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• OE_TO_maths_en_COMP / train

  Given a positive integer n, determine the largest real number μ satisfying…

  Answer published by the source

  Identifiers

  upstream_id

  1610

  task_id

  427b6feb-796f-52d0-b386-be09e60b5c67

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1610

• OE_TO_maths_en_COMP / train

  For a positive integer a, define a sequence of integers x₁,x₂,……

  Answer published by the source

  Identifiers

  upstream_id

  1631

  task_id

  273cf6cd-be5d-51a6-806c-29a475905f3f

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1631

• OE_TO_maths_en_COMP / train

  Find all positive integers n, for which the numbers in the set …

  Answer published by the source

  Identifiers

  upstream_id

  1664

  task_id

  41362e25-d82d-5fc9-8a4f-71b02ee05e95

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1664

• OE_TO_maths_en_COMP / train

  Determine all pairs (f,g) of functions from the set of positive integers to…

  Answer published by the source

  Identifiers

  upstream_id

  1687

  task_id

  240ca69e-8fc5-551a-8166-224534bb617e

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1687

• OE_TO_maths_en_COMP / train

  Determine the greatest positive integer k that satisfies the following…

  Answer published by the source

  Identifiers

  upstream_id

  1694

  task_id

  055e365c-6619-5097-b445-57e73dfc6cc3

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1694

• OE_TO_maths_en_COMP / train

  On a square table of 2011 by 2011 cells we place a finite number of napkins that…

