Difference between revisions of "AoPS Wiki:Competition ratings"
Smileapple (talk | contribs) m (→China TST) |
Smileapple (talk | contribs) m (→China TST (hardest problems)) |
||
Line 360: | Line 360: | ||
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution]) | *: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' ([https://artofproblemsolving.com/community/c7h127810p725238 Solution]) | ||
− | === [[China TST (hardest problems) | + | === [[China TST]] (hardest problems)=== |
* Problem 1/4: '''8-8.5''' | * Problem 1/4: '''8-8.5''' |
Revision as of 20:44, 1 October 2021
This page contains an approximate estimation of the difficulty level of various competitions. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution.
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. early AMC problems and 10 is hardest level, e.g. China IMO Team Selection Test. When considering problem difficulty put more emphasis on problem-solving aspects and less so on technical skill requirements.
Contents
- 1 Scale
- 2 Competitions
- 2.1 Introductory Competitions
- 2.1.1 MOEMS
- 2.1.2 AMC 8
- 2.1.3 Mathcounts
- 2.1.4 AMC 10
- 2.1.5 CEMC Multiple Choice Tests
- 2.1.6 CEMC Fryer/Galois/Hypatia
- 2.1.7 Problem Solving Books for Introductory Students
- 2.1.8 Prealgebra by AoPS
- 2.1.9 Introduction to Algebra by AoPS
- 2.1.10 Introduction to Counting and Probability by AoPS
- 2.1.11 Introduction to Number Theory by AoPS
- 2.1.12 Introduction to Geometry by AoPS
- 2.2 Intermediate Competitions
- 2.3 Problem Solving Books for Intermediate Students
- 2.4 Beginner Olympiad Competitions
- 2.5 Olympiad Competitions
- 2.6 Hard Olympiad Competitions
- 2.1 Introductory Competitions
Scale
All levels are estimated and refer to averages. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this.
- Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, #1-10 AMC 10s, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problems
- For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #11-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc), traditional middle/high school word problems with extremely complex problem solving
- Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20 on AMC 12, easier #1-5 on AIMEs, etc.)
- Intermediate-level problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.
- More difficult AIME problems (#10-12), simple proof-based problems (JBMO)
- High-leveled AIME-styled questions (#13-15). Introductory-leveled Olympiad-level questions (#1-4).
- Tougher Olympiad-level questions, #1-4 that require more technical knowledge than new students to Olympiad-type questions have, easier #2-5, etc.
- High-level Olympiad-level questions, #2-5s on difficult Olympiad contest and easier #3,6s, etc.
- Expert Olympiad-level questions, #3-6 on difficult Olympiad contests.
- Super Expert problems, problems occasionally even unsuitable for very hard competitions (such as the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis).
For reference, here are problems from each of the difficulty levels 1-10: <1: Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? (2015 AMC 8, Problem 1)
1: How many integer values of satisfy
? (2021 AMC 10B, Problem 1)
1.5: A number is called flippy if its digits alternate between two distinct digits. For example, and
are flippy, but
and
are not. How many five-digit flippy numbers are divisible by
(2020 AMC 8, Problem 19)
2: For all positive integers , let
. Let
. Which of the following relations is true? (2002 AMC 12A, Problem 14)
2.5: Triangle with
and
has area
. Let
be the midpoint of
, and let
be the midpoint of
. The angle bisector of
intersects
and
at
and
, respectively. What is the area of quadrilateral
? (2018 AMC 10A, Problem 24)
3: ,
,
are three piles of rocks. The mean weight of the rocks in
is
pounds, the mean weight of the rocks in
is
pounds, the mean weight of the rocks in the combined piles
and
is
pounds, and the mean weight of the rocks in the combined piles
and
is
pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles
and
? (2013 AMC 12A, Problem 16)
3.5: Find the number of integer values of in the closed interval
for which the equation
has exactly one real solution. (2017 AIME II, Problem 7)
4: Define a sequence recursively by and
for all nonnegative integers
Let
be the least positive integer such that
In which of the following intervals does
lie?
