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