## Problem List

1. Prove that the center of gravity of a finite convex region halves at least three chords of the region.

Miklos Schweitzer Contest

2. Let be a sequence of numbers. For a prime we call the sequence to be -balanced if

.

Suppose we have the sequence which is -balanced for every . Prove that all the terms of our sequence are 0’s.

3. Given a sequence of positive integers, we define . Prove that we can find using if and only if is not a power of 2.

This means that if is a power of 2 then there exists a sequence of integers, let’s say , such that and the ‘s are not a permutation of ‘s. Conversely, if is not a power of 2 then implies that the ‘s are a permutation of ‘s.

4. Prove that in a sequence of numbers there exist an increasing subsequence of elements or a decreasing sequence of elements.

5. For any function there exist two functions such that and have the **Darboux property**.

A function has the **Darboux property** if for any interval we have is also an interval. This is slightly different from continuity and intermediate value property. Cotinuity implies Darboux and Darboux implies Intermediate value property.

6. There exist functions with Darboux property, such that there exist sets with and .

*Problem proposed in the “Traian Lalescu” contest for undergraduate students in Romania in 2004.*

7. Prove that if is a two variable polynomial which vanishes in finitely many points, then does not change it’s sign.

*(old Romanian TST problem)*

8. Is it possible to find 5 vectors in space such that the angle between any two of them is greater than ?

9. A simple curve splits the circle into two regions of equal areas. Prove that the length of the curve is at least the diameter of the circle.

Prove that if a simple curve lying on a sphere divides the area of the sphere into two equal parts, then the length of the curve is at least the length of a great circle of the sphere.

**Simple curve** means a curve which does not intersect itself.

10. We have a function , times differentiable such that . Prove that .

Here means the derivative.

11. Prove that there exists a continuous function for which the equation has exactly solutions () forall in the image of if and only if is odd.

12. Prove that for any integer there exists a prime number .

13. We have bounded a sequence of real numbers such that . Prove that the set of accumulation points of such a sequence is a closed interval.

14. Prove that a normed vector space is a Banach space if and only if any absolutely convergent series converges.

15. Prove that in any convex polygon we can inscribe an equilateral triangle. Moreover, if the polygon has a symmetry axis, then we can inscribe a square in it.

A polygon is inscribed in another polygon if the vertices of the first are on the edges of the second one.

16. Prove that in any convex polyhedra there exists the following relation:

(no. of vertices) + (no. of faces) = (no. of edges) + 2

17. We denote by the Lebesgue measure on and a Lebesgue measurable set. Prove that if for any we have , where , then we have .

18. For a triangle we consider and the greatest and the smallest equilateral triangles such that . Prove that we have the following relation .

You can consider proving that and do exist. Moreover, prove that given any polygon, we can find one greatest equilateral triangle inscribed in it.

19. Suppose we have a matrix such that . Prove that .

As a generalisation, we can state the following:

Given a matrix with , where is a real polynomial with distinct roots, we have , and are the distinct roots of .

20. Suppose we have a set such that whenever . Prove that is countable.

21. 1. Prove that given any triangle, we can inscribe a square in it.

2.0 Prove that there exists a square with maximum area inscribed in a given triangle.

2.1 Prove that if a square which lies inside a triangle has maximum area then two of its vertices lie on the same edge of the triangle.

2.2 Prove that if a square lies inside a triangle , then .

3. Now, we have a square inscribed in a triangle such that two of the vertices of lie on the same edge of . Prove that the incenter of lies inside .

22. Prove that , where are positive integers (i.e. ).

23. Find a closed form (i.e. an explicit formula in terms of ) for the -th number which is not a perfect square.

24. If a simple closed plane curve has length and encloses area , then , and the equality holds if and only if is a circle.

25. Suppose is a sequence of continuous functions such that there exists satisfying .

Prove there exists and an interval such that .

26. Prove that if is a convex figure in plane which can be represented as a finite union of disks, then is also a disk.

27. Given a convex polygon prove that there exist two perpendicular lines which divide the area of into four equal parts.

