2003 AIME II problem 1 | AIME | Math for fun and glory | Khan Academy
A1 A2 B1 B2 C1 C2
problem
US / ˈprɒbləm /
•UK / ˈprɑbləm /
- adjecitve
- Causing trouble
- noun
- Something difficult to deal with or causes trouble
- Question to show understanding of a math concept
invitational
US / ˌɪnvɪˈteɪʃənl /
•UK / ˌɪnvɪˈteʃənəl /
- other
- Pertaining to or characteristic of an invitation
could
US / kəd /
•UK / kʊd /
- other
- Past form of 'can' to mean have ability
- Used to ask for something politely, or to offer
know
US / nəʊ /
•UK / noʊ /
- verb
- To be familiar with a person or place, thing
- To feel sure or be convinced about something
- To discover or be aware of something
- To be able to distinguish
- To have knowledge of things
with
US / wɪð /
•UK / wɪð, wɪθ /
- preposition
- As an employee of a certain company
- Against something or someone
- By using something as a means
- Due to; because of
- Including
- Possessing as a feature
- Related to
- (Of the manner/attitude by which you do something)
- Used to express that people or things are together
negative
US / 'neɡətɪv /
•UK / ˈnɛɡətɪv /
- adjecitve
- Being harmful, unwanted or unhelpful
- In mathematics, being less than zero
- Focusing on the bad aspects; pessimistic
- noun
- The opposite to a positive electrical charge
- In grammar, containing words such as 'no' or 'not'
- Reply to a question or statement that means 'no'
- Image on camera film used to make a photo
see
US / si: /
•UK / si /
- verb
- To go on a date with someone
- To predict whether something is, or will be true
- To understand what someone says
- To use your eyes to look at something
- To visit a place often for sightseeing or pleasure
- To visit or meet with someone
- To find out by waiting or looking
- To watch (a game, movie or TV show)
have
US / həv /
•UK / hæv /
- verb
- To drink, smoke, eat or use something
- To experience the effects of something
- To organize an event, or join in an activity
- To own, possess, or hold something
- To cause to happen or produce a particular effect
last
US / lɑ:st /
•UK / læst /
- adjecitve
- Most recent; the latest
- The one before
- adverb
- Final; with nothing following; after all others
- Most recent; the latest
- determiner
- The one at the end
- verb
- To remain or continue for a certain period of time
information
US / ˌɪnfəˈmeɪʃn /
•UK / ˌɪnfɚˈmeʃən /
- noun
- Collection of facts and details about something
should
US / ʃʊd /
•UK / ʃʊd /
- other
- Used to indicate what is proper or reasonable
plus
US / plʌs /
•UK / plʌs /
- noun
- A benefit or advantage
- preposition
- In addition; add
- In addition to; Indicates a number above
- Having a value greater than zero
reason
US / 'ri:zn /
•UK / ˈrizən /
- noun
- Ability to think about facts and form a judgment
- Explanation for why something occurred or was done
- verb
- To think and make conclusions in a logical manner
side
US / saɪd /
•UK / saɪd /
- adjecitve
- In addition to the central or most important thing
- Next to, or very close to, the right or left of
- noun
- Position or opinion that is opposite to another
- Right or left part of a person's body
- Feature of someone's personality
- Straight edge of an object
- One of two or more groups fighting each other
- verb
- To decide to agree with one point, not the other
actually
US / ˈæktʃuəli /
•UK / ˈæktʃuəli /
- adverb
- Used to add new (often different) information
write
US / raɪt /
•UK / raɪt /
- verb
- To compose letters and words on paper or a screen
- To invent or create a computer program
- To create a song or piece of music
divide
US / dɪ'vaɪd /
•UK / dɪˈvaɪd /
- verb
- To split numbers by another number, e.g. 6 / 2 = 3
- To separate something into equal pieces
matter
US / 'mætə(r) /
•UK / ˈmætɚ /
- noun
