High school problems from JHMT 2022
Algebra & Number Theory
1
The graph of 
, where 
is a constant, is drawn on the interval ![$[0,\pi]$](//latex.artofproblemsolving.com/e/c/e/ece27ba559bd93cdf2416aa0f88fc8d7d85e5791.png)
. Suppose that there exists a point 
on the graph such that the triangle with vertices 
, 
, and is equilateral. Find 
.
, where 
is a constant, is drawn on the interval ![$[0,\pi]$](http://latex.artofproblemsolving.com/e/c/e/ece27ba559bd93cdf2416aa0f88fc8d7d85e5791.png)
. Suppose that there exists a point 
on the graph such that the triangle with vertices 
, 
, and is equilateral. Find 
.
has roots 
,
,
for some angle 
.3
Let 
be a positive integer. Find the sum of all possible values of 
such that the product of the distinct divisors of is 
.
be a positive integer. Find the sum of all possible values of 
such that the product of the distinct divisors of is 
.4
For an integer 
and positive integers 
and 
, let 
be the remainder when 
is divided by . Find the largest composite integer 
that guarantees the infinite sequence
![\[ f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots \]](//latex.artofproblemsolving.com/1/7/c/17c2bb240258b9e3888b1e8fb4430b8d383d27a4.png)
to be periodic for all integers (i.e., for each choice of , there is some positive integer 
such that 
for all ).
and positive integers 
and 
, let 
be the remainder when 
is divided by . Find the largest composite integer 
that guarantees the infinite sequence![\[ f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots \]](http://latex.artofproblemsolving.com/1/7/c/17c2bb240258b9e3888b1e8fb4430b8d383d27a4.png)
to be periodic for all integers (i.e., for each choice of , there is some positive integer 
such that 
for all ).5
Let 
be a quadratic polynomial satisfying the following conditions:
be the set of all such possible , and let 
be the polynomial obtained upon summing all the elements of . Find the sum of the roots of .
be a quadratic polynomial satisfying the following conditions:
- has leading coefficient
. - has nonnegative integer roots that are at most
. - the set of the roots of is a subset of the set of the roots of
.
.
be the set of all such possible , and let 
be the polynomial obtained upon summing all the elements of . Find the sum of the roots of .6
Let 
denote the greatest integer less than or equal to 
. Find the number of positive integers 
between and inclusive such that
![\[ \left\lfloor \frac{3^m}{11} \right\rfloor \]](//latex.artofproblemsolving.com/5/7/0/570b557824c0da93dfe9f57dc63957c2cfc7552a.png)
is even.
denote the greatest integer less than or equal to 
. Find the number of positive integers 
between 
and 
inclusive such that![\[ \left\lfloor \frac{3^m}{11} \right\rfloor \]](http://latex.artofproblemsolving.com/5/7/0/570b557824c0da93dfe9f57dc63957c2cfc7552a.png)
is even.7
Find the least positive integer such that there exist positive real numbers 
such that
![\[ \sum_{k=1}^{N}ka_k=1 \quad \text{and} \quad \sum_{k=1}^{N}\frac{a_k^2}{k}\leq \frac{1}{2022^2}. \]](//latex.artofproblemsolving.com/b/5/4/b54ccac3d1c809721acc846d226cead980c0396e.png)
such that there exist positive real numbers 
such that![\[ \sum_{k=1}^{N}ka_k=1 \quad \text{and} \quad \sum_{k=1}^{N}\frac{a_k^2}{k}\leq \frac{1}{2022^2}. \]](http://latex.artofproblemsolving.com/b/5/4/b54ccac3d1c809721acc846d226cead980c0396e.png)
8
Let 
be a complex number satisfying 
and 
. Find the unique ordered pair of nonnegative integers 
satisfying
![\[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]](//latex.artofproblemsolving.com/d/2/7/d27141fcdc99e2d7d35298324816dbc1795d933e.png)
be a complex number satisfying 
and 
. Find the unique ordered pair of nonnegative integers 
satisfying![\[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]](http://latex.artofproblemsolving.com/d/2/7/d27141fcdc99e2d7d35298324816dbc1795d933e.png)
9
Let 
and 
be sequences of real numbers. Suppose 
, 
, and for all integers 
,
If 
, then determine the value of 
.
and 
be sequences of real numbers. Suppose 
, 
, and for all integers 
,
If 
, then determine the value of 
.10
Compute the exact value of
![\[ \sum_{a=1}^{\infty}\sum_{b=1}^{\infty} \frac{\gcd(a, b)}{(a + b)^3}. \]](//latex.artofproblemsolving.com/3/e/f/3ef1ed7bcae07acbf574a1d17389a5bbec82a7fd.png)
If necessary, you may express your answer in terms of the Riemann zeta function, 
, for integers 
.
