Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# KöMaL Problems in Mathematics, March 2018

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## Problems with sign 'K'

Deadline expired on April 10, 2018.

K. 583. An integer is said to be a prime-rose if its first digit is a prime, the sum of the first two digits is also a prime, the sum of the first three digits is also a prime, and so on. Find the largest prime-rose number in which all digits are different.

(6 pont)

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K. 584. Santa Claus is very strong, but he can only carry a maximum of 100 kg of presents in his sack. In a large apartment building, he was to deliver three different kinds of presents: A, B and C. The mass of each type of present is a whole number of kilograms. He can carry eight A and eight B at the same time, but in that case he cannot take any further piece (neither A, nor B or C) in that round. Similarly, he cannot take anything further if he carries ten A, four B and four C. How much may each of the presents A, B and C weigh in kilograms?

(6 pont)

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K. 585. Andrew wrote three (not necessarily different) positive integers on a blackboard, each of them smaller than $\displaystyle 2018$. Then he erased these numbers ($\displaystyle A$, $\displaystyle B$ and $\displaystyle C$), and replaced them with

$\displaystyle \frac{A+B}2,\quad \frac{B+C}2, \quad \frac{A+C}2.$

He repeated this procedure $\displaystyle 11$ times altogether. As a result, one of the three numbers on the board is $\displaystyle 100$. What are the other two numbers?

(6 pont)

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K. 586. The distances of an interior point of a regular hexagon from three consecutive vertices are 4, 4 and 8 units. How long are the sides of the hexagon?

(6 pont)

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K. 587. How many of the numbers 2014, 2015, 2016 and 2017 can be expressed as a sum of squares of six not necessarily different odd numbers?

(6 pont)

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K. 588. Let $\displaystyle A > B$ be four-digit numbers such that $\displaystyle B$ is obtained by writing the digits of $\displaystyle A$ in reverse order. What are the smallest and largest possible values of $\displaystyle A-B$?

(6 pont)

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## Problems with sign 'C'

Deadline expired on April 10, 2018.

C. 1469. The foot of the altitude drawn from vertex $\displaystyle C$ of a triangle $\displaystyle ABC$ is $\displaystyle T$, an interior point of side $\displaystyle AB$. The angle bisector drawn from $\displaystyle C$ intersects $\displaystyle AB$ at $\displaystyle R$. Given that $\displaystyle AB=10$, $\displaystyle AT=3$ and $\displaystyle AR=4$, find the lengths of the sides of the triangle.

(5 pont)

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C. 1470. What is the radius of two touching congruent spheres centred at the centres of two adjacent faces of a unit cube?

(5 pont)

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C. 1471. Prove that every power of two greater than four can be expressed as the difference of two odd square numbers. For example, $\displaystyle 32=81-49$.

(5 pont)

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C. 1472. A certain game involves collecting cards with various things on them. Each card has exactly two of the following 9 things: colours (red, white, or green), elements (air, earth, fire, or water) and animals (rabbit or sheep). A card shows at most one of each category. In how many different ways is it possible to select four cards such that there are eight different things on them, provided that the game contains all possible combinations?

(5 pont)

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C. 1473. The number $\displaystyle abc$ is expressed in base $\displaystyle 2a$ notation. What is the base if $\displaystyle c-b=b-a=1$, and the value of $\displaystyle abc$ equals $\displaystyle 29a^{2}+9a+9$ in decimal notation?

(5 pont)

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C. 1474. Let $\displaystyle P$, $\displaystyle Q$ and $\displaystyle R$ denote the feet of the altitudes of the acute-angled triangle $\displaystyle ABC$. Given that $\displaystyle BP:PA=1:2$ and $\displaystyle AQ:QC=3:1$, find the proportion of the pieces formed by $\displaystyle R$ on side $\displaystyle BC$.

(5 pont)

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C. 1475. What is the largest possible area of the lateral surface of a cylinder inscribed in a unit sphere?

(5 pont)

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## Problems with sign 'B'

Deadline expired on April 10, 2018.

B. 4939. Show that a convex 2018-sided polygon cannot be dissected into triangles in which the angles in degrees are all integers.

(3 pont)

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B. 4940. What may be the value of the sum $\displaystyle x+y+z$ if

$\displaystyle x^4 + 4y^4 + 16 z^4 + 64 = 32xyz?$

(3 pont)

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B. 4941. The centre $\displaystyle O$ of the circumscribed circle of an acute-angled triangle $\displaystyle ABC$ is reflected in the feet of the altitudes. Prove that the circle formed by the three reflections has the same radius as the circumscribed circle of the triangle.