- Material all things are made of that fills space
- Problem or reason for concern
- verb
- To be of great importance; to count
once
US / wʌns /
•UK / wʌns /
- adverb
- One time; one instance
- At a time in the past
- conjunction
- When; as soon as
always
US / 'ɔ:lweɪz /
•UK / ˈɔlwez,-wɪz,-wiz /
- adverb
- For a very long time
- At all times; in every situation
- For ever
many
US / 'menɪ /
•UK / ˈmɛni /
- determiner
- Used to refer a large number of things
will
US / wɪl /
•UK / wɪl /
- noun
- Desire or choice of someone
- Document saying who gets your money when you die
- Strong desire to do something; determination
- Desire to do something; strong wish to do
- other
- Used with verbs to express the future
- verb
- To leave property to someone after your death
- To use mental effort to make something happen
- Am (is, are) likely to do
- To influence someone to do something
- To make something happen by strongly wanting it
here
US / hɪə(r) /
•UK / hɪr /
- adverb
- In this place; at this location, position or time
try
US / traɪ /
•UK / traɪ /
- noun
- Act of putting the ball over the rugby goal line
- verb
- To make an effort, to attempt to do something
- To examine someone or something in a court of law
- To test to see if it is good or you like it
- To make a person go through hardship; test
- To examine something to see if it works
answer
US / 'ɑ:nsə(r) /
•UK / ˈænsɚ /
- noun
- Reply to a question someone asks
- Solution to a problem or test question
- verb
- To reply to a question someone asks
- To solve a test question or a problem
less
US / les /
•UK / lɛs /
- adjecitve
- Smaller in amount or degree
- adverb
- To a smaller degree
- noun
- A smaller amount
- preposition
- With the subtraction of
let
US / let /
•UK / lɛt /
- verb
- To allow someone to do something
- Introducing a suggestion to do something together
- To rent a house, etc. to others
little
US / 'lɪtl /
•UK / ˈlɪtl /
- adjecitve
- Being short in length of time
- Not great in size, amount, or degree; small
- Being very young or younger than others
- adverb
- By an amount or degree that is not very much
- Not often; rarely
- noun
- A person's name
the
US / ðə /
•UK / ðə /
- article
- Used to refer to something already mentioned
- Used to show there is only one of something
- determiner
- Used to describe all of a family
three
US / θri: /
•UK / θri /
- number
- Number 3
verify
US / 'verɪfaɪ /
•UK / ˈvɛrəˌfaɪ /
- verb
- To prove that something is right, accurate or true
other
US / 'ʌðə(r) /
•UK / ˈʌðər /
- adjecitve
- A thing different from that mentioned
- determiner
- (Something) else; not the first (one)
- pronoun
- Being the one and only thing of that category
one
US / wʌn /
•UK / wʌn /
- pronoun
- You; we; people in general; someone
- The thing or person previously referred to
- adjecitve
- Used to refer to people generally
- Being a strong example of (something mentioned)
- number
- Number 1
than
US / ðən /
•UK / ðən /
- conjunction
- Used to compare two things that are not equal
kind
US / kaɪnd /
•UK / kaɪnd /
- adjecitve
- In a caring and helpful manner
- friendly and considerate
- noun
- One type of thing
you
US / jʊ /
•UK / ju /
- pronoun
- Person someone is speaking or writing to
- Person or people in general
stand
US / stænd /
•UK / stænd /
- noun
- Effort to fight against something
- Piece of furniture; device to hold or support
- Place in a stadium where spectators watch
- Table where things are sold at a fair or market
- verb
- To be a candidate in an election
- To be able to accept someone, something; bear
- To last for a long time
- To be upright; not be sitting or lying down
- To place something in a particular position; place
- To be (as stated)
- To still be valid
- To have an opinion about something