![\[ \sum_{a=1}^{\infty}\sum_{b=1}^{\infty} \frac{\gcd(a, b)}{(a + b)^3}. \]](http://latex.artofproblemsolving.com/3/e/f/3ef1ed7bcae07acbf574a1d17389a5bbec82a7fd.png)
If necessary, you may express your answer in terms of the Riemann zeta function, 
, for integers 
.Geometry
1
The side lengths of an equiangular octagon alternate between 
and 
. Find its area.
and 
. Find its area.2
Four mutually externally tangent spherical apples of radius 
are placed on a horizontal flat table. Then, a spherical orange of radius 
is placed such that it rests on all the apples. Find the distance from the center of the orange to the table.
are placed on a horizontal flat table. Then, a spherical orange of radius 
is placed such that it rests on all the apples. Find the distance from the center of the orange to the table.3
Triangle 
has side lengths 
, 
, and 
. Points 
and 
lie on 
such that 
. Let the angle bisector of 
intersect 
and 
at points 
and , respectively. Find the area of quadrilateral 
.
has side lengths 
, 
, and 
. Points 
and 
lie on 
such that 
. Let the angle bisector of 
intersect 
and 
at points 
and , respectively. Find the area of quadrilateral 
.4
Hexagon 
has side lengths 
and 
. Moreover, the vertices 
, 
, , , 
, and 
lie on a circle 
. Find the area of .
has side lengths 
and 
. Moreover, the vertices 
, 
, , , 
, and 
lie on a circle 
. Find the area of .5
Suppose 
satisfies 
and 
. There is a unique circle passing through 
that is tangent to 
at its midpoint; let this circle intersect 
and 
again at points 
and 
, respectively. Let 
be the unique point on such that 
. Compute the perimeter of quadrilateral 
.
satisfies 
and 
. There is a unique circle passing through 
that is tangent to 
at its midpoint; let this circle intersect 
and 
again at points 
and 
, respectively. Let 
be the unique point on such that 
. Compute the perimeter of quadrilateral 
.6
Triangle 
has side lengths 
, 
, and 
. Let be the incenter of 
, and let 
be the parabola with focus and directrix 
. Suppose that intersects 
and 
at points 
and 
, respectively. Find 
.
has side lengths 
, 
, and 
. Let be the incenter of 
, and let 
be the parabola with focus and directrix 
. Suppose that intersects 
and 
at points 
and 
, respectively. Find 
.7
Two rays emanate from the origin 
and form a 
angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers 
, 
, and , the ray with the larger slope passes through point 
, and the ray with the smaller slope passes through point 
. If 
, then determine the maximum possible area of 
.
and form a 
angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers 
, 
, and , the ray with the larger slope passes through point 
, and the ray with the smaller slope passes through point 
. If 
, then determine the maximum possible area of 
.8
In equilateral 
, point lies on such that the radius of the circumcircle 
of 
is 
and the radius of the incircle 
of 
is 
. Suppose that and intersect at points and . Find 
.
, point lies on such that the radius of the circumcircle 
of 
is 
and the radius of the incircle 
of 
is 
. Suppose that and intersect at points and . Find 
.9
In convex quadrilateral 
, angles 
, 
, and 
measure 
, 
, and 
, respectively. Given that 
and that 
and 
intersect at point , compute the value of 
.
, angles 
, 
, and 
measure 
, 
, and 
, respectively. Given that 
and that 
and 
intersect at point , compute the value of 
.10
In 
, 
, 
, and 
. Let and be the incenter and orthocenter of , respectively. The circumcircle of 
intersects 
at a point 
, and 
. Find 
.
