Difference between revisions of "2023 AIME II Problems"
MRENTHUSIASM (talk | contribs) (→Problem 13) |
MRENTHUSIASM (talk | contribs) (The original file uses English words for Problem 2.) |
||
(16 intermediate revisions by 8 users not shown) | |||
Line 94: | Line 94: | ||
==Problem 14== | ==Problem 14== | ||
− | + | ||
+ | A cube-shaped container has vertices <math>A,</math> <math>B,</math> <math>C,</math> and <math>D,</math> where <math>\overline{AB}</math> and <math>\overline{CD}</math> are parallel edges of the cube, and <math>\overline{AC}</math> and <math>\overline{BD}</math> are diagonals of faces of the cube, as shown. Vertex <math>A</math> of the cube is set on a horizontal plane <math>\mathcal{P}</math> so that the plane of the rectangle <math>ABDC</math> is perpendicular to <math>\mathcal{P},</math> vertex <math>B</math> is <math>2</math> meters above <math>\mathcal{P},</math> vertex <math>C</math> is <math>8</math> meters above <math>\mathcal{P},</math> and vertex <math>D</math> is <math>10</math> meters above <math>\mathcal{P}.</math> The cube contains water whose surface is parallel to <math>\mathcal{P}</math> at a height of <math>7</math> meters above <math>\mathcal{P}.</math> The volume of water is <math>\frac{m}{n}</math> cubic meters, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
+ | <asy> | ||
+ | //Made by Djmathman (orz) | ||
+ | size(250); | ||
+ | defaultpen(linewidth(0.6)); | ||
+ | pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y; | ||
+ | pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W; | ||
+ | pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8); | ||
+ | filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2)); | ||
+ | fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9)); | ||
+ | draw(A--B--Z--X--A--Y--C--X^^C--D--Z); | ||
+ | draw(P1--P2--P3--P4--cycle^^D--P4); | ||
+ | dot("$A$",A,S); | ||
+ | dot("$B$",B,S); | ||
+ | dot("$C$",C,N); | ||
+ | dot("$D$",D,N); | ||
+ | label("$\mathcal P$",(-13,4.5)); | ||
+ | </asy> | ||
[[2023 AIME II Problems/Problem 14|Solution]] | [[2023 AIME II Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
− | + | ||
+ | For each positive integer <math>n</math> let <math>a_n</math> be the least positive integer multiple of <math>23</math> such that <math>a_n \equiv 1 \pmod{2^n}.</math> Find the number of positive integers <math>n</math> less than or equal to <math>1000</math> that satisfy <math>a_n = a_{n+1}.</math> | ||
[[2023 AIME II Problems/Problem 15|Solution]] | [[2023 AIME II Problems/Problem 15|Solution]] |
Latest revision as of 08:51, 13 June 2024
2023 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is Find the greatest number of apples growing on any of the six trees.
Problem 2
Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than that is a palindrome both when written in base ten and when written in base eight, such as
Problem 3
Let be an isosceles triangle with
There exists a point
inside
such that
and
Find the area of
Problem 4
Let and
be real numbers satisfying the system of equations
Let
be the set of possible values of
Find the sum of the squares of the elements of
Problem 5
Let be the set of all positive rational numbers
such that when the two numbers
and
are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of
can be expressed in the form
where
and
are relatively prime positive integers. Find
Problem 6
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points and
are chosen independently and uniformly at random from inside the region. The probability that the midpoint of
also lies inside this L-shaped region can be expressed as
where
and
are relatively prime positive integers. Find
Problem 7
Each vertex of a regular dodecagon (-gon) is to be colored either red or blue, and thus there are
possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.
Problem 8
Let where
Find the value of the product
Problem 9
Circles and
intersect at two points
and
and their common tangent line closer to
intersects
and
at points
and
respectively. The line parallel to
that passes through
intersects
and
for the second time at points
and
respectively. Suppose
and
Then the area of trapezoid
is
where
and
are positive integers and
is not divisible by the square of any prime. Find
Problem 10
Let be the number of ways to place the integers
through
in the
cells of a
grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by
One way to do this is shown below. Find the number of positive integer divisors of
Problem 11
Find the number of collections of distinct subsets of
with the property that for any two subsets
and
in the collection,
Problem 12
In with side lengths
and
let
be the midpoint of
Let
be the point on the circumcircle of
such that
is on
There exists a unique point
on segment
such that
Then
can be written as
where
and
are relatively prime positive integers. Find
Problem 13
Let be an acute angle such that
Find the number of positive integers
less than or equal to
such that
is a positive integer whose units digit is
Problem 14
A cube-shaped container has vertices
and
where
and
are parallel edges of the cube, and
and
are diagonals of faces of the cube, as shown. Vertex
of the cube is set on a horizontal plane
so that the plane of the rectangle
is perpendicular to
vertex
is
meters above
vertex
is
meters above
and vertex
is
meters above
The cube contains water whose surface is parallel to
at a height of
meters above
The volume of water is
cubic meters, where
and
are relatively prime positive integers. Find
Problem 15
For each positive integer let
be the least positive integer multiple of
such that
Find the number of positive integers
less than or equal to
that satisfy
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2023 AIME I |
Followed by 2024 AIME I | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.