this
US / ðɪs /
•UK / ðɪs /
- adjecitve
- Used to indicate something already been discussed
- Used to indicate something happening around now
- adverb
- An indication by a physical motion or gesture
- A specific amount of
- determiner
- Person, thing, or idea near you
- pronoun
- Person, thing that has already been discussed
from
US / frəm /
•UK / frʌm, frɑm,frəm /
- preposition
- Starting at a particular place, time or level
mathematic
US / ˌmæθɪ'mætɪk /
•UK / ˌmæθə'mætɪk /
- other
out
US / aʊt /
•UK / aʊt /
- adverb
- No longer stylish (e.g. fashion)
- Away from home, business, or the usual place
- In a direction/movement away from inside or center
- noun
- Act when a batter cannot continue playing
- verb
- To tell people a secret
over
US / 'əʊvə(r) /
•UK / ˈovɚ /
- adverb
- Finished; no longer happening
- Towards the ground/away from a standing position
- noun
- A set of six consecutive balls bowled in cricket
- other
- Finished
- preposition
- Showing the relation when a thing is above another
- On or covering something
- Above or across something
- More than you need
- From one side to the other
- In the direction of; toward
any
US / 'enɪ /
•UK / ˈɛni /
- adjecitve
- One, or some, no matter which
- pronoun
- One (thing) of many; some
think
US / θɪŋk /
•UK / θɪŋk /
- verb
- To have an idea about something without certainty
- To have an idea, opinion or belief about something
name
US / neɪm /
•UK / nem /
- noun
- What something or someone is called
- verb
- To state what someone is called, e.g. John
version
US / ˈvə:ʃən /
•UK / ˈvɚʒən, -ʃən /
- noun
- Different way that someone interprets something
- New or different form of something
- An updated form of some software
possibility
US / ˌpɒsə'bɪlətɪ /
•UK / ˌpɑsəˈbɪlɪti /
- noun
- Something with a chance of happening or being true
only
US / ˈəunli /
•UK / ˈonli /
- adverb
- Just one; just that amount or thing; not more than
- Not very important or serious
- Just that type of thing, person
- Short time ago; not before a certain time
- conjunction
- Except; but
two
US / tu: /
•UK / tu /
- number
- 2
large
US / lɑ:dʒ /
•UK / lɑ:rdʒ /
- adjecitve
- Big; of great size; broad, tall, wide, or fat
and
US / ənd /
•UK / ənd, ən,ænd /
- conjunction
- Used to refer to two or more things
- Plus; in addition; on top of that
- Used to introduce an action that follows another
call
US / kɔ:l /
•UK / kɔl /
- noun
- A order or request for action
- The sound an animal makes, often when in danger
- A correct prediction, e.g. a decision in a game
- A short visit to a place or person
- A decision by a referee or umpire in a game
- verb
- To make a request or order for action
- (Animal) to make sounds to warn of danger
- To announce the date of an election
- To make a judgment about something
- To give someone or something a name
- To guess in advance the result or outcome
- To visit a place or person for a short time
- To phone someone
- To speak, announce or shout in a loud voice
integer
US / ˈɪntɪdʒə(r) /
•UK / ˈɪntɪdʒɚ /
- noun
- Math a whole number
where
US / hwɛə /
•UK / hwɛr, wɛr /
- conjunction
- At or in the place which
- In whatever place
- pronoun
- Word you use to ask the location of something
another
US / əˈnʌðə(r) /
•UK / əˈnʌðɚ /
- adjecitve
- One more, but not this
- determiner
- One more added
- pronoun
- One more (thing)
they
US / ðeɪ /
•UK / ðe /
- pronoun
- Two or more people, animals, or things
all
US / ɔ:l /
•UK / ɔl /
- adverb
- Completely; totally
- determiner
- Being every one of something
- pronoun
- 100% of something; the complete amount
bit
US / bɪt /
•UK / bɪt /
- noun
- Device put in a horse's mouth to control it
- Small piece of something
- verb
- (E.g. of fish) to take bait and be caught
multiply
US / ˈmʌltɪplaɪ /