(4 pont)

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B. 4942. The one hundred mathematicians participating in an international combinatorial conference were all housed in the same hotel. The receptionist was originally planning to place them in the order of their arrival in the rooms numbered 1 to 100. However, he forgot to give that instruction to the guest arriving first, who has thus chosen a room at random. So the receptionist instructed all the other guests to take the room with their number in the order of arrivals, or, if that room has already been taken, to select any other room they like. How many possible arrangements of the guests in the rooms are there?

Proposed by A. Faragó and T. Káspári, Paks

(4 pont)

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B. 4943. There is an ant at each corner of a given face of a rectangular brick. Each ant wants to get to the opposite vertex of the cuboid, that is, to the other endpoint of the space diagonal drawn from his vertex of the cuboid. Is it possible for the ants to crawl to the opposite vertices along the surface of the brick, so that they follow the shortest possible paths and their paths do not intersect?

Proposed by M. E. Gáspár, Budapest

(4 pont)

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B. 4944. Let $\displaystyle t$ denote the area of (some) triangle of maximum area inscribed in a convex plane figure $\displaystyle \mathcal{S}$, and let $\displaystyle T$ denote the area of (some) triangle of minimum area circumscribed about $\displaystyle \mathcal{S}$. What is the maximum of the ratio $\displaystyle \frac{T}{t}$?

(5 pont)

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B. 4945. Find all positive integers $\displaystyle n$ for which

$\displaystyle 1\cdot 2^0+2\cdot 2^1+ 3\cdot 2^2+ \ldots + n\cdot 2^{n-1}$

is a perfect square.

Based on the idea of L. Németh, Fonyód

(5 pont)

solution (in Hungarian), statistics

B. 4946. Let $\displaystyle f(x)$ be a polynomial of real coefficients such that $\displaystyle f(k)$ is an integer for every positive integer $\displaystyle k$ that ends in $\displaystyle 5$ or $\displaystyle 8$ in decimal notation.

$\displaystyle a)$ Prove that $\displaystyle f(0)$ is an integer.

$\displaystyle b)$ Give an example of a polynomial $\displaystyle f(x)$ that meets the above conditions, but $\displaystyle f(1)$ is not an integer.

(6 pont)

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B. 4947. Prove that there is exactly one way of dissecting a cube into five tetrahedra. (Two dissections are not considered different if the resulting pieces are congruent.)

(6 pont)

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## Problems with sign 'A'

Deadline expired on April 10, 2018.

A. 719. Let $\displaystyle ABC$ be a scalene triangle with circumcenter $\displaystyle O$ and incenter $\displaystyle I$. The $\displaystyle A$-excircle, $\displaystyle B$-excircle, and $\displaystyle C$-excircle of triangle $\displaystyle ABC$ touch $\displaystyle BC$, $\displaystyle CA$, and $\displaystyle AB$ at points $\displaystyle A_1$, $\displaystyle B_1$, and $\displaystyle C_1$, respectively. Let $\displaystyle P$ be the orthocenter of $\displaystyle AB_1C_1$ and $\displaystyle H$ be the orthocenter of $\displaystyle ABC$. Show that if $\displaystyle M$ is the midpoint of $\displaystyle PA_1$, then lines $\displaystyle HM$ and $\displaystyle OI$ are parallel.

Proposed by: Michael Ren, Andover, Massachusetts, USA

(5 pont)

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A. 720. We call a positive integer lively if it has a prime divisor greater than $\displaystyle 10^{10^{100}}$. Prove that if $\displaystyle S$ is an infinite set of lively positive integers, then it has an infinite subset $\displaystyle T$ with the property that the sum of the elements in any finite nonempty subset of $\displaystyle T$ is a lively number.

(5 pont)

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A. 721. Let $\displaystyle n\ge 2$ be a positive integer, and suppose $\displaystyle a_1,a_2,\dots,a_n$ are positive real numbers whose sum is $\displaystyle 1$ and whose squares add up to $\displaystyle S$. Prove that if $\displaystyle b_i=\frac{a_i^2}{S}$ ($\displaystyle i=1,\dots,n$), then for every $\displaystyle r>0$, we have

$\displaystyle \sum_{i=1}^n \frac{a_i}{{(1-a_i)}^r}\le \sum_{i=1}^n \frac{b_i}{{(1-b_i)}^r}.$

(5 pont)

solution (in Hungarian), statistics

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