  Answer published by the source

  Identifiers

  upstream_id

  1697

  task_id

  0fef6719-2e9e-5693-995e-dc3350b967f3

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1697

• OE_TO_maths_en_COMP / train

  On some planet, there are 2^(N) countries (N≥4). Each country has a…

  Answer published by the source

  Identifiers

  upstream_id

  1723

  task_id

  052eaa5e-5724-536b-8b90-6dc65e62bbf9

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1723

• OE_TO_maths_en_COMP / train

  Find all pairs (m,n) of nonnegative integers for which

  Answer published by the source

  Identifiers

  upstream_id

  1737

  task_id

  1c060b08-9779-5e35-b2ca-6f2277d52f44

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1737

• OE_TO_maths_en_COMP / train

  Determine the largest N for which there exists a table T of integers with…

  Answer published by the source

  Identifiers

  upstream_id

  1789

  task_id

  330125de-819a-5df5-91ee-84dfc4c7ca99

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1789

• OE_TO_maths_en_COMP / train

  Let n be a positive integer. Determine the smallest positive integer k with…

  Answer published by the source

  Identifiers

  upstream_id

  1854

  task_id

  0011c201-f74e-5d66-83b6-ff469a75c813

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1854

• OE_TO_maths_en_COMP / train

  Define P(n)=n²+n+1. For any positive integers a and b, the set

  Answer published by the source

  Identifiers

  upstream_id

  1866

  task_id

  2609a701-0625-5c4f-8898-d69ce3d0eb38

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1866

• OE_TO_maths_en_COMP / train

  Denote by N the set of all positive integers. Find all functions …

  Answer published by the source

  Identifiers

  upstream_id

  1869

  task_id

  3221b6a2-b5ae-5b43-a6ba-73d00ae80a1e

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1869

• OE_TO_maths_en_COMP / train

  Players A and B play a game on a blackboard that initially contains 2020…

  Answer published by the source

  Identifiers

  upstream_id

  1887

  task_id

  270c94a9-394f-5f77-bc95-a0ccae247439

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1887

• OE_TO_maths_en_COMP / train

  There are 60 empty boxes B₁,…,B₆₀ in a row on a table and an…

  Answer published by the source

  Identifiers

  upstream_id

  1918

  task_id

  0e028db9-fc29-599c-b313-e9e122dd2bd0

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1918

• OE_TO_maths_en_COMP / train

  For any two different real numbers x and y, we define D(x,y) to be the…

  Answer published by the source

  Identifiers

  upstream_id

  1920

  task_id

  0c045671-fbe8-5ee7-8b03-ed87a8f43027

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1920

• OE_TO_maths_en_COMP / train

  For a finite set A of positive integers, we call a partition of A into two…

  Answer published by the source

  Identifiers

  upstream_id

  1945

  task_id

  42df09d2-ee90-57c0-986d-31693b48ebba

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1945

• OE_TO_maths_en_COMP / train

  Determine all triples (a,b,c) of positive integers for which ab−c,bc−a,…

  Answer published by the source

  Identifiers

  upstream_id

  1962

  task_id

  43ae9a5e-2f50-501c-9f9e-b6fdd46c9534

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1962

• OE_TO_maths_en_COMP / train

  Determine all functions f:Z→Z satisfying

  Answer published by the source

  Identifiers

  upstream_id

  1969

  task_id

  0de3e128-9dfc-56e9-82fe-f8ff503c3c94

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1969

• OE_TO_maths_en_COMP / train

  Consider all polynomials P(x) with real coefficients that have the following…

  Answer published by the source

  Identifiers

  upstream_id

  1970

  task_id

  2ca12d04-d771-5127-9235-1b94c111f613

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--1970

• OE_TO_maths_en_COMP / train

  A number is called Norwegian if it has three distinct positive divisors whose…

  Answer published by the source

  Identifiers

  upstream_id

  2021

  task_id

  1416e0e4-e337-5b57-971f-0cd456c77091

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2021

• OE_TO_maths_en_COMP / train

  Find all positive integers n>2 such that

  Answer published by the source

  Identifiers

  upstream_id

  2022

  task_id

  41f5987e-59da-501c-b869-81965a744e6a

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2022

• OE_TO_maths_en_COMP / train

  Let k be a positive integer. The organising committee of a tennis tournament…

  Answer published by the source

  Identifiers

  upstream_id

  2040

  task_id

  1ae7525e-5684-524a-b691-a69576b2a0bd

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2040

• OE_TO_maths_en_COMP / train

  Let Z_(>0) be the set of positive integers. Find all functions …

  Answer published by the source

  Identifiers

  upstream_id

  2075

  task_id

  20eeb793-7e0e-572c-b444-2d2f683b153f

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2075

• OE_TO_maths_en_COMP / train

  Let ABC be a triangle with AB=AC. The angle bisectors of A and B…

  Answer published by the source

  Identifiers

  upstream_id

  2099

  task_id

  35b972d5-76f0-5722-aa30-b21ccb554093

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2099

• OE_TO_maths_en_COMP / train

  In the plane we consider rectangles whose sides are parallel to the coordinate…

  Answer published by the source

  Identifiers

  upstream_id

  2124

  task_id

  0145c384-f2d4-5ddb-b392-c22c01ba5c9a

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2124

• OE_TO_maths_en_COMP / train

  Let n≥1 be an integer. What is the maximum number of disjoint pairs of…

  Answer published by the source

  Identifiers

  upstream_id

  2151

  task_id

  0cc2c95a-d9c1-5869-b615-3ed61178355b

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2151

• OE_TO_maths_en_COMP / train

  In a 999×999 square table some cells are white and the remaining ones…

  Answer published by the source

  Identifiers

  upstream_id

  2152

  task_id

  0a3f91c7-b79f-576f-af3c-e3471c2c2660

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2152

• OE_TO_maths_en_COMP / train

  Players A and B play a game with N≥2012 coins and 2012 boxes arranged…

  Answer published by the source

  Identifiers

  upstream_id

  2153

  task_id

  061e6849-1c3e-57f2-ba1d-e4e7aea2d8b3

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2153

• OE_TO_maths_en_COMP / train

  Let m be a positive integer. Consider a 4m×4m array of square unit…

  Answer published by the source

  Identifiers

  upstream_id

  2185

  task_id

  0b492756-499d-55a2-b31b-7a3855819018

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2185

• OE_TO_maths_en_COMP / train

  Find all triples (a,b,c) of real numbers such that ab+bc+ ca=1 and

  Answer published by the source

  Identifiers

  upstream_id

  2203

  task_id

  0a4d3a15-6eae-575f-8d23-b57fcbe2f307

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2203

• OE_TO_maths_en_COMP / train

  Determine all integer values of x such that…

  Answer published by the source

  Identifiers

  upstream_id

  2238

  task_id

  0cf71184-ced4-57de-a486-ea040e5fda05

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2238

• OE_TO_maths_en_COMP / train

  What is the value of x such that log₂(log₂(2x−2))=2 ?