•UK / ˈmʌltəˌplaɪ /
- verb
- To increase in number by reproducing/having babies
- To increase something by a large quantity
- To have increased by a large quantity
- To calculate two numbers e.g. 2 x 4
add
US / æd /
•UK / æd /
- verb
- To include as well
- To put two things or numbers together, e.g. 2+2
take
US / teɪk /
•UK / tek /
- noun
- Filming of a single scene for TV or the movies
- verb
- To accept something that is offered
- To use a method of traveling, e.g. a bus or train
- To use a camera to make (a photo)
- To pick up something and go away with it
- To study a subject in school
- To attempt to pass (an exam)
- To require a certain amount of time, money, space
both
US / bəuθ /
•UK / boθ /
- determiner
- Used to refer to two things at the same time
- pronoun
- Used to refer to two things at the same time
their
US / ðɛː /
•UK / ðer /
- determiner
- Belonging to them
expression
US / ɪk'spreʃn /
•UK / ɪkˈsprɛʃən /
- noun
- Act of making your thoughts and feelings known
- Group of words that have a specific meaning
pretty
US / 'prɪtɪ /
•UK / ˈprɪti /
- adjecitve
- Being attractive to the eye in a simple way
- adverb
- Very; to a great degree
possible
US / ˈpɒsəbl /
•UK / ˈpɑsəbəl /
- adjecitve
- Being able or appropriate for a task or purpose
- Having a chance of happening, or being true
work
US / wɜ:k /
•UK / wɚk /
- adjecitve
- Activity completed in order to make money
- noun
- The product of some artistic or literary endeavor
- Everything created by an author, artist, musician
- Activity you do in order to make money
- Place of employment
- Final product or result of a some labor or effort
- verb
- To bring into a specific state of success
- To be functioning properly, e.g. a car
- To do your job in your company or workplace
- To make an effort to do something
- To operate a machine or device
again
US / ə'ɡen /
•UK / əˈɡɛn /
- adverb
- One more time, once more
about
US / ə'baʊt /
•UK / əˈbaʊt /
- adverb
- To be ready to; going to
- Nearly; approximately; roughly
- Near; next to; close to
- In every direction; in the area of
- preposition
- Concerning or of (a topic)
constraint
US / kənˈstreɪnt /
•UK / kənˈstrent /
- noun
- Thing preventing you moving or acting; restriction
care
US / keə(r) /
•UK / kɛr /
- noun
- Treatment you receive for your health or safety
- Something that a person worries about
- verb
- To feel interest, concern, or worry
- To look after someone or something
that
US / ðæt /
•UK / ðæt, ðət /
- adjecitve
- The person or thing being talked about
- Used to identify something both the speakers know
- adverb
- Used to reinforce adjectives and adverbs
- To such a degree
- conjunction
- Used to connect a noun clause
- Used to introduce an adverbial clause
- Used to introduce a noun clause
- determiner
- Used as a noun to refer to something
- pronoun
- Referring an object far away
- Used to refer to the relative pronoun ‘which’
what
US / wɒt /
•UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /
- pronoun
- Thing or things that; the same kind as
- Used to emphasize something you are about to say
- adjecitve
- Used to refer to a particular amount, or number
- Used to place emphasis on a particular quality
- adverb
- To a certain degree
- determiner
- Question words used for asking for information
- exclamation
- Used to express excitement, shock, or surprise
- Used to ask people to repeat something they said
first
US / fɜ:st /
•UK / fɚst /
- adjecitve
- 1st; Coming before all others in time or place
- noun
- Something that has not happened before
- Position of the winner of a competition
get
US / ɡet /
•UK / ɡɛt /
- verb
- To become affected by illness or disease
- To begin to feel or understand an emotion or idea
- To send or bring someone or something home, etc.
