schweitzer competition, 2014

(5 pont)

solution, statistics

A. 627. Let $\displaystyle n\ge1$ be a fixed integer. Calculate the distance $\displaystyle \inf_{p,f} \max_{0\le x\le 1} \big|f(x)-p(x)\big|$, where $\displaystyle p$ runs over polynomials of degree less than $\displaystyle n$ with real coefficients and $\displaystyle f$ runs over functions $\displaystyle f(x) = \sum_{k=n}^\infty c_k x^k$ defined on the closed interval $\displaystyle [0,1]$, where $\displaystyle c_k\ge0$ and $\displaystyle \sum_{k=n}^\infty c_k=1$.

Miklós Schweitzer competition, 2014

(5 pont)

solution, statistics

A. 628. Is it true that for every infinite sequence $\displaystyle x_1,x_2,\ldots$ of integers satisfying $\displaystyle |x_{k+1}-x_k|=1$ for every positive integer $\displaystyle k$, there exists a sequence $\displaystyle k_1<k_2<\ldots<k_{2014}$ of positive integers such that as well the indices $\displaystyle k_1,k_2,\ldots,k_{2014}$ as the numbers $\displaystyle x_{k_1},x_{k_2},\ldots,x_{k_{2014}}$ (in this order) form arithmethic progressions?

Proposed by: E. Csóka, Warwick and Ben Green, Oxford

(5 pont)

solution, statistics