28. We say that a function satisfies the **Cauchy** functional equation if .

1. Prove that . ( There fore, is a linear application, of we consider considered a vector space over .)

2. The following statements are equivalent:

i) .

ii) is continuous.

iii) there exists a point such that is continuous in .

iv) is non-decreasing/non-increasing.

v) is bounded on some interval.

vi) is positive/negative when is positive/negative.

**Solutions of the type above are called “trivial” solutions of the Cauchy functional equation.**

3. Prove that there exist non-trivial solutions of the Cauchy functional equation.

4. Prove that if is a non-trivial solution of the Cauchy functional equation then , for any .

5. Prove that there exist solutions of the Cauchy functional equation which have the Darboux property, which means that is an interval for any interval .

29. Prove that if is an infinite cardinal number, then and .

30. Prove that if is a vector space over and the dimension of is , an infinite cardinal number and then .

As an easy application of this proposition, prove that the dimension of over is .

31. Suppose is a “subset” of the integers (the elements of can repeat themself ). What is the minimum number of elements of such that we can be sure that there exist of the elements of such that their sum is divisible by ?

32. Prove that is prime if and only if any echiangular polygon having edges with rational lengths must be regular.

33. Suppose is a non-increasing sequence of positive real numbers and such that is convergent.

Prove that .

34. Prove that if is a differentiable function on , then the set of continuity points of is dense in .

35. Take . We define the sequence of real numbers in the following way: . Prove that the given sequence doesn’t contain two equal numbers if and only if .

36. Prove that if we have a function which is uniformly continuous on with then .

37. Let be a holomorphic function on a domain containing . Prove that we have the following identity:

.

38. Prove that if is an entire function and then is constant or there exists with and some positive integer such that .

39. Suppose are four points in the same plane. Prove that these points lie on a circle if and only if there exist constants , not all of them equal to , such that for any point in our plane.

40. Two circles intersect each other in and . Through we draw the parallel line to the closest common tangent of the two circles. This line intersects circles for the second time in and , respectively. Suppose is the center of the circle which is exterior tangent to and interior tangent to the circumcircle of . Prove that is the angle bisector of .

41. Consider a triangle and its circumcircle. Further, consider the circle tangent to (interior), which intersects in respectively. Prove that if is the incenter of then is the midpoint of .

42. Freddy Flinstone has a strange car, with square wheels. He wants you to design a road for him, such that he could ride his car perfectly smooth (i.e. the center of his wheels travel in a horizontal line). Can you design such a road?

43. A closed, planar curve is said to have **constant breadth ** if the distance between parallel tangent lines to is always . needn’t be a circle. See the Wankel engine design.

a) If we call two points with parallel tangent lines **opposite**, prove that the chord joining opposite points is normal to the curve at both points.

b) Prove that the sum of reciprocals of the curvature at opposite points is equal to .

c) (easy application) Prove (using b) ) that the circle of radius has constant curvature .

d) Prove that the length of is .

44. Outside triangle we construct triangles such that , .

Prove that and .

*IMO 1975*

45. Determine which is the greatest number which can be written as a product of positive integer factors whose sum is .

*Generalisation of an IMO 1976 problem*

46. There is a forest having a square shape with edge of 1000 meters. This forest contains 4500 oak trees having 0.5 meters in diameter. The owner of the forest wants to know if he can find a place having a rectangular place of dimensions 20 meters by 10 meters which does not intersect any of the oak trees, because he wants to build a house there. He meets a mathematician, and asks him if he can deduce the existence of such a place without going out in the field to measure things on the spot. After a few minutes and calculations, the mathematician says that he can find such a place with certainty. How did he do that?

47. A triangle and a square are circumscribed around the unit circle. Prove that the common area of these two figures is at least 3.4.

48. If the angles between the three pairs of opposite sides of a tetrahedron are equal, prove that these angles are right.