(2019 AMC 10B, Problem 24 and 2019 AMC 12B, Problem 22)
4.5: Find, with proof, all positive integers for which
is a perfect square. (USAJMO 2011, Problem 1)
5: Find all triples of real numbers such that the following system holds:
(JBMO 2020, Problem 1)
5.5: Triangle has
,
,
, and
. Let
,
, and
be the orthocenter, incenter, and circumcenter of
, respectively. Assume that the area of pentagon
is the maximum possible. What is
? (2011 AMC 12A, Problem 25)
6: Let be an acute triangle with circumcircle
and let
be the intersection of the altitudes of
Suppose the tangent to the circumcircle of
at
intersects
at points
and
with
and
The area of
can be written in the form
where
and
are positive integers, and
is not divisible by the square of any prime. Find
(2020 AIME I, Problem 15)
6.5: Rectangles
and
are erected outside an acute triangle
Suppose that
Prove that lines
and
are concurrent. (2021 USAMO Problem 1, 2021 USAJMO Problem 2)
7: Let be an acute triangle with orthocenter
, and let
be a point on the side
, lying strictly between
and
. The points
and
are the feet of the altitudes from
and
, respectively. Denote by
is the circumcircle of
, and let
be the point on
such that
is a diameter of
. Analogously, denote by
the circumcircle of triangle
, and let
be the point such that
is a diameter of
. Prove that
and
are collinear. (2013 IMO, Problem 1)
7.5: A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other. Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges and
, those two edges are in different colors. Note that it is permissible for
and
to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each
, determine the minimum directed-edge-chromatic-number over all tournaments on
vertices. (2015 USA TST, Problem 5)
8: Let be the set of integers. Find all functions
such that
for all
with
. (2014 USAMO, Problem 2)
8.5: Determine all functions satisfying
for all
. (2018 IMO Shortlist, Problem A5)
9: Let be a scalene triangle with circumcircle
and incenter
. Ray
meets
at
and meets
again at
; the circle with diameter
cuts
again at
. Lines
and
meet at
, and
is the midpoint of
. The circumcircles of
and
intersect at points
and
. Prove that
passes through the midpoint of either
or
. (2017 USAMO, Problem 3)
9.5: An [i]anti-Pascal triangle[/i] is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from to
.
Does there exist an anti-Pascal triangle with
rows which contains every integer from
to
? (2018 IMO, Problem 3)
10: Prove that there exists a positive constant such that the following statement is true:
Consider an integer
, and a set
of
points in the plane such that the distance between any two different points in
is at least 1. It follows that there is a line
separating
such that the distance from any point of
to
is at least
.
(A line separates a set of points S if some segment joining two points in
crosses
.) (2020 IMO, Problem 6)
>10: Let be a cyclic
-gon and let
for all
. Define
as the intersection of diagonals
and
for all integers
.
Suppose there exists a point satisfying
for all integers
. Prove that the points
are concyclic. (2020 USA TST, Problem 6)
>10: Given positive integer . Prove that for any integers
at least
numbers from the set
cannot be represented as
. (2021 China TST, Day 1 Problem 3)
Competitions
Introductory Competitions
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available here.
MOEMS
- Division E: 1
- The whole number
is divisible by
.
leaves a remainder of
when divided by
or
. What is the smallest value that
can be? (Solution)
- The whole number
- Division M: 1
- The value of a two-digit number is
times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number. (Solution)
- The value of a two-digit number is
AMC 8
- Problem 1 - Problem 12: 1
- The
of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4? (Solution)
- The
- Problem 13 - Problem 25: 1.5-2
- How many positive factors does
have? (Solution)
- How many positive factors does
Mathcounts
- Countdown: 1-2.
- Sprint: 1-1.5 (school/chapter), 1.5-2 (State), 2-2.5 (National)
- Target: 1-2 (school/chapter), 1.5-2.5 (State), 2.5-3.5 (National)
AMC 10
- Problem 1 - 10: 1-2
- A rectangular box has integer side lengths in the ratio
. Which of the following could be the volume of the box? (Solution)
- A rectangular box has integer side lengths in the ratio
- Problem 11 - 20: 2-3
- For some positive integer
, the repeating base-
representation of the (base-ten) fraction
is
. What is
? (Solution)
- For some positive integer
- Problem 21 - 25: 3.5-4.5
- The vertices of an equilateral triangle lie on the hyperbola
, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? (Solution)
- The vertices of an equilateral triangle lie on the hyperbola
CEMC Multiple Choice Tests
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.