, 
, 
, and 
. Let and be the incenter and orthocenter of , respectively. The circumcircle of 
intersects 
at a point 
, and 
. Find 
.Probability & Combinatorics
1
Daredevil Darren challenges Forgetful Fred to spell "Johns Hopkins." Forgetful Fred will spell it correctly except for the 's's; there is a 
and 
chance that he will omit the 's' in the first and last names, respectively, with his mistakes being independent of each other. If Forgetful Fred spells the name correctly, then he is happy; otherwise, Daredevil Darren will present him with a dare, and there is a 
chance that Forgetful Fred will not be happy. Find the probability that Forgetful Fred will be happy.
and 
chance that he will omit the 's' in the first and last names, respectively, with his mistakes being independent of each other. If Forgetful Fred spells the name correctly, then he is happy; otherwise, Daredevil Darren will present him with a dare, and there is a 
chance that Forgetful Fred will not be happy. Find the probability that Forgetful Fred will be happy.2
Erica intends to construct a subset of 
, but if she is unsure about including an element of in , she will write in bold and include it in . For example, 
and 
are valid examples of , while 
is not. Find the total number of such subsets that Erica can construct.
of 
, but if she is unsure about including an element of in , she will write in bold and include it in . For example, 
and 
are valid examples of , while 
is not. Find the total number of such subsets that Erica can construct.3
Dr. G has a bag of five marbles and enjoys drawing one marble from the bag, uniformly at random, and then putting it back in the bag. How many draws, on average, will it take Dr. G to reach a point where every marble has been drawn at least once?
4
For a nonempty set of integers, let 
. Find the number of subsets of
![\[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]](//latex.artofproblemsolving.com/6/1/b/61b7259425c5e88da0e4bd9e382b75efc60d6082.png)
such that 
is an element of .
of integers, let 
. Find the number of subsets of![\[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]](http://latex.artofproblemsolving.com/6/1/b/61b7259425c5e88da0e4bd9e382b75efc60d6082.png)
such that 
is an element of .5
Consider an array of white unit squares arranged in a rectangular grid with 
rows of unit squares and 
columns of unit squares, for some positive integer . What is the smallest possible value of such that, if we shade exactly 
unit squares in each column black, then there must necessarily be some row with at least 
black unit squares?
rows of unit squares and 
columns of unit squares, for some positive integer . What is the smallest possible value of such that, if we shade exactly 
unit squares in each column black, then there must necessarily be some row with at least 
black unit squares?6
Let be the number of arrangements of the letters in JOHNS HOPKINS such that no two Os are adjacent, no two Hs are adjacent, no two Ns are adjacent, and no two Ss are adjacent. Find 
.
be the number of arrangements of the letters in JOHNS HOPKINS such that no two Os are adjacent, no two Hs are adjacent, no two Ns are adjacent, and no two Ss are adjacent. Find 
.7
A spider sits on the circumference of a circle and wants to weave a web by making several passes through the circle's interior. On each pass, the spider starts at some location on the circumference, picks a destination uniformly at random from the circumference, and travels to that destination in a straight line, laying down a strand of silk along the line segment they traverse. After the spider does of these passes (with each non-initial pass starting where the previous one ended), what is the expected number of points in the circle's interior where two or more non-parallel silk strands intersect?
of these passes (with each non-initial pass starting where the previous one ended), what is the expected number of points in the circle's interior where two or more non-parallel silk strands intersect?8
Find the number of ways to completely cover a 
rectangular grid of unit squares with 
rectangles and 
- - triangles such that the following all hold:
![[asy]
unitsize(1cm);
fill((0,0)--(10,0)--(10,2)--(0,2)--cycle, grey);
draw((0,0)--(10,0)--(10,2)--(0,2)--cycle);
draw((1,0)--(1,2));
draw((1,2)--(3,0));
draw((1,0)--(3,2));
draw((3,2)--(5,0));
draw((3,0)--(5,2));
draw((2,1)--(4,1));
draw((5,0)--(5,2));
draw((7,0)--(7,2));
draw((5,1)--(7,1));
draw((8,0)--(8,2));
draw((8,0)--(10,2));
draw((8,2)--(10,0));
[/asy]](//latex.artofproblemsolving.com/a/1/8/a18f4645195ff4051e3629e41969a6da14f91105.png)
rectangular grid of unit squares with 
rectangles and 
- - triangles such that the following all hold:
- a placement of must have all of its sides parallel to the grid lines,
- a placement of must have its longest side parallel to a grid line,
- the tiles are non-overlapping, and
- no tile extends outside the boundary of the grid.