49. Prove that and are both transcendental, which means neither one of them is root for a polynomial with rational coefficients.

50. Take such that . Prove that if and and also if or .

Prove that for any fixed in the unit disk , the mapping has the following properties:

1) maps to and is holomorphic.

2) interchanges and .

3) .

4) is an involution on and thus is bijective.

5) Construct a bijective, holomorphic function, which maps the unit disk to itself, takes the unit circle into itself and interchanges two given complex numbers with .

51. Amazing property of entire functions.

52. Power series and number theory.

53. Every expansion has a zero.

54. Characterization of an affine mapping.

55. Projections and Thales

56. Isometry of an affine space

57. Injective entire functions

58. Mascheroni Theorem

59. Twice differentiable function

60. Matrices with zero and one elements and their determinants

61. Intersection of maximal subspaces

62. Jensen convexity

63. Linear maximal subspaces and linear functionals

64. Weird inequality

65. Regular polygons and lattice points

66. Line segments (Dilworth theorem)

67. Disjoint convex sets

68. Irreducible polynomial

69. Position of roots of a polynomial

70. Switch the coefficients of a polynomial (what happens to its roots ?)

71. Irreducible polynomial 1

72. Measurable sets (Miklos Schweitzer problem)

73. Strange Congruency (Miklos Schweitzer problem)

74. Hard/Easy Inequality

75. Problem with sets (combinatorics)

76. Matrix with

77. Rectangles in the plane

78. Geometry problem with a right triangle

79. disks in the plane and equilateral triangles

80. Irreducible polynomials

81. Equilateral locus

82. Problem with sets (combinatorics)

83. Partition of an equilateral triangle

84. Not a square (IMO problem)

85. Find the angle

86. Probability

87. Equation roots

88. Convex function limit

89. Egorov’s theorem

90. Lusin’s theorem

91. Approximation of a measurable function by polynomials

92. Cavalieri’s Principle

93. Interesting problem concerning rational numbers as radicals

94. Impossible relation

95. Uniform limit of polynomials and their degrees

96. Critical points and extremal points

97. Fixed point theorem

98. Strange functional equation

99. Fixed point for an operator

100. Dual of spaces

101. Inequality in a triangle

102. Fuglede and Putnam theorem

103. An irreducible polynomial problem from the Romanian Mathematical Gazette

104. Measurable finite valued function

105. Integral formula (particular case of coarea formula)

106. Hahn-Banach theorem (real version)

107. Hahn-Banach theorem (complex version)

108. Banach spaces of operators

109. The dual is not trivial (Hahn-Banach application)

110. Zero on a subspace, non-zero on the outside (Hahn-Banach application)

111. Nice problem with functions

112. Putnam 2009

113. Putnam 2009 solutions part A

114. Density, Kronecker’s theorem

115. Divergence of an interesting series

116. Van der Waerden’s Theorem

117. IMO 2001 inequality

118. A theorem of Minkowski about the area of a convex polygon which is symmetric but does not contain any laticeal point inside it