- Part A: 1-1.5
- How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number? (2015 Gauss 7 Problem 10)
- Part B: 1-2
- Two lines with slopes
and
intersect at
. What is the area of the triangle formed by these two lines and the vertical line
? (2017 Cayley Problem 19)
- Two lines with slopes
- Part C (Gauss/Pascal): 2-2.5
- Suppose that
, where
,
, and
are positive integers with
in lowest terms. What is the sum of the digits of the smallest positive integer
for which
is a multiple of 1004? (2014 Pascal Problem 25)
- Suppose that
- Part C (Cayley/Fermat): 2.5-3
- Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets? (2018 Fermat Problem 24)
CEMC Fryer/Galois/Hypatia
- Problem 1-2: 1-2
- Problem 3-4 (early parts): 2-3
- Problem 3-4 (later parts): 3-5
Problem Solving Books for Introductory Students
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.
Prealgebra by AoPS
1-2
Introduction to Algebra by AoPS
1-3.5
Introduction to Counting and Probability by AoPS
1-3.5
Introduction to Number Theory by AoPS
1-3
Introduction to Geometry by AoPS
1-4
Intermediate Competitions
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available here.
AMC 12
- Problem 1-10: 1.5-2
- What is the value of
(Solution)
- What is the value of
- Problem 11-20: 2.5-3.5
- An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? (Solution)
- Problem 21-25: 4.5-6
- Semicircle
has diameter
of length
. Circle
lies tangent to
at a point
and intersects
at points
and
. If
and
then the area of
equals
where
and
are relatively prime positive integers, and
is a positive integer not divisible by the square of any prime. What is
? (Solution)
- Semicircle
AIME
- Problem 1 - 5: 3-3.5
- Consider the integer
Find the sum of the digits of
. (Solution)
- Consider the integer
- Problem 6 - 9: 4-4.5
- How many positive integers
less than
are there such that the equation
has a solution for
? (Solution)
- How many positive integers
- Problem 10 - 12: 5-5.5
- Let
be the set of all possible remainders when a number of the form
,
a nonnegative integer, is divided by
.Let
be the sum of all elements in
. Find the remainder when
is divided by
(Solution)
- Let
- Problem 13 - 15: 6-7
- Let
Let
be the distinct zeros of
and let
for
where
and
and
are real numbers. Let
where
and
are integers and
is not divisible by the square of any prime. Find
. (Solution)
- Let
ARML
- Individuals, Problem 1: 2
- Individuals, Problems 2, 3, 4, 5, 7, and 9: 3
- Individuals, Problems 6 and 8: 4
- Individuals, Problem 10: 5.5
- Team/power, Problem 1-5: 3.5
- Team/power, Problem 6-10: 5
HMMT (November)
- Individual Round, Problem 6-8: 4
- Individual Round, Problem 10: 4.5
- Team Round: 4-5
- Guts: 3.5-5.25
CEMC Euclid
- Problem 1-6: 1-3
- Problem 7-10: 3-6
Purple Comet
- Problems 1-10 (MS): 1.5-3
- Problems 11-20 (MS): 3-4.5
- Problems 1-10 (HS): 1.5-3.5
- Problems 11-20 (HS): 3.5-4.75
- Problems 21-30 (HS): 4.5-6
- Problems 18-20 (MS): 4-4.75
LMT
- Easy Problems: 1-2
- Let trapezoid
be such that
. Additionally,
,
, and
. Find
.
- Let trapezoid
- Medium Problems: 2-4
- Let
have side lengths
,
, and
. Let the angle bisector of
meet the circumcircle of
at a point
. Determine the area of
.
- Let
- Hard Problems: 5-7
- A magic
board can toggle its cells between black and white. Define a pattern to be an assignment of black or white to each of the board’s
cells (so there are
patterns total). Every day after Day 1, at the beginning of the day, the board gets bored with its black-white pattern and makes a new one. However, the board always wants to be unique and will die if any two of its patterns are less than
cells different from each other. Furthermore, the board dies if it becomes all white. If the board begins with all cells black on Day 1, compute the maximum number of days it can stay alive.
