KöMaL Problems in Mathematics, April 2023
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Problems with sign 'K'Deadline expired on May 10, 2023. |
K. 764. In the Seventh Kingdom in the back of beyond, a week lasts one seventh as many days as on the Earth, a day consists of 42 hours, and there are 77 minutes in an hour and 33 seconds in a minute. How many seconds elapse during the course of two weeks there?
Proposed by K.\(\displaystyle \,\)A. Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
K. 765. The midpoint of side \(\displaystyle AB\) of a triangle \(\displaystyle ABC\) is \(\displaystyle D\), and the midpoint of \(\displaystyle CD\) is \(\displaystyle E\). Which point of the line segment \(\displaystyle CD\) should be marked \(\displaystyle F\) so that the sum of the areas of triangles \(\displaystyle AEC\) and \(\displaystyle BFC\) is exactly \(\displaystyle 40\%\) of the area of triangle \(\displaystyle ABC\)?
Proposed by B. Bálint, Eger
(5 pont)
solution (in Hungarian), statistics
K. 766. Alpha, Lambda and Zeta each have more than 1000 forints (HUF, Hungarian currency) on their bank accounts. Lambda's money equals 35 percent of Alpha's money, and Zeta's money equals \(\displaystyle \frac{12}{7}\) of Lambda's money. How much money do Alpha, Lambda and Zeta have altogether if Zeta has \(\displaystyle 10\,110\) forints more than Lambda?
Proposed by K.\(\displaystyle \,\)A. Kozma, Győr
(5 pont)
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Problems with sign 'K/C'Deadline expired on May 10, 2023. |
K/C. 767. The circle \(\displaystyle k\) passes through vertices \(\displaystyle A\), \(\displaystyle B\) of a given square \(\displaystyle ABCD\) in the plane, and touches the side \(\displaystyle CD\). Let \(\displaystyle M\) denote the intersection of circle \(\displaystyle k\) and side \(\displaystyle BC\) which is different from \(\displaystyle B\). Find the exact value of the ratio \(\displaystyle \frac{CM}{BM}\).
Proposed by I. Keszegh, Révkomárom
(5 pont)
solution (in Hungarian), statistics
K/C. 768. The number \(\displaystyle 2023\) has exactly one digit of \(\displaystyle 0\). How many four-digit positive odd numbers are there for which this property does not hold?
Proposed by K.\(\displaystyle \,\)A. Kozma, Győr
(5 pont)
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Problems with sign 'C'Deadline expired on May 10, 2023. |
C. 1763. Prove that the number \(\displaystyle 4^{52}+52^{2023}+2023^{52}\) is divisible by \(\displaystyle 15\).
(5 pont)
solution (in Hungarian), statistics
C. 1764. Solve the simultaneous equations
$$\begin{align*} x(2x+6)(3x+5y) & =64;\\ 2x^2+9x+5y & =16 \end{align*}$$where \(\displaystyle x\), \(\displaystyle y\) are positive real numbers.
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1765. The base of a regular four-sided pyramid \(\displaystyle ABCDE\) is the square \(\displaystyle ABCD\), and each edge of the pyramid is \(\displaystyle 32\) units long. A snail starts at vertex \(\displaystyle E\) and crawls to vertex \(\displaystyle A\) as follows: first moves along edge \(\displaystyle EA\) to the point \(\displaystyle P\) for which \(\displaystyle EP=2\). Then continues to cross the face \(\displaystyle ABE\) and arrives at point \(\displaystyle Q\) of edge \(\displaystyle EB\), where \(\displaystyle EQ=4\). Then crosses the face \(\displaystyle BCE\) to reach that point \(\displaystyle R\) of edge \(\displaystyle EC\) for which \(\displaystyle ER=8\), and continues along face \(\displaystyle CDE\) to point \(\displaystyle S\) on edge \(\displaystyle ED\), with \(\displaystyle ES=16\). Finally, it crawls on the surface of face \(\displaystyle DAE\), from \(\displaystyle S\) to point \(\displaystyle A\). What is the minimum distance covered by the snail altogether?