![[asy]
unitsize(1cm);
fill((0,0)--(10,0)--(10,2)--(0,2)--cycle, grey);
draw((0,0)--(10,0)--(10,2)--(0,2)--cycle);
draw((1,0)--(1,2));
draw((1,2)--(3,0));
draw((1,0)--(3,2));
draw((3,2)--(5,0));
draw((3,0)--(5,2));
draw((2,1)--(4,1));
draw((5,0)--(5,2));
draw((7,0)--(7,2));
draw((5,1)--(7,1));
draw((8,0)--(8,2));
draw((8,0)--(10,2));
draw((8,2)--(10,0));
[/asy]](http://latex.artofproblemsolving.com/a/1/8/a18f4645195ff4051e3629e41969a6da14f91105.png)
9
Let 
and be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point (this unit sphere is the set of all points in 
a distance of away from ). Compute the expected value of 
.
and be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point (this unit sphere is the set of all points in 
a distance of away from ). Compute the expected value of 
.10
Let be the rectangle in the coordinate plane with corners 
, 
, 
, and 
, and partition into a 
grid of unit squares. For a given line in the coordinate plane, let its pixelation be the set of grid squares in that contain part of the line in their interior. If is a point chosen uniformly at random in , then compute the expected number of sets of grid squares that are pixelations of some line through .
be the rectangle in the coordinate plane with corners 
, 
, 
, and 
, and partition into a 
grid of unit squares. For a given line in the coordinate plane, let its pixelation be the set of grid squares in that contain part of the line in their interior. If is a point chosen uniformly at random in , then compute the expected number of sets of grid squares that are pixelations of some line through .Calculus
![\[ \frac{d}{dx}\int_{1}^{10} x^3\,dx. \]](http://latex.artofproblemsolving.com/5/6/e/56e275b88ca713a9d3259281ab9bb804dd1cc313.png)
2
Suppose that 
is a differentiable function such that 
and 
for all real numbers . Let and 
be real numbers such that every such function satisfies 
. Find the smallest possible value of 
.
is a differentiable function such that 
and 
for all real numbers . Let and 
be real numbers such that every such function satisfies 
. Find the smallest possible value of 
.3
Let be a variable that can take any positive real value. For certain positive real constants 
and 
, the value of 
is minimized at 
, and the value of 
is minimized at 
. Compute the ordered pair 
.
be a variable that can take any positive real value. For certain positive real constants 
and 
, the value of 
is minimized at 
, and the value of 
is minimized at 
. Compute the ordered pair 
.4
Consider the rectangle in the coordinate plane with corners , 
, 
, and 
. For a constant ![$x_0 \in [0, 16]$](//latex.artofproblemsolving.com/f/b/e/fbee3afe88f5cf911e3651e4cab735b190b7c401.png)
, the curves
![\[ \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\} \quad \text{and} \quad \{(x_0, y) : 0 \leq y \leq 4\} \]](//latex.artofproblemsolving.com/0/2/c/02c18fca1e16b13f8720e07505ad268fa6035a79.png)
partition this rectangle into four 2D regions. Over all choices of 
, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition.
(The bottom-left region is
, and the top-right region is 
.)
, 
, 
, and 
. For a constant ![$x_0 \in [0, 16]$](http://latex.artofproblemsolving.com/f/b/e/fbee3afe88f5cf911e3651e4cab735b190b7c401.png)
, the curves![\[ \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\} \quad \text{and} \quad \{(x_0, y) : 0 \leq y \leq 4\} \]](http://latex.artofproblemsolving.com/0/2/c/02c18fca1e16b13f8720e07505ad268fa6035a79.png)
partition this rectangle into four 2D regions. Over all choices of 
, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition.(The bottom-left region is
, and the top-right region is 
.)5
A point 
is chosen uniformly at random from the ball of radius centered at the origin (i.e., the set 
). Compute the probability that the inequalities 
and 
simultaneously hold.