119. Round garden and visibility

120. Two hunters and probabilities

121. Recurrent sequence is constant (L. Panaitopol)

122. Some points inside a polygon (AMM problem)

123. AMM inequality with radicals

124. Constant function AMM problem

125. Brocard angle of 30 degrees implies equilateral triangle

126. Irreducible polynomial property inspired from an AMM article

127. Black and white polyhedra

128. Fibonacci and twin primes

129. Sum of integer parts via quadratic residues.

130. Functional equation AMM problem

131. Intermediate value AMM problem

132. A point in a triangle and some angles (AMM)

133. Heronian triangles and embeddings in a lattice

134. Functional equation with cubes

135. Thebault’s Theorem

136. Sum of integer parts involving prime number

137. Quadratic residue divisor

138. Multiplicative function

139. Shortest path on a sphere

140. Finite ring problem

141. Last non-zero digit of a factorial

142. Polynomial degree

143. Greatest roots of two polynomials

144. Interesting sequence, IMO shortlist

145. Increasing sequence of real numbers

146. Subset with sum equal to zero

147. The integer matrix which is equal to any power

148. Arithmetic progression IMO 2009

149. Division problem IMO 2009

150. Romanian TST division problem by L. Panaitopol

151. Diophantine equation Romanian TST

152. Combinatorial identity, Romanian TST

153. A triangle with area greater than four (Romanian TST)

154. Not an integer. Romanian TST, L. Panaitopol

155. Function defined on polygonal surfaces

156. Regular polygon exists on union of arcs

157. Trigonometric inequality, Romanian TST

158. Butterfly problem

159. Morera type theorems

160. Enumeration of the rationals

161. Set of measure zero with interesting property

162. Borel Cantelli lemma

163. Recurrent function sequence SEEMOUS 2010

164. Circles in a square SEEMOUS 2010

165. Permutations with distinct distances

166. Polyhedron problem Romanian TST 2005

167. Functional equation with divisibility

168. Coloring combinatorics

169. Another problem about entire functions

170. Function defined on subsets. Miklos Schweitzer

171. Another function defined on subsets

172. Harmonic function in the plane and semilines

173. Sum of squared distances. Miklos Schweitzer

174. Permutation of . Miklos Schweitzer

175. Harmonic functions and holomorphic functions

176. Inequality between remainders

177. Orthogonal polynomial on subspace

178. Continuous function and two variable relation

179. Sequence of continuous functions

180. A theorem of Sylvester and Schur

181. -adic surprises

182. Integral recurrence

183. Average on circles is constant

184. Numbers placed in lattice points

185. Condition for an integer to be the power of another integer

186. BMO 2010 strip cover problem

187. Convex function limit Traian Lalescu contest 2010

188. Cool coloring of

189. Borsuk Ulam theorem

190. Impossibility of partitioning a square. Monsky theorem

191. Determinant divisible by a prime number

192. Even degree polynomial

193. Ants moving a bread crumb

194. Inequality related to a group

195. Closed cover for the dimensional unit sphere

196. Characterization of continuous functions

197. Matrix power limit

198. BMO 2010 geometry problem

199. Titeica 5 lei problem

200. Proving Euler’s relation using inversions

201. Rational polynomial and images of .

202. Cauchy problem with two solutions

203. Number of planes, IMO 2007 problem 6

204. Consecutive positive integers and some properties

205. Variable points in plane and the maximal area of a triangle

206. Sum independent of triangulation

207. Various geometry problems

208. Measure zero subset of the reals can be translated into the irrationals

209. Circles inside a square

210. Coloured subarcs

211. Terms of the harmonic sequence

212. Least common multiple problem

213. Zero matrix product

214. Line through the incircle

215. Invisible points

216. Application to the isoperimetric inequality

217. IMO 2010 Problem 1

218. IMO 2010 Problem 2