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1766. Show that in every triangle (with conventional notations),
\(\displaystyle \sqrt{a\sin{\alpha}}+\sqrt{b\sin{\beta}}+\sqrt{c\sin{\gamma}}=\sqrt{(a+b+c)(\sin{\alpha}+\sin{\beta}+\sin{\gamma})}. \)
Proposed by G. Holló, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1767. Given are \(\displaystyle 2\) coins of \(\displaystyle 7\), \(\displaystyle 3\) coins of \(\displaystyle 17\), \(\displaystyle 5\) coins of \(\displaystyle 119\), \(\displaystyle 7\) coins of \(\displaystyle 289\), \(\displaystyle 11\) coins of \(\displaystyle 2023\) and \(\displaystyle n\) coins of \(\displaystyle 1\). Two coins are selected at random, and their values are multiplied together, and a result of \(\displaystyle 2023\) is obtained. Find the value of \(\displaystyle n\), given that the probability of obtaining that result is \(\displaystyle \frac{12}{55}\).
Proposed by O. Teleki, Tököl
(5 pont)
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Problems with sign 'B'Deadline expired on May 10, 2023. |
B. 5310. A game of strategy is played by four teams on a map represented by an \(\displaystyle n \times n\) square grid (\(\displaystyle n \ge 3\)). Every square field is either sea or land. The bases of the four teams are in the four corners of the map, on land. Given that there is one large connected region of sea on the map and no two bases are connected by a path on land, determine the minimum possible number of fields that represent the sea. (A path consists of adjacent fields. Two fields are adjacent if they have an edge in common. The region of sea is connected in this sense: any two sea squares can be connected by a path consisting of sea squares.)
Proposed by K. Williams, Cambridge
(4 pont)
solution (in Hungarian), statistics
B. 5311. Is it true that if the sine of each angle of a triangle is rational then the cosine of each angle is rational, too?
Proposed by M. Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5312. Let \(\displaystyle F_k\) denote the \(\displaystyle k\)th Fibonacci number (\(\displaystyle F_1=F_2=1\), \(\displaystyle F_{k+1}=F_k+F_{k-1}\)). Prove that
\(\displaystyle 2\sum_{k=1}^nF_k^2F_{k+1}=F_nF_{n+1}F_{n+2} \)
for all positive integers \(\displaystyle n\).
Proposed by M. Bencze, Brassó
(3 pont)
solution (in Hungarian), statistics
B. 5313. Triangle \(\displaystyle ABC\) is acute angled, and \(\displaystyle AC<AB<BC\). The centre of the circumscribed circle is \(\displaystyle O\), the orthocentre is \(\displaystyle M\). The perpendicular bisector of side \(\displaystyle AB\) intersects line \(\displaystyle AM\) at point \(\displaystyle P\), and the circle \(\displaystyle OMP\) intersects line \(\displaystyle BM\) again at a point \(\displaystyle Q\), different from \(\displaystyle M\). Prove that line \(\displaystyle BC\) is tangent to the circle \(\displaystyle ABQ\).
Proposed by G. Kós, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5314. Let \(\displaystyle S\) be an \(\displaystyle n\)-element set, and let \(\displaystyle 1\le k\le n\) be an odd integer. What is the largest number of subsets of \(\displaystyle S\) that can be selected so that the symmetric difference of no pair of subsets should have exactly \(\displaystyle k\) elements?
Proposed by P.\(\displaystyle \,\)P. Pach, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5315. Consider a triangle \(\displaystyle ABC\). Let \(\displaystyle B'\) be a point on the extension of side \(\displaystyle AB\) beyond \(\displaystyle B\), and let \(\displaystyle C'\) be the point on the extension of side \(\displaystyle AC\) beyond \(\displaystyle C\) such that \(\displaystyle BB'=CC'\). Let \(\displaystyle k\) and \(\displaystyle k'\) denote the circumscribed circles of triangles \(\displaystyle ABC'\) and \(\displaystyle AB'C\), respectively. Prove that the common chord of \(\displaystyle k\) and \(\displaystyle k'\) lies on the angle bisector drawn from \(\displaystyle A\).