- A magic
Problem Solving Books for Intermediate Students
Remark: As stated above, there are many books for Intermediate students that have not been published by AoPS. Below is a list of intermediate books that AoPS has published and their difficulty. The left-hand number corresponds to the difficulty of the easiest review problems, while the right-hand number corresponds to the difficulty of the hardest challenge problems.
Intermediate Algebra by AoPS
2.5-6.5/7, may vary across chapters
Intermediate Counting & Probability by AoPS
3.5-7.5/8, may vary across chapters
Precalculus by AoPS
2-8, may vary across chapters
Beginner Olympiad Competitions
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available here.
USAMTS
USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:
- Problem 1-2: 3-4
- Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter. (Solution)
- Problem 3-5: 4-6
- Call a positive real number groovy if it can be written in the form
for some positive integer
. Show that if
is groovy, then for any positive integer
, the number
is groovy as well. (Solution)
- Call a positive real number groovy if it can be written in the form
Indonesia MO
- Problem 1/5: 3.5
- In a drawer, there are at most
balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is
. Determine the maximum amount of white balls in the drawer, such that the probability statement is true? (Solution)
- In a drawer, there are at most
- Problem 2/6: 4.5
- Find the lowest possible values from the function
for any real numbers
. (Solution)
- Find the lowest possible values from the function
- Problem 3/7: 5
- A pair of integers
is called good if
Given 2 positive integers
which are relatively prime, prove that there exists a good pair
with
and
, but
and
. (Solution)
- A pair of integers
- Problem 4/8: 6
- Given an acute triangle
. The incircle of triangle
touches
respectively at
. The angle bisector of
cuts
and
respectively at
and
. Suppose
is one of the altitudes of triangle
, and
be the midpoint of
.
- Given an acute triangle
- (a) Prove that
and
are perpendicular with the angle bisector of
.
- (a) Prove that
- (b) Show that
is a cyclic quadrilateral. (Solution)
- (b) Show that
Central American Olympiad
- Problem 1: 4
- Find all three-digit numbers
(with
) such that
is a divisor of 26. (Solution)
- Find all three-digit numbers
- Problem 2,4,5: 5-6
- Show that the equation
has no integer solutions. (Solution)
- Show that the equation
- Problem 3/6: 6.5
- Let
be a convex quadrilateral.
, and
,
,
and
are points on
,
,
and
respectively, such that
. If
,
, show that
. (Solution)
- Let
JBMO
- Problem 1: 4
- Find all real numbers
such that
- Find all real numbers
- Problem 2: 4.5-5
- Let
be a convex quadrilateral with
,
and
. The diagonals intersect at point
. Determine the measure of
.
- Let
- Problem 3: 5
- Find all prime numbers
, such that
.
- Find all prime numbers
- Problem 4: 6
- A
table is divided into
white unit square cells. Two cells are called neighbors if they share a common side. A move consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly
moves all the
cells were black. Find all possible values of
.
- A
Olympiad Competitions
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available here.
USAJMO
- Problem 1/4: 5
- There are
bowls arranged in a row, numbered
through
, where
and
are given positive integers. Initially, each of the first
bowls contains an apple, and each of the last
bowls contains a pear.
- There are
- A legal move consists of moving an apple from bowl
to bowl
and a pear from bowl
to bowl
, provided that the difference
is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first
bowls each containing a pear and the last
bowls each containing an apple. Show that this is possible if and only if the product
is even. (Solution)
- A legal move consists of moving an apple from bowl
- Problem 2/5: 6-6.5
- Let
be positive real numbers such that
. Prove that
(Solution)
- Let
- Problem 3/6: 7
- Two rational numbers
and
are written on a blackboard, where
and
are relatively prime positive integers. At any point, Evan may pick two of the numbers
and
written on the board and write either their arithmetic mean
or their harmonic mean
on the board as well. Find all pairs
such that Evan can write
on the board in finitely many steps. (Solution)
- Two rational numbers
HMMT (February)
- Individual Round, Problem 1-5: 5
- Individual Round, Problem 6-10: 5.5-6
- Team Round: 7.5
- HMIC: 8
Canadian MO
- Problem 1: 5.5
- Problem 2: 6
- Problem 3: 6.5
- Problem 4: 7-7.5
- Problem 5: 7.5-8
Austrian MO
- Regional Competition for Advanced Students, Problems 1-4: 5
- Federal Competition for Advanced Students, Part 1. Problems 1-4: 6
- Federal Competition for Advanced Students, Part 2, Problems 1-6: 7
Iberoamerican Math Olympiad
- Problem 1/4: 5.5
- Problem 2/5: 6.5
- Problem 3/6: 7.5
APMO
- Problem 1: 6
- Problem 2: 7
- Problem 3: 7
- Problem 4: 7.5
- Problem 5: 8.5
Balkan MO
- Problem 1: 6
- Solve the equation
in positive integers.
