In the World of Mathematics

More documents

Recommendations

Info

323. Let AA 1 and CC 1 be angle bisectors of triangle ABC (A 1 ∈ BC, C 1 ∈ AB). Straight line A 1 C 1 intersects ray AC at point D. Prove that angle ABD is obtuse. (I. Nagel, Evpatoria) 324. Let H be the orthocenter of acute-angled triangle ABC. Circle ω with diameter AH and circumcircle of triangle BHC intersect at point P ≠ H. Prove that the straight line AP pass through the midpoint of BC. 325. Solve the inequality |x − 1| + 3|x − 3|+5|x − 5| + . . . + 2009|x − 2009| ≥ (Yu. Biletskyy, Kyiv) ≥2|x − 2| + 4|x − 4| + 6|x − 6| + . . . + 2008|x − 2008|. (O. Kukush, Kyiv) 326. Let P be arbitrary point inside the triangle ABC, ω A , ω B and ω C be the circumcircles of triangles BP C, AP C and AP B respectively. Denote by X, Y, Z the intersection points of straight lines AP, BP, CP with circles ω A , ω B , ω C respectively (X, Y, Z ≠ P ). Prove that AP AX + BP BY + CP CZ = 1. (O. Manzjuk, Kyiv) 327. Some cities of the country are connected by air flights in both directions. It is known that it is possible to reach every city from any another (probably, with changes) and there are exactly 100 flights from each city. Some m flights have been canceled because of bad weather conditions. For which maximum m it is still possible to travel between each two cities (probably, with changes)? 328. Let a and b be rational numbers such that Volume 14(2008) Issue 3 2 2a + b = 2a b + b 2a − 1 Prove that 1 − 2ab is a square of rational number. (here a ≠ 0, b ≠ 0, b ≠ −2a). (A. Prymak, Kyiv) (I. Nagel, Evpatoria) 329. Construct triangle ABC given points O A and O B , which are symmetric to its circumcenter O with respect to BC and AC, and the straight line h A , which contains its altitude to BC. (G. Filippovskyy, Kyiv) 330. Let O be the midpoint of the side AB of triangle ABC. Points M and K are chosen at sides AC and BC respectively such that ∠MOK = 90 ◦ . Find angle ACB, if AM 2 + BK 2 = CM 2 + CK 2 . (I. Fedak, Ivano-Frankivsk) 331. Do there exist positive integers a and b such that a) 4a 3 − 53a − 1 = 10 2b ; b) 4a 3 − 53a − 1 = 10 2b−1 ? (O. Makarchuk, Kirovograd) 2
332. Function f : (0; +∞) → (0; +∞) satisfies the inequality f(3x) ≥ f ( 1 2 f(2x)) + 2x for every x > 0. Prove that f(x) ≥ x for every x > 0. (V. Yasinskyy, Vinnytsya) 333. Let circle ω touches the sides of angle ∠A at points B and C, B ′ and C ′ are the midpoints of AB and AC respectively. Points M and Q are chosen at the straight line B ′ C ′ and point K is chosen at bigger ark BC of the circle ω. Line segments KM and KQ intersect ω at points L and P. Find ∠MAQ, if the intersection point of line segments MP and LQ belongs to circle ω. Volume 14(2008) Issue 4 334. Let x, y, z be pairwise distinct real numbers such that k = 1 + xy x − y , l = 1 + yz 1 + zx and m = y − z z − x are integers. Prove that k, l and m are pairwise relatively prime. (I. Nagel, Evpatoria) (L. Orydoroga, Donetsk) 335. A point O is chosen at the side AC of triangle ABC so that the circle ω with center O touches the side AB at point K and BK = BC. Prove that the altitude that is perpendicular to AC bisects the tangent line from the point C to ω. (I. Nagel, Evpatoria) 336. Find all sequences {a n , n ≥ 1} of positive integers such that for every positive integers m and n the number n + m2 a n + a 2 m is an integer. (V. Yasinskyy, Vinnytsya) 337. In a school class with 3n pupils, any two of them make a common present to exactly one other pupil. Prove that for all odd n it is possible that the following holds: for any three pupils A, B and C in the class, if A and B make a present to C then A and C make a present to B. (Baltic Way) 338. A circle ω 1 touches sides of angle A at points B and C. A straight line AD intersects ω 1 at points D and Q, AD < AQ. The circle ω 2 with center A and radius AB intersects AQ at a point I and intersects some line passing through the point D at points M and P. Prove that I is the incenter of triangle MP Q. (I. Nagel, Evpatoria) 339. The insphere of triangular pyramid SABC is tangent to the faces SAB, SBC and SAC at points G, I and O respectively. Let G be the intersection point of medians in the triangle SAB, I be the incenter of triangle SBC and O be the circumcenter of triangle SAC. Prove that the straight lines AI, BO and CG are concurrent. (V. Yasinskyy, Vinnytsya) 3
Page 1: In the World of Mathematics Problem

In the World of Mathematics

Create successful ePaper yourself

Delete template?

Save as template?