Proposed by M. Hujter, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5316. Prove that if \(\displaystyle 0<a,b<1\) then
\(\displaystyle (a+b-ab)\big(a^b+b^a\big) > a+b. \)
Proposed by M. Bencze, Brassó
(6 pont)
solution (in Hungarian), statistics
B. 5317. An ellipse lies in the closed positive orthant, its foci are \(\displaystyle (x_1;y_1)\) and \(\displaystyle (x_2;y_2)\), and it touches the coordinate axes at the points of abscissa \(\displaystyle p\), and ordinate \(\displaystyle q\), respectively. Show that the point \(\displaystyle (p;q)\) is collinear with the origin and the centre of the ellipse, and calculate the numerical eccentricity of the ellipse.
Proposed by L. László, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on May 10, 2023. |
A. 851. Let \(\displaystyle k\), \(\displaystyle l\) and \(\displaystyle m\) be positive integers. Let \(\displaystyle ABCDEF\) be a hexagon that has a center of symmetry whose angles are all \(\displaystyle 120^{\circ}\) and let its sidelengths be \(\displaystyle AB=k\), \(\displaystyle BC=l\) and \(\displaystyle CD=m\). Let \(\displaystyle f(k,l,m)\) denote the number of ways we can partition hexagon \(\displaystyle ABCDEF\) into rhombi with unit sides and an angle of \(\displaystyle 120^{\circ}\).
Prove that by fixing \(\displaystyle l\) and \(\displaystyle m\), there exists polynomial \(\displaystyle g_{l,m}\) such that \(\displaystyle f(k,l,m)=g_{l,m}(k)\) for every positive integer \(\displaystyle k\), and find the degree of \(\displaystyle g_{l,m}\) in terms of \(\displaystyle l\) and \(\displaystyle m\).
Submitted by Zoltán Gyenes, Budapest
(7 pont)
A. 852. Let \(\displaystyle (a_i,b_i)\) be pairwise distinct pairs of positive integers for \(\displaystyle 1 \le i \le n\). Prove that
\(\displaystyle (a_1+a_2+\dots+a_n) (b_1+b_2+\dots+b_n)>\frac{2}{9} n^3, \)
and show that the statement is sharp, i.e. for an arbitrary \(\displaystyle c>\frac{2}{9}\) it is possible that
\(\displaystyle (a_1+a_2+\dots+a_n) (b_1+b_2+\dots+b_n)<cn^3. \)
Submitted by Péter Pál Pach, Budapest, based on an OKTV problem
(7 pont)
A. 853. Let points \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\), \(\displaystyle A'\), \(\displaystyle B'\), \(\displaystyle C'\) be chosen in the plane such that no three of them are collinear, and let lines \(\displaystyle AA'\), \(\displaystyle BB'\), \(\displaystyle CC'\) be tangent to a given equilateral hyperbola at points \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\), respectively. Assume that the circumcircle of \(\displaystyle A'B'C'\) is the same as the nine-point circle of triangle \(\displaystyle ABC\). Let \(\displaystyle s(A')\) be the Simson line of point \(\displaystyle A'\) with respect to the pedal triangle of \(\displaystyle ABC\). Let \(\displaystyle A^*\) be the intersection of line \(\displaystyle B'C'\) and the perpendicular of \(\displaystyle s(A')\) through point \(\displaystyle A\). Points \(\displaystyle B^*\) and \(\displaystyle C^*\) are defined in a similar manner. Prove that points \(\displaystyle A^*\), \(\displaystyle B^*\) and \(\displaystyle C^*\) are collinear.
Submitted by Áron Bán-Szabó, Budapest
(7 pont)
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