- To go somewhere to obtain something
- To (cause to) do a particular thing
- To obtain, receive or be given something
- To currently have
- To prepare for use
- To understand something being said or read
find
US / faɪnd /
•UK / faɪnd /
- noun
- Object discovered by chance
- Things that have been discovered from searching
- verb
- To become aware of something that is happening
- To discover or meet by chance
- To decide if someone is guilty of a crime, or not
- To learn through study, research, or effort
- To discover something by looking for it
boil
US / bɔɪl /
•UK / bɔɪl /
- verb
- To become extremely angry or excited
- To cook in very hot water
- To make liquids hot so they turn to gas
aime
US / ˈeɪlməntz /
•UK / ˈeɪlməntz /
essentially
US / ɪˈsenʃəli /
•UK / ɪˈsenʃəli /
- adverb
- Basically; (said when stating the basic facts)
fun
US / fʌn /
•UK / fʌn /
- adjecitve
- Amusing and enjoyable
- noun
- Enjoyable or amusing quality or feeling
everywhere
US / ˈevriweə(r) /
•UK / ˈevriwer /
- adverb
- Every place
right
US / raɪt /
•UK / raɪt /
- adjecitve
- Correct or true
- The south side of you looking at the rising sun
- adverb
- Exactly in a place or time
- Being appropriate for a particular event
- In a straight or direct manner
- noun
- Something you are legally or morally allowed to do
- Ideal of what is just and good
- Political view based on conservatism
- verb
- To fix something previously wrong or unjust
- To make upright, as a boat that has turned over
value
US / 'vælju: /
•UK / ˈvælju /
- noun
- Importance you attach to something
- Some information expressed as a number
- Stated price of something; worth
- verb
- To place a high importance on something
- To say how much money something is worth
interesting
US / ˈɪntrəstɪŋ /
•UK / ˈɪntrɪstɪŋ, -tərɪstɪŋ, -təˌrɛstɪŋ /
- adjecitve
- Taking your attention; making you want to know
- verb
- To make someone want to know about something
- To persuade to do, become involved with something
number
US / ˈnʌmbə(r) /
•UK / ˈnʌmbɚ /
- noun
- Symbols such as 1, 2, 56, 793
- Particular song or dance performed during a show
- Total quantity or amount of things
- verb
- To put numbers on things
- To assign a sequence within a group, series, set
- To claim to be part of a larger group, series, set
down
US / daʊn /
•UK / daʊn /
- adjecitve
- Being sad, sick, or unhappy
- Not operating, working, or functioning as expected
- adverb
- In a sitting or lying position
- Towards or on the ground, floor, or bottom
- Written or copied onto a piece of paper or book
- noun
- A period of play in American Football
- The soft, smooth hairs that cover a bird's body
- preposition
- Along something's path, course, or direction
- Going from a higher position to a lower position
- verb
- To eat or drink very quickly and without stopping
then
US / ðen /
•UK / ðɛn /
- adverb
- A time that is not now
- Following another thing in time or arrangement
- Following on reasonably from what was just stated
- At that time, not now
for
US / fə(r) /
•UK / fɔr,fə /
- preposition
- Used to show the purpose, or need of something
time
US / taɪm /
•UK / taɪm /
- noun
- Speed at which music is played; tempo
- Point as shown on a clock, e.g. 3 p.m
- Number of hours, minutes needed to do something
- Occasion when something happens
- Period or occasion that something occurred
- Period in history or the past
- Something measured in minutes, hours, days, etc.
- How long an event takes; duration
- verb
- To check speed at which music is performed
- To choose a specific moment to do something
- To measure how long an event takes, e.g. a race
- To schedule something to occur at a specific moment
through
US / θru: /
•UK / θru /
- adjecitve
- having finished using or doing something
- adverb
- From the beginning of something until the end
- preposition
- Over, in, across an entire thing or place
- Allowing you to pass between, or to
- By a particular way or process; by the use of
why
US / waɪ /
•UK / hwaɪ, waɪ /
- adverb
- Question word to ask for a reason
- exclamation
- (Used to express surprise or approval)
definitely
US / ˈdefɪnətli /
•UK / ˈdefɪnətli /
- adverb
- Without question; beyond doubt
column
US / ˈkɒləm /
•UK / ˈkɑ:ləm /
- noun
- Regular series of articles in a newspaper/magazine
- Strong vertical stone pillar that holds up a roof
- Vertical line in a table or chart
replace
US / rɪ'pleɪs /
•UK / rɪˈples /
- verb
- To use instead of something else
- To return an item to its original location
those
US / ðəʊz /
•UK / ðoz /
- determiner