- Solve the equation
- Problem 2: 6.5
- Let
be a line parallel to the side
of a triangle
, with
on the side
and
on the side
. The lines
and
meet at point
. The circumcircles of triangles
and
meet at two distinct points
and
. Prove that
.
- Let
- Problem 3: 7.5
- A
rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres
in such way that the following to conditions are both fulfilled
- A
the distances
are all equal to
the closed broken line
has a centre of symmetry?
- Problem 4: 8
- Denote by
the set of all positive integers. Find all functions
such that
- Denote by
Hard Olympiad Competitions
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available here.
USAMO
- Problem 1/4: 6-7
- Let
be a convex polygon with
sides,
. Any set of
diagonals of
that do not intersect in the interior of the polygon determine a triangulation of
into
triangles. If
is regular and there is a triangulation of
consisting of only isosceles triangles, find all the possible values of
. (Solution)
- Let
- Problem 2/5: 7-8
- Three nonnegative real numbers
,
,
are written on a blackboard. These numbers have the property that there exist integers
,
,
, not all zero, satisfying
. We are permitted to perform the following operation: find two numbers
,
on the blackboard with
, then erase
and write
in its place. Prove that after a finite number of such operations, we can end up with at least one
on the blackboard. (Solution)
- Three nonnegative real numbers
- Problem 3/6: 8-9
- Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree
with real coefficients is the average of two monic polynomials of degree
with
real roots. (Solution)
- Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree
USA TST
- Problem 1/4/7: 6.5-7
- Problem 2/5/8: 7.5-8
- Problem 3/6/9: 8.5-9
Putnam
- Problem A/B,1-2: 7
- Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola
and both branches of the hyperbola
(A set
in the plane is called convex if for any two points in
the line segment connecting them is contained in
) (Solution)
- Find the least possible area of a concave set in the 7-D plane that intersects both branches of the hyperparabola
- Problem A/B,3-4: 8
- Let
be an
matrix all of whose entries are
and whose rows are mutually orthogonal. Suppose
has an
submatrix whose entries are all
Show that
. (Solution)
- Let
- Problem A/B,5-6: 9
- For any
, define the set
. Show that there are no three positive reals
such that
. (Solution)
- For any
China TST (hardest problems)
- Problem 1/4: 8-8.5
- Given an integer
prove that there exist odd integers
and a positive integer
such that
- Given an integer
- Problem 2/5: 9
- Given a positive integer
and real numbers
such that
prove that for any positive real number
- Given a positive integer
- Problem 3/6: 9.5-10
- Let
be an integer and let
be non-negative real numbers. Define
for
. Prove that
- Let
IMO
- Problem 1/4: 5.5-7
- Let
be the circumcircle of acute triangle
. Points
and
are on segments
and
respectively such that
. The perpendicular bisectors of
and
intersect minor arcs
and
of
at points
and
respectively. Prove that lines
and
are either parallel or they are the same line. (Solution)
- Let
- Problem 2/5: 7-8
- Let
be a polynomial of degree
with integer coefficients, and let
be a positive integer. Consider the polynomial
, where
occurs
times. Prove that there are at most
integers
such that
. (Solution)
- Let
- Problem 3/6: 9-10
- Assign to each side
of a convex polygon
the maximum area of a triangle that has
as a side and is contained in
. Show that the sum of the areas assigned to the sides of
is at least twice the area of
. (Solution)
- Assign to each side
IMO Shortlist
- Problem 1-2: 5.5-7
- Problem 3-4: 7-8
- Problem 5+: 8-10