In the World of Mathematics

In the World of Mathematics
Problems 316 —339
Volume 14(2008) Issue 1
316. Let S > 0 and n ≥ 3 be fixed. Find the minimum value of the expression
a 1 (1 + a 2 a 3 ) + a 2 (1 + a 3 a 4 ) + . . . + a n (1 + a 1 a 2 )
( 3√ a 1 a 2 a 3 + 3√ a 2 a 3 a 4 + . . . + 3√ a n a 1 a 2 ) 3 ,
where a 1 , a 2 , . . . , a n are arbitrary positive numbers such that a 1 + a 2 + . . . + a n = S.
(D. Mitin, Kyiv)
317. Let AB be a diameter of circle ω. Points M, C and K are chosen at circle ω in such a way
that the tangent line to the circle ω at point M and the secant line CK intersect at point
Q and points A, B, Q are collinear. Let D be the projection of point M to AB. Prove that
DM is the angle bisector of angle CDK.
(I. Nagel, Evpatoria)
318. Ten pairwise distinct points T 1 , T 2 , . . . , T 10 are chosen in the space and some of them are
connected by segments without intersections. A beetle sitting at the point T 1 can move along
the segments to the point T 10 . Prove that at least one of the following statements is true:
(i) there exist a route of the beetle from T 1 to T 10 which pass through at most two points
distinct from T 1 and T 10 ;
(ii) there exist points T i and T j (2 ≤ i < j ≤ 10) such that any route of the beetle from T 1
to T 10 pass through the point T i or through the point T j .
(V. Yasinskyy, Vinnytsya)
319. Circles ω 1 and ω 2 intersect at points A and B. Diameter BP of ω 2 intersects the circle ω 1 at
point C and diameter BK of the circle ω 1 intersects the circle ω 2 at point D. The straight
line CD intersects the circle ω 1 at point S ≠ C and the circle ω 2 at point T ≠ D. Prove that
BS = BT.
(I. Fedak, Ivano-Frankivsk)
320. Let k be a positive integer. Prove that there exist polynomials P 0 (n), P 1 (n), . . . , P k−1 (n)
(which may depend on k) such that for any integer n,
[ n
] k
k = P0 (n) + P 1 (n) [ ]
n
k + . . . + Pk−1 (n) [ ]
n k−1
k .
([a] means the largest integer ≤ a.)
(William Lowell Putnam Math. Competition)
321. Let ω 1 be the circumcircle of triangle A 1 A 2 A 3 , let W 1 , W 2 , W 3 be the midpoints of arcs
A 2 A 3 , A 1 A 3 , A 1 A 2 and let the incircle ω 2 of triangle A 1 A 2 A 3 touches the sides A 2 A 3 ,
A 1 A 3 , A 1 A 2 at points K 1 , K 2 , K 3 respectively. Prove that
where R, r are the radii of ω 1 and ω 2 .
W 1 K 1 + W 2 K 2 + W 3 K 3 ≥ 2R − r,
Volume 14(2008) Issue 2
322. Find the minimum possible ratio of 5-digit number to the sum of its digits.
(A. Prymak, Kyiv)
(Yu. Rabinovych, Kyiv)
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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
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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)
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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)
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