  Answer published by the source

  Identifiers

  upstream_id

  2241

  task_id

  066a4dec-d740-5fb1-a121-5db4401cbd82

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2241

• OE_TO_maths_en_COMP / train

  Let f(x)=2^(kx)+9, where k is a real number. If f(3):f(6)=1:3,…

  Answer published by the source

  Identifiers

  upstream_id

  2242

  task_id

  031a202f-3778-57b0-8b8b-17f35df68209

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2242

• OE_TO_maths_en_COMP / train

  Determine, with justification, all values of k for which y=x²−4 and…

  Answer published by the source

  Identifiers

  upstream_id

  2243

  task_id

  38c3b7c1-9937-50ed-bc0a-95cc9be57c49

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2243

• OE_TO_maths_en_COMP / train

  Determine all values of x for which …

  Answer published by the source

  Identifiers

  upstream_id

  2256

  task_id

  218ab57a-0c0b-5e51-afcf-8fa1db1f8196

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2256

• OE_TO_maths_en_COMP / train

  Let ⌊x⌋ denote the greatest integer less than or equal to x.…

  Answer published by the source

  Identifiers

  upstream_id

  2261

  task_id

  148fe168-6722-5706-98aa-0a4af55f00de

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2261

• OE_TO_maths_en_COMP / train

  A goat starts at the origin (0,0) and then makes several moves. On move 1 , it…

  Answer published by the source

  Identifiers

  upstream_id

  2265

  task_id

  4037d698-987d-575b-b8aa-20526067a4f5

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2265

• OE_TO_maths_en_COMP / train

  If f(x)=sin²x−2sinx+2, what are the minimum and maximum values of…

  Answer published by the source

  Identifiers

  upstream_id

  2268

  task_id

  0a3373db-954d-5656-9e04-1a1adca47751

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2268

• OE_TO_maths_en_COMP / train

  Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average…

  Answer published by the source

  Identifiers

  upstream_id

  2277

  task_id

  3fbf9328-6949-576a-95ef-f0d2ca44e435

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2277

• OE_TO_maths_en_COMP / train

  For some real numbers m and n, the list 3n²,m²,2(n+1)²…

  Answer published by the source

  Identifiers

  upstream_id

  2282

  task_id

  3ba0b427-fadf-5d41-85f6-3072c715a4be

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2282

• OE_TO_maths_en_COMP / train

  Chinara starts with the point (3,5), and applies the following three-step…

  Answer published by the source

  Identifiers

  upstream_id

  2283

  task_id

  1bdd1760-ef05-5047-9e74-a4a23f3d6eac

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2283

• OE_TO_maths_en_COMP / train

  Suppose that n>5 and that the numbers …

  Answer published by the source

  Identifiers

  upstream_id

  2286

  task_id

  28d37915-a8e8-5ada-818b-b78313d789ba

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2286

• OE_TO_maths_en_COMP / train

  Suppose that a=5 and b=4. Determine all pairs of integers (K,L) for which…

  Answer published by the source

  Identifiers

  upstream_id

  2292

  task_id

  221a1403-4140-535d-b152-258b8c9ef3bb

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2292

• OE_TO_maths_en_COMP / train

  Consider the following system of equations in which all logarithms have base 10:

  Answer published by the source

  Identifiers

  upstream_id

  2303

  task_id

  25a89b47-7f7f-5c48-a56b-c03a918dff46

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2303

• OE_TO_maths_en_COMP / train

  A Skolem sequence of order n is a sequence …

  Answer published by the source

  Identifiers

  upstream_id

  2319

  task_id

  2bd88933-8ead-575c-85f4-8890032a70ea

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2319

• OE_TO_maths_en_COMP / train

  Determine all integers k, with k≠0, for which the parabola with…

  Answer published by the source

  Identifiers

  upstream_id

  2326

  task_id

  2975736c-5486-5dcf-93ee-1e86c9e089e1

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2326

• OE_TO_maths_en_COMP / train

  A geometric sequence has first term 10 and common ratio (1)/(2).

  Answer published by the source

  Identifiers

  upstream_id

  2328

  task_id

  31f7c49c-5b0f-5b25-bc90-e37de24ab3e3

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2328

• OE_TO_maths_en_COMP / train

  Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in…

  Answer published by the source

  Identifiers

  upstream_id

  2333

  task_id

  212625a1-31f3-520b-b671-bb0a93a403ea

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2333

• OE_TO_maths_en_COMP / train

  A bag contains 40 balls, each of which is black or gold. Feridun reaches into…

  Answer published by the source

  Identifiers

  upstream_id

  2343

  task_id

  0649ffd5-6852-5f59-8df3-03ea41f09793

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2343

• OE_TO_maths_en_COMP / train

  If log₁₀x=3+log₁₀y, what is the value of (x)/(y) ?

  Answer published by the source

  Identifiers

  upstream_id

  2351

  task_id

  162a40d5-8b08-5567-bd05-9daa3d19eee3

  task_key

  OE~5fTO~5fmaths~5fen~5fCOMP--train--2351