219. IMO 2010 Problem 3

220. IMO 2010 Problem 4

221. IMO 2010 Problem 5

222. IMO 2010 Problem 6

223. Interresting recurrence

224. Geometric transformation solutions

225. Planets

226. Sets with an interesting property

227. discs

228. Positive integer partitioned by translating the same set

229. Partition of a set of sets

230. Sufficient condition for a sequence to be an enumeration of the integers

231. IMC 2010 Day 1 Problem 1

232. IMC 2010 Day 1 Problem 2

233. IMC 2010 Day 1 Problem 3

234. IMC 2010 Day 1 Problem 4

235. IMC 2010 Day 1 Problem 5

236. IMC 2010 Day 2 Problem 1

237. IMC 2010 Day 2 Problem 2

238. IMC 2010 Day 2 Problem 3

239. IMC 2010 Day 2 Problem 4

240. IMC 2010 Day 2 Problem 5

241. Good rectangles

242. The 15 Puzzle

243. Cover of the unit segment

244. 7 Points and isosceles triangles

245. Helly’s Theorem Application

246. Martingales and Applications

247. Easy functional equation

248. Grasshopper

249. Curve with normal passing through a fixed point

250. Characterization of a subset of the line or of the circle

251. The set of orthogonal matrices is a differential manifold

252. 167 Sets

253. Nice inequality (brute forced…)

254. Snails in a triangular array

255. Halfplanes cover

256. Hidden functionals

257. Limit points

258. Divergent integral and harmonic functions

259. Almost everywhere convergence implies strong convergence

260. Poincare’s inequality

261. Generalized Jordan Curve theorem

262. Euclidean spaces of different dimension are not homeomorphic

263. Condition for distinct eigenvalues

264. p-integrable product

265. Frobenius coin problem

266. Surface with normal passing through a fixed point

267. Irreducible polynomial TST 2003

268. Even cycles permutations

269. tilling of a square board

270. Reverse Riesz Problem

271. Separable spaces

272. Non-separable space example 1

273. Separable space 2

274. Bilinear continuous operator

275. Closure and span

276. l-infinity is not separable

277. Properties of distinct eigenvalues operators

278. Cayley Hamilton Theorem alternative proof

279. Functional is continuous if and only if the kernel is closed

280. Family of rectangles

281. Geometric mean for complex numbers

282. Useful polynomial roots position lemma

283. Positive measure set.

284. Conditions for an operator to be bounded

285. Set distance construction and properties

286. PUTNAM 2010 A1

287. PUTNAM 2010 A2

288. PUTNAM 2010 A3

289. PUTNAM 2010 A4

290. PUTNAM 2010 A5

291. PUTNAM 2010 A6

292. Lamps and switches

293. Euler homogeneous formula and applications

294. Pi and e are irrational

295. Own problem involving regular polygons

296. The complement of a kernel intersection

297. Useful continuity property

298. Zero trace implies zero product

299. Brouwer fixed point theorem

300. Dini’s theorem

301. Measurable set has nice property

302. Weakly compact means bounded

303. Compact and Hausdorff spaces

304. SEEMOUS 2011 Problem 1

305. SEEMOUS 2011 Problem 2

306. SEEMOUS 2011 Problem 3

307. SEEMOUS 2011 Problem 4

308. Lamps and switches 2

309. Intersection of diagonals in regular polygons

310. Arithmetic progressions which cover the integers

311. Weird sequence

312. Irreducible polynomial with paired roots.

313. Permutable line (Romanian National Mathematical Olympiad)

314. Romanian TST 2011 Problem 1

315. Romanian TST 2011 Problem 2

316. Romanian TST 2011 Problem 3

317. Romanian TST 2011 Problem 4

318. Equidistribution

319. Integrals with matrices

320. Sobolev functions and vanishing at infinity

321. Weakly harmonic implies harmonic

322. Traian Lalescu 2008 Problem 1

323. Traian Lalescu 2008 Problem 2

324. Traian Lalescu 2008 Problem 3

325. Traian Lalescu 2008 Problem 4

326. Traian Lalescu 2009 Problem 1

327. Traian Lalescu 2009 Problem 2

328. Traian Lalescu 2009 Problem 3

329. BMO 2011 Problem 1

330. BMO 2011 Problem 2

331. BMO 2011 Problem 3

332. BMO 2011 Problem 4

333. Increasing sequence of trace terms (Traian Lalescu 2011 Problem 1)

