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Problems in Mathematics, January 2006

Please read the rules of the competition.


Problems sign 'A'

Deadline expired on 15 February 2006.

A. 389. Point P lies in the interior of the acute triangle ABC. The circumcenters of triangles ABC, BCP, CAP and ABP are O, A1, B1 and C1, respectively. Prove that


\frac{\mathrm{area}\,(\Delta A_1B_1O)}{\mathrm{area}\,(\Delta ABP)} =
\frac{\mathrm{area}\,(\Delta B_1C_1O)}{\mathrm{area}\,(\Delta BCP)} =
\frac{\mathrm{area}\,(\Delta C_1A_1O)}{\mathrm{area}\,(\Delta CAP)}.

(5 points)

Statistics

A. 390. Find all functions f\colon \mathbb{R}\to\mathbb{R} such that (f(x)+f(y))(f(z)+1)=f(xz-y)+f(x+yz) for all real x, y, z.

(5 points)

Solution (in Hungarian)

A. 391. Construct a sequence a1,a2,...,aN of positive reals such that n1an0+n2an1+...+nkank-1>2.7(a1+a2+...+aN) for arbitrary integers 1=n0<n1<...<nk=N.

(5 points)

Solution, statistics


Problems sign 'B'

Deadline expired on 15 February 2006.

B. 3872. The angle A of a triangle ABC is obtuse. Let D denote an arbitrary point on side AB, and let E be an arbitrary point on side AC. Show that CD+BE>BD+DE+EC.

(3 points)

Solution (in Hungarian)

B. 3873. The inscribed circle of the right-angled triangle ABC touches the leg AC at P, the leg BC at Q and the hypotenuse AB at R. Let M denote the orthocentre of the triangle PQR. Prove that RM=PQ.

Suggested by L. Gerőcs, Budapest

(3 points)

Solution (in Hungarian)

B. 3874. Define the sequence an (n is a natural number) as follows:


a_0=2 \quad\text{and} \quad a_n=a_{n-1}- \frac{n}{(n+1)!}, \quad\text{if} \quad n>0.

Express an in terms of n.

(National Mathematics Competition for Secondary Schools, 2005)

(3 points)

Solution (in Hungarian)

B. 3875. There were 31 people at a party. For any 15 of them, there is a further member of the group who knows all of them. (Acquaintances are mutual.) Prove that there is a guest of the party who knows all the others.

(5 points)

Solution (in Hungarian)

B. 3876. Find the distance between the (non-intersecting) diagonals of two adjacent faces of a unit cube.

Suggested by S. Kiss, Szatmárnémeti

(4 points)

Solution (in Hungarian)

B. 3877. The centroid of the triangle ABC is S, and the midpoint of AB is F. For an interior point P of the line segment AF, consider that point Q of the line PS for which QC and AB are parallel. Let R be the intersection of the lines QA and BC. Prove that the line segment PR halves the area of the triangle ABC.

(4 points)

Solution (in Hungarian)

B. 3878. Prove that the simultaneous equations x+y+z=0, x2+y2+z2=100 have no rational solution.

(4 points)

Solution (in Hungarian)

B. 3879. An escribed circle of a convex polygon is defined as a circle that touches one side of the polygon externally, and also touches the extensions of the two adjacent sides. Let k denote the sum of the areas of the escribed circles of a polygon, and let t denote the total area of the circles drawn over the sides of the polygon as diameters. Assume that the polygon has an inscribed circle, and let K and T denote the perimeter and area of the inscribed circle, respectively. Show that \frac{k}{t}\le \frac{K}{T}.

(4 points)

Solution (in Hungarian)

B. 3880. Prove that every positive integer n has a (positive) multiple smaller than n2, such that in decimal notation it does not contain all the ten different digits.

(5 points)

Solution (in Hungarian)

B. 3881. Let a, b, c denote rational numbers, such that \big(a+b\root3\of{2} + c\root3\of{4}\,
\big)^3 is also rational. Prove that at least two of the numbers a, b, c must be zero.

Suggested by E. Fried, Budapest

(5 points)

Solution (in Hungarian)


Problems sign 'C'

Deadline expired on 15 February 2006.

C. 835. How many solutions does the equation x+y+z=100 have on the set of positive integers?

(5 points)

Solution (in Hungarian)

C. 836. In a Fun Shop selling party novelties, a packet of streamers costs p percent more than a packet of confetti. In other words, a packet of confetti is q percent cheaper than a packet of streamers. The difference of p and q is 90. How many packets of streamers can one buy for the price of 10 packets of confetti?

(5 points)

Solution (in Hungarian)

C. 837. At most how many concave interior angles may a 2006-sided polygon have?

(5 points)

Solution (in Hungarian)

C. 838. Mihály Tímár (the hero of a famous Hungarian novel) is in a trouble. The red crescent that marked the sack containing the treasure has come off. He knows, however, that the heaviest of the four sacks hides the treasure. From three measurements, he has found out that the first sack together with the second one are lighter than the other two, the first and third together are equal in weight to the other two, and the first and fourth sacks together are heavier than the other two. Which sack hides the treasure?

(5 points)

Solution (in Hungarian)

C. 839. Three sides of a convex polygon are 1 cm, 4 cm and 8 cm long, and its diagonals are perpendicular to each other. How long may the fourth side be?

(5 points)

Solution (in Hungarian)


Problems sign 'K'

Deadline expired on 10 February 2006.

K. 67. In a television quiz, 10 people are asked questions. They all answer simultaneously by pressing the button assigned to the answer they think correct. Those who answer correctly get as many points each as the number of incorrect answers. After the fifth question answered, the players had 116 points altogether. 32 of these were collected by Géza. Show that Géza answered all the five questions correctly.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 68. Connect an interior point of a parallelogram to each vertex. Show that it is possible to construct a quadrilateral with the four segments hence obtained as sides such that its vertices are lying on the sides of the parallelogram.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 69. Out of the digits of a three-digit number of different digits, all the possible two-digit numbers of different digits are formed, and these two-digit numbers are added. Given that the sum is equal to the original three-digit number, find all such three-digit numbers.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 70. A merchant paid 6000 Ft for a product. What price should he write on it so that he having reduced this price by 10 percent he still make 20% profit.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 71. The diameter of the wheels on Jack's bike is 28'' (inches), and the diameter of the wheels of Jill's bike is 14'' (inches). Jack's wheel turns around twice while he makes 3 turns on the pedals. Jill's wheels turns around 3 times while she makes two turns on the pedals. Jack turns the pedals twice as many times a minute as Jill does. Cycling at a uniform speed, Jill covers 2 kilometers in 20 minutes. How long does Jack take to cover the same distance?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 72. The lengths of the edges in cm of a square-based prism are whole numbers. With a plane perpendicular to the base and parallel to one lateral face, a 4-cm-thick part is cut off the prism. The volume of the remaining solid is 126 cm3. How long are the edges of the original prism?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)


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