- The people or things we are talking about
- pronoun
- The objects far from the speaker
just
US / dʒʌst /
•UK / dʒʌst /
- adjecitve
- Proper or appropriate; as is deserved
- Doing or being what is right or fair
- Legally correct; having enough evidence
- adverb
- Almost; nearly
- Exactly at the time of
- Exactly (the same as, like)
positive
US / ˈpɒzətɪv /
•UK / ˈpɑzɪtɪv /
- adjecitve
- Showing agreement or support for something
- Being sure about something; knowing the truth
- Having the charge produced by electrons
- Being good or useful
- Showing that a chemical is present in a test
fraction
US / ˈfrækʃn /
•UK / ˈfrækʃən /
- noun
- Small part of something
since
US / sɪns /
•UK / sɪns /
- adverb
- At a point before the present time
- conjunction
- For this reason; because
- preposition
- From the time in the past that
undefined
US / ˌʌndɪˈfaɪnd /
•UK / ˌʌndɪˈfaɪnd /
- adjecitve
- Not precisely limited or described
well
US / wel /
•UK / wɛl /
- adjecitve
- Being in a fortunate, good condition or state
- Healthy and in good condition
- adverb
- In an acceptable, proper, or successful manner
- In a complete, total, or full manner
- A great and considerable degree
- In a successful way, in a good or satisfactory way
- exclamation
- Used to begin or continue a particular statement
- Used to show you want to start saying something
- Used to express that you are waiting for a reply
- You say this to change the topic of a conversation
- Used to indicate you are making a final remark
- noun
- Long deep hole in the ground to get water
- verb
- To cause your eyes to fill with tears
become
US / bɪ'kʌm /
•UK / bɪˈkʌm /
- verb
- To begin to be; grow to be; develop into
- To be suitable for a particular job or role
same
US / seɪm /
•UK / sem /
- adjecitve
- Being equal in quality and amount
- Identical; not different
- Not changing; constant
pick
US / pɪk /
•UK / pɪk /
- noun
- Sharp tool used for breaking the ground
- Item selected from a group of choices
- verb
- To take fruit or vegetables from a plant or tree
- To decide on a thing from various choices; select
- To remove (flower, etc.) with your fingers
would
US / wʊd /
•UK / wʊd /
- other
- Used to indicate something imagined
- Used to ask or request something politely
- Verb used to indicate possibility, request etc.
- Used to indicate a preference
- Used to indicate a reaction
- Indicates something that occurred repeatedly
- Used to indicate willingness or ability to do
but
US / bət /
•UK / bʌt,bət /
- conjunction
- Used before you say something different, opposite
give
US / ɡɪv /
•UK / ɡɪv /
- noun
- Degree of flexibility in something, a material
- verb
- To hand over or present something to someone
- To cause someone to have or experience something
can
US / kæn /
•UK / kən,kæn /
- noun
- A metal container for storing food
- other
- Able to; have the skill to; have the time to
- verb
- To have permission to
- To be possible to do
- To preserve food by sealing it in a metal container
deal
US / di:l /
•UK / dil /
- verb
- To cope with something - usually troubles
- To give (something bad e.g. news) to
- To buy and sell illegal drugs
- To give out (cards, etc.) to; distribute
- To do business with someone or to sell products
equal
US / 'i:kwəl /
•UK / ˈikwəl /
- adjecitve
- Same in shape, size, or number
- verb
- To add up or be the same as
sum
US / sʌm /
•UK / sʌm /
- noun
- Amount when all is added together; total
- verb
- To add two or more numbers together; total
want
US / wɔnt /
•UK / wɑnt, wɔnt /
- noun
- Something you desire or wish for
- State or condition of poverty
- verb
- To desire or wish for something; hope for a thing
color
US / 'kʌlə(r) /
•UK / ˈkʌlɚ /
- noun
- Quality of things you can see, e.g. red, blue
- Pink or red in your face, e.g. after being ill
- The shade of someone's skin e.g. black, or yellow
- verb
- To change or affect someone's opinion
- To make something colorful using colored pencils
manipulation
US / məˌnɪpjʊ'leɪʃn /
•UK / məˌnɪpjʊˈleʃən /
- noun
- Act of controlling to gain an advantage
- Act of moving something using your hands
maybe
US / ˈmeɪbi /
•UK / ˈmebi /
- adverb
- Possibly but not certainly; perhaps
not
US / nɒt /
•UK / nɑt /
- adverb
- Word indicating the negative
- (Used to form the negative of verbs)
our
US / 'aʊə(r) /
•UK / aʊr /
- determiner
- Belonging to us
relationship
US / rɪˈleɪʃnʃɪp /
•UK / rɪˈleʃənˌʃɪp /
- noun
- Connection between two or more people or things
- Way people interact or live with each other
- Romantic or loving friendship between two people