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Problems in Mathematics, December 2016

Please read the rules of the competition.


Problems with sign 'A'

Deadline expired on 10 January 2017.

A. 683. Let \(\displaystyle K=(V,E)\) be a finite, simple, complete graph. Let \(\displaystyle \phi\colon E\to\mathbb{R}^2\) be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set \(\displaystyle V\), and the points assigned to the edges of any triangle are collinear. Show that the range of \(\displaystyle \phi\) is contained in a line.

(Based on a problem of the Miklós Schweitzer competition)

(5 points)

Statistics

A. 684. Show that there are no rational numbers \(\displaystyle x\) and \(\displaystyle y\) such that

\(\displaystyle x - \frac1x + y - \frac1y = 4. \)

(Korean problem)

(5 points)

Statistics

A. 685. Let \(\displaystyle AB\) and \(\displaystyle CD\) be two chords in a circle \(\displaystyle \Omega\). Choose a circle \(\displaystyle \omega\) tangent to the segments \(\displaystyle AB\) and \(\displaystyle CD\) at their interior points \(\displaystyle M\) and \(\displaystyle N\) respectively, and let \(\displaystyle \omega\) intersect \(\displaystyle \Omega\) at \(\displaystyle P\) and \(\displaystyle Q\). Suppose that a third circle \(\displaystyle \omega'\), distinct from \(\displaystyle \Omega\) and \(\displaystyle \omega\), and also passing through \(\displaystyle P\) and \(\displaystyle Q\), meets the line \(\displaystyle MN\) at \(\displaystyle M'\) and \(\displaystyle N'\). Prove that the points \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\), \(\displaystyle D\), \(\displaystyle M'\), and \(\displaystyle N'\) lie on a conic section that is tangent to \(\displaystyle \omega'\) at \(\displaystyle M'\) and \(\displaystyle N'\).

(Proposed by: Ilya Bogdanov and Pavel Kozhevnikov, Moscow)

(5 points)

Solution, statistics


Problems with sign 'B'

Deadline expired on 10 January 2017.

B. 4831. We have a ,,pocket calculator'' that can only be used for addition, subtraction and taking the reciprocal of a number. Starting with the number \(\displaystyle \sqrt{20}+16\), is it possible to get 1 as a result? (During the calculation, we can store the original number, as well as any intermediate result in separate memory caches, and we may access them as many times as we wish.)

(Proposed by S. Kiss, Nyíregyháza)

(3 points)

Solution (in Hungarian)

B. 4832. Let \(\displaystyle a\), \(\displaystyle b\), and \(\displaystyle c\) denote some positive integers. Prove that there exist two relatively prime positive numbers \(\displaystyle r\) and \(\displaystyle s\) such that \(\displaystyle ar+bs\) is divisible by \(\displaystyle c\).

(5 points)

Solution (in Hungarian)

B. 4833. For a sector of a circle of radius \(\displaystyle R\), smaller than a semicircle, the radius of the inscribed circle is \(\displaystyle r\), and the length of the chord connecting the endpoints of the boundary arc of the sector is \(\displaystyle 2a\). Prove that

\(\displaystyle \frac 1r= \frac 1R+ \frac 1a. \)

(3 points)

Solution (in Hungarian)

B. 4834. Let \(\displaystyle K\) denote the centre of the circumscribed circle of an acute-angled triangle \(\displaystyle ABC\). The line drawn through \(\displaystyle K\), parallel to \(\displaystyle AB\), intersects line \(\displaystyle BC\) at \(\displaystyle D\), and line \(\displaystyle CA\) at \(\displaystyle E\). Similarly, let the line drawn through \(\displaystyle K\), parallel to \(\displaystyle BC\) intersect line \(\displaystyle CA\) at \(\displaystyle F\), and line \(\displaystyle AB\) at \(\displaystyle G\). Prove that the circle of radius \(\displaystyle EA\) centred at \(\displaystyle E\), the circle of radius \(\displaystyle FC\) centred at \(\displaystyle F\), and the circumscribed circle of the triangle \(\displaystyle ABC\) are concurrent.

(Proposed by Sz. Miklós, Herceghalom)

(4 points)

Solution (in Hungarian)

B. 4835. Solve the following simultaneous equations:

\(\displaystyle x+y+z =3,\)

\(\displaystyle x^{2}+y^{2}+z^{2} =7,\)

\(\displaystyle x^{3}+y^{3}+z^{3} =15.\)

(Proposed by B. Kovács, Szatmárnémeti)

(4 points)

Solution (in Hungarian)

B. 4836. In a parallelogram \(\displaystyle ABCD\), \(\displaystyle BC=\lambda\, AB\). The intersection of the interior angle bisectors drawn from \(\displaystyle A\) and from \(\displaystyle B\) intersect at \(\displaystyle M\). What fraction of the parallelogram \(\displaystyle ABCD\) is covered by the triangle \(\displaystyle ABM\)?

(Proposed by J. Kozma, Szeged)

(5 points)

Solution (in Hungarian)

B. 4837. Find all functions \(\displaystyle f\colon \mathbb{R}\to \mathbb{R}\) satisfying \(\displaystyle (x+1)\cdot f(x+2)-2(x+2)\cdot f(-x-1)=3x^2+8x+3\).