334. A subset of divisors (Traian Lalescu 2011 Problem 2)

335. Integral inequality and boundedness (Traian Lalescu 2011 Problem 3)

336. Some PDE problems (Traian Lalescu 2011 Problem 4)

337. Squares and inclusion (Romanian TST 2 2011 Problem 1)

338. Concurrency (Romanian TST 2 2011 Problem 2)

339. Graphs and cycles of even length (Romanian TST 2 2011 Problem 3)

340. Sum of squares of consecutive positive integers is again a square (Romanian TST 2 2011 Problem 4)

341. Number of even images is very small

342. IMO 2011 Problem 1

343. IMO 2011 Problem 2

344. IMO 2011 Problem 3

345. IMO 2011 Problem 4

346. IMO 2011 Problem 5

347. IMO 2011 Problem 6

348. IMC 2011 Day 1 Problem 1

349. IMC 2011 Day 1 Problem 2

350. IMC 2011 Day 1 Problem 3

351. IMC 2011 Day 1 Problem 4

352. IMC 2011 Day 1 Problem 5

353. IMC 2011 Day 2 Problem 1

354. IMC 2011 Day 2 Problem 2

355. IMC 2011 Day 2 Problem 3

356. IMC 2011 Day 2 Problem 4

357. IMC 2011 Day 2 Problem 5

358. Infinitely countable sigma algebra

359. Condition for boundedness of an operator

360. Maximum dimension for a linear space of singular matrices

361. Inequality involving Banach spaces and an operator

362. Generalized Poincare inequality

363. Compact operator maps weakly convergenc sequences onto strongly convergent sequences

364. Nice characterization of convergence

365. Sobolev space impossible extension

366. A lemma of J. L. Lions

367. Pointwise convergence implies convergence

368. Arcs on a circle

369. Functional equation on positive integers

370. Non-existence of a function with some properties

371. Miklos Schweitzer 2011

372. SEEMOUS 2012 Problem 1

373. SEEMOUS 2012 Problem 2

374. SEEMOUS 2012 Problem 3

375. SEEMOUS 2012 Problem 4

376. Torus as zero of a polynomial

377. Subharmonic functions

378. Characterization of a normal matrix

379. The symmetric difference is associative

380. Hausdorff measure and Hausdorff dimension

381. Slick proof that the harmonic series diverges

382. Two dimensional Barbalat’s Lemma

383. Approximation by polynomials of a function defined on the entire real line

384. BMO 2012 Problem 1

385. BMO 2012 Problem 2

386. BMO 2012 Problem 3

387. BMO 2012 Problem 4

388. Caratheodory Theorem on convex hulls

389. Darboux functions with no iterated fixed points

390. IMO 1995 Day 1

391. IMO 1995 Day 2

392. IMO 1996 Day 1

393. IMC 2012 Day 1 Problem 1

394. IMC 2012 Day 1 Problem 2

395. IMC 2012 Day 1 Problem 3

396. IMC 2012 Day 1 Problem 4

397. IMC 2012 Day 1 Problem 5

398. IMC 2012 Day 2 Problem 1

399. IMC 2012 Day 2 Problem 2

400. IMC 2012 Day 2 Problem 3

401. IMC 2012 Day 2 Problem 4

402. IMC 2012 Day 2 Problem 5

403. Agregation 2001 dual of convex sets

404. One dimensional trace of a Sobolev function

405. Characterization of sigma-finite measure spaces

406. Krein Milman theorem

407. The space of real vector measures is complete

408. Lax Milgram application

409. Characterizations of Sobolev spaces

410. Nice inequality similar to IMO 2001

Foarte frumoasa aceasta lista de probleme! M-ar interesa care este sursa Problemei 33.

Ma bucur ca va place lista de probleme. Problema 33 am discutat-o impreuna cu niste prieteni la un concurs Traian Lalescu in urma cu cativa ani. Nu i-am gasit solutie pana de curand, prin intermediul unui site de intrebari. Problema este rezolvata in linkul urmator.

Can I plan to enter in future competitions with my diagrams?

I don’t know what you mean. You want to participate yourself in competitions or submit one of your puzzles to the competition? Anyway, you have some interesting work on your website. 🙂

Greeeeeeeeat Blog Love the Infomation you have provided me .

In #5: “This is slightly different from continuity and intermediate value property. Cotinuity implies Darboux and Darboux implies Intermediate value property.” Someone posted a question about this comment at Mathematics Stackexchange: http://math.stackexchange.com/questions/1851413/is-intermediate-value-property-equivalent-to-darboux-property