Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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KöMaL Problems in Mathematics, May 2017

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Problems with sign 'C'

Deadline expired on June 12, 2017.

C. 1420. A circle is centred at the centroid of a regular triangle of unit side. The total length of the part of the circumference of the triangle inside the circle equals the total length of the part outside. What is the radius of the circle?

(5 pont)

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C. 1421. Prove that if $\displaystyle n\in \mathbb{N}^+$ then there exist $\displaystyle a,b\in \mathbb{N}^+$ such that $\displaystyle a^2+b^2=13^n$.

Based on a problem by F. Olosz, Szatmárnémeti

(5 pont)

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C. 1422. In a single-storey rectangular apartment, there is at most one door connecting any pair of rooms. In addition, in each room there is at most one exit from the apartment. If the apartment consists of four rooms, what is the maximum possible number of doors altogether?

(5 pont)

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C. 1423. Given five different circles in the plane such that any four have a point in common, prove that there is a point that lies on all the circles.

(5 pont)

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C. 1424. What is the smallest positive value in the range of the function $\displaystyle x\mapsto \frac{16x^2-96x+153}{x-3}$?

(5 pont)

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C. 1425. Triangle $\displaystyle ABC$ is right-angled at $\displaystyle C$, and its Fermat point (see https://en.wikipedia.org/wiki/Fermat_point) is $\displaystyle I$. Given that $\displaystyle IC=12$ mm and $\displaystyle IB=16$ mm, find the length of the line segment $\displaystyle IA$.

(5 pont)

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C. 1426. The equation

$\displaystyle x^4+ax^3+bx^2+cx+d=0$

has four real solutions, and its coefficients (in this order) form an arithmetic sequence of positive integers. Prove that the roots cannot all be integers.

Proposed by Á. Kertész

(5 pont)

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Problems with sign 'B'

Deadline expired on June 12, 2017.

B. 4876. Find the largest positive integer $\displaystyle n$ for which there exists exactly one integer $\displaystyle k$ such that

$\displaystyle \frac{10}{11}<\frac{n}{k+n}<\frac{11}{12}?$

(3 pont)

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B. 4877. Points $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ and $\displaystyle D$, in this order, lie on a straight line. Point $\displaystyle E$ does not lie on the line, and

$\displaystyle AEB\sphericalangle=BEC\sphericalangle =CED\sphericalangle = 45^\circ.$

Let $\displaystyle F$ and $\displaystyle G$ be the midpoints of $\displaystyle AC$ and $\displaystyle BD$, respectively. What is the measure of angle $\displaystyle FEG$?

Proposed by the class 11C of Fazekas Gimnázium, Budapest

(3 pont)

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B. 4878. What is the maximum possible value of the sum $\displaystyle PA+PB+PC+PD$ if $\displaystyle P$ is a point of the unit square $\displaystyle ABCD$?

(4 pont)

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B. 4879. $\displaystyle a)$ Is it true that for any irrational number $\displaystyle a$ there exists an irrational number $\displaystyle x$ such that $\displaystyle a+x$ is rational and $\displaystyle ax$ is irrational?

$\displaystyle b)$ Is it true that for any irrational number $\displaystyle a$ there exists an irrational number $\displaystyle y$ such that $\displaystyle a+y$ is irrational and $\displaystyle ay$ is rational?

Proposed by S. Róka, Nyíregyháza

(4 pont)

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B. 4880. In the sequence $\displaystyle a_1$, $\displaystyle a_2$, $\displaystyle a_3$, ... of positive integers, $\displaystyle a_n\cdot a_{n+1} = a_{n+2}\cdot a_{n+3}$ for all positive integers $\displaystyle n$. Show that the sequence is eventually periodic.

Proposed by M. E. Gáspár, Budapest

(4 pont)

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B. 4881. Let $\displaystyle d(n)$ denote the number of (positive) factors of the number $\displaystyle n$. Prove that

$\displaystyle n+d(1)+d(2)+\cdots +d(n)\le d(n+1)+d(n+2)+\dots +d(2n).$

(5 pont)

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B. 4882. What is the maximum number of faces of a convex polyhedron if it is possible to select three faces such that every edge of the polyhedron lies on a selected face?

(5 pont)

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B. 4883. Define the sequence $\displaystyle a_1, a_2, \dots$ with the following recurrence relation:

$\displaystyle a_1=4, \quad a_2=2 \quad\text{and}\quad a_{n+1}=\frac{na_n^2}{na_n^2-(n+1)a_n+n+1}, \text{ if } n\ge 2.$

Prove that

$\displaystyle a_1+2\cdot a_2+3\cdot a_3+\cdots +n\cdot a_n=a_1\cdot a_2\cdot a_3\cdot\,\cdots\, \cdot a_n$

for all $\displaystyle n\ge 1$.

Proposed by B. Kovács, Szatmárnémeti

(6 pont)

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B. 4884. What is the minimum possible area of a triangle that contains a unit square?

Proposed by P. Bogár, Békéscsaba

(6 pont)

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Problems with sign 'A'

Deadline expired on June 12, 2017.

A. 698. Let $\displaystyle m$ and $\displaystyle n$ be positive integers, and let $\displaystyle H$ denote a subset of the set $\displaystyle \{1,2,\ldots,m\}\times\{1,2,\ldots,n\}$. Show that if $\displaystyle |H|>m+(m+n)\log_2n$, then there exist integers $\displaystyle 1\le u<v\le m$ and $\displaystyle 1\le x<y<z\le n$ such that the pairs $\displaystyle (u,x)$, $\displaystyle (u,y)$, $\displaystyle (v,x)$ and $\displaystyle (v,z)$ are elements of $\displaystyle H$.

(5 pont)

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A. 699. A circle $\displaystyle \omega$ lies in a circle $\displaystyle \Omega$ such that their common center is the point $\displaystyle O$. Fix a point $\displaystyle A\ne O$ inside $\displaystyle \omega$. Let $\displaystyle X$ denote an arbitrary point on the circumference of $\displaystyle \Omega$, and let $\displaystyle Y$ denote the second intersection point of $\displaystyle \Omega$ and the line $\displaystyle AX$. Let $\displaystyle Z$ denote the intersection of $\displaystyle \omega$ and the line segment $\displaystyle AX$. Let $\displaystyle M$ denote the point on the line segment $\displaystyle AZ$ for which $\displaystyle MX\cdot MZ\cdot AY = MA\cdot MY\cdot XZ$. Let $\displaystyle x$ and $\displaystyle y$ denote the tangents of the circle $\displaystyle \Omega$ at the points $\displaystyle X$ and $\displaystyle Y$, respectively. Let $\displaystyle t$ denote the line that passes through $\displaystyle M$ and either also passes through the intersection of $\displaystyle x$ and $\displaystyle y$, or is parallel to both $\displaystyle x$ and $\displaystyle y$. Finally, let $\displaystyle T$ denote the intersection of $\displaystyle t$ and the line $\displaystyle OZ$.

Show that the locus of the points $\displaystyle T$, as $\displaystyle X$ is varied, is an ellipse, and the line $\displaystyle t$ is a tangent of this ellipse.

(5 pont)

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A. 700. A positive integer $\displaystyle n$ satisfies the following: it is possible to select some integers such that if we randomly choose two different integers from this list, say, $\displaystyle i$ and $\displaystyle j$, then $\displaystyle i+j$ $\displaystyle \mathrm{mod\ } n$ is equal to one of the numbers $\displaystyle 0,1,\dots,n-1$ with equal probability. Find all numbers $\displaystyle n$ with this property.

(5 pont)

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