(Proposed by G. Szöllősy, Máramarossziget)

(4 points)

Solution (in Hungarian)

B. 4838. A centrally symmetrical convex polyhedron \(\displaystyle \mathcal{P}\) has the following property: any two vertices \(\displaystyle X\) and \(\displaystyle Y\) of \(\displaystyle \mathcal{P}\) are either centrally opposite or there exists a face of \(\displaystyle \mathcal{P}\) that contains both. Show that \(\displaystyle \mathcal{P}\) is either a parallelepiped, or the vertices of \(\displaystyle \mathcal{P}\) are the centres of the faces of a parallelepiped.

(6 points)

Solution (in Hungarian)

B. 4839. Solve the following equation in the set of positive integers:

\(\displaystyle n!= 2^a - 2^b. \)

(Proposed by K. Williams, Szeged)

(6 points)

Solution (in Hungarian)


Problems with sign 'C'

Deadline expired on 10 January 2017.

C. 1385. A man has five chequered shirts, out of which two are brown and one is blue. He has seven brown shirts, three of which are striped. He has three blue shirts, none of which is striped. In addition, he has got eight red shirts, half of which are striped. He regularly wears three pairs of trousers: one pair has blue stripes on it, one is brown with a checquered pattern, and one is a striped gray pair. He never wears blue and brown together, and he never wears striped with checquered either. How many different outfits (of one shirt and one pair of trousers) may he wear?

(5 points)

This problem is for grade 1 - 10 students only.

Solution (in Hungarian)

C. 1386. How many different convex figures can be put together out of four congruent isosceles right-angled triangles?

(5 points)

This problem is for grade 1 - 10 students only.

Solution (in Hungarian)

C. 1387. Determine the base \(\displaystyle x\) in which the following equation holds:

\(\displaystyle 2016_x=x^3+2x+342. \)

(Matlap, Kolozsvár)

(5 points)

Solution (in Hungarian)

C. 1388. The front and back edges of a rectangular billiards table are 90 cm long, while the left and right edges are 180 cm long. A ball situated 10 cm from the front edge and 15 cm from the left edge is hit. Rebounding from the front edge, then from the right edge, it rolls along the shortest possible path into the hole of the back left corner. What is the total distance covered by the ball until it sinks in the hole? (Ignore the size of the ball and of the hole.)

(Proposed by F. Olosz, Szatmárnémeti)

(5 points)

Solution (in Hungarian)

C. 1389. Find the common points of the curves of equations \(\displaystyle y={(x-1)}^2\) and \(\displaystyle y=1-\sqrt x\,\).

(5 points)

Solution (in Hungarian)

C. 1390. Triangle \(\displaystyle ABC\) is right-angled at vertex \(\displaystyle C\). A central similitude of centre \(\displaystyle C\) and negative scale factor is applied, to obtain a diminished triangle \(\displaystyle A'B'C\). Given that \(\displaystyle BC=5\), \(\displaystyle B'C=0.2\) and \(\displaystyle \tan \alpha =2.4\), prove that the interior angle bisector from \(\displaystyle B\) and the exterior angle bisector from \(\displaystyle B'\) intersect at the midpoint of \(\displaystyle AA'\).

(5 points)

This problem is for grade 11 - 12 students only.

Solution (in Hungarian)

C. 1391. Ever since his birth, Martin gets as many decks of 32 cards for Christmas as the number of Christmas days he has seen, including the current one. One year, on the second day of Christmas (that is, on his birthday) he decided to build a specific castle out of the cards he had received. The lowermost level consisted of 216 cards, and every consecutive level had 8 cards less in it than the level below. How old was Martin if he managed to build 16 levels?

(5 points)

This problem is for grade 11 - 12 students only.

Solution (in Hungarian)


Problems with sign 'K'

Deadline expired on 10 January 2017.

K. 523. Given a two-digit positive integer, we multiply its digits to get a new number. If the new number has two digits, then we multiply the digits again. This process is continued until a one-digit number is obtained. How many two-digit positive integers are there for which the process will terminate at the number 8?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 524. Andy's washing machine keeps the door locked for an additional 5 minutes after the washing program is finished: it can only be opened after this delay. Andy knows this, so he includes this period when he calculates washing times. When the machine displays that 90% of the program time has elapsed, Andy knows that 20% of the calculated washing time is yet to follow. How long is the washing program in minutes?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 525. Six numbers are written on the circumference of a circle. Then every number smaller than the mean of the two adjacent numbers is circled. How many numbers can be circled this way?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 526. In the diagram, each letter or number stands for a different prime number, and the sum of the four numbers in every ``straight line'' of triangles is the same. What is the smallest possible value of this sum?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 527. Calculate the area of the regular dodecagon inscribed in a circle of radius 6 cm.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 528. The numbers from 1 to 14 are written next to the vertices of a regular heptagon such that the sum is the same on each side (that is, for the two vertices lying on the side).

\(\displaystyle a)\) What is this sum? Show a valid construction.

\(\displaystyle b)\) May there be a vertex without any number written next to it?

\(\displaystyle c)\) What may be the sum of the numbers written next to a particular vertex?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)


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