is chosen uniformly at random from the ball of radius centered at the origin (i.e., the set 
). Compute the probability that the inequalities 
and 
simultaneously hold.6
There is a unique choice of positive integers , , and such that is not divisible by the square of any prime and the infinite sums
![\[ \sum_{n=0}^{\infty} \left(\left(\frac{a - b\sqrt{c}}{10}\right)^{n-10}\cdot\prod_{k=0}^{9} (n - k)\right) \quad \text{and} \quad \sum_{n=0}^{\infty} \left((a - b\sqrt{c})^{n+1}\cdot\prod_{k=0}^{9} (n - k)\right) \]](//latex.artofproblemsolving.com/9/c/8/9c8d08f594dbc2fbe4c2b1c6d2d6b63ddc4e4662.png)
are equal (i.e., converging to the same finite value). Compute 
.
, , and such that is not divisible by the square of any prime and the infinite sums![\[ \sum_{n=0}^{\infty} \left(\left(\frac{a - b\sqrt{c}}{10}\right)^{n-10}\cdot\prod_{k=0}^{9} (n - k)\right) \quad \text{and} \quad \sum_{n=0}^{\infty} \left((a - b\sqrt{c})^{n+1}\cdot\prod_{k=0}^{9} (n - k)\right) \]](http://latex.artofproblemsolving.com/9/c/8/9c8d08f594dbc2fbe4c2b1c6d2d6b63ddc4e4662.png)
are equal (i.e., converging to the same finite value). Compute 
.7
Let be the unique real number satisfying 
. Find a closed-form expression for
![\[ \int_{a}^{\infty} \frac{x + 1}{x\sqrt{(xe^x)^{11} - 1}}\,dx. \]](//latex.artofproblemsolving.com/4/d/7/4d731a1a8e6d1fffd434644fc0ac21a99bd24804.png)
You may express your answer in terms of elementary operations, functions, and constants.
be the unique real number satisfying 
. Find a closed-form expression for![\[ \int_{a}^{\infty} \frac{x + 1}{x\sqrt{(xe^x)^{11} - 1}}\,dx. \]](http://latex.artofproblemsolving.com/4/d/7/4d731a1a8e6d1fffd434644fc0ac21a99bd24804.png)
You may express your answer in terms of elementary operations, functions, and constants.8
Let 
and 
be two points on the -axis of the Cartesian coordinate plane, and let and be points on the -axis and 
-axis, respectively, such that over all on line 
, the perimeter of 
has a minimum value of . What is the smallest possible value of 
?
and 
be two points on the -axis of the Cartesian coordinate plane, and let and be points on the -axis and 
-axis, respectively, such that over all on line 
, the perimeter of 
has a minimum value of . What is the smallest possible value of 
?9
There is a unique continuous function over the positive real numbers satisfying 
and
![\[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \]](//latex.artofproblemsolving.com/a/f/1/af16bce9ec8a5282f0c39a987eb647c2c839e18e.png)
for all positive . Compute the value of 
.
over the positive real numbers satisfying 
and![\[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \]](http://latex.artofproblemsolving.com/a/f/1/af16bce9ec8a5282f0c39a987eb647c2c839e18e.png)
for all positive . Compute the value of 
.10
The maximum value of
![\[ 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} \]](//latex.artofproblemsolving.com/b/6/5/b6548c4725ce0f4074f2294088b96b88f4ffaf7e.png)
over all real numbers 
can be expressed as a common fraction 
. Compute 
.
![\[ 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} \]](http://latex.artofproblemsolving.com/b/6/5/b6548c4725ce0f4074f2294088b96b88f4ffaf7e.png)
over all real numbers 
can be expressed as a common fraction 
. Compute 
.General
1
If three of the roots of the quartic polynomial 
are 
, , and , and the sum of , , and is at most 
, then find the largest possible value of 
.
are 
, , and , and the sum of , , and is at most 
, then find the largest possible value of 
.2
Find the number of ordered pairs of positive integers 
, where 
, such that 
is a multiple of 
.
, where 
, such that 
is a multiple of 
.3
Andy, Bella, and Chris are playing in a trivia contest. Andy has 
points, Bella has 
points, and Chris has 
points. They have reached the final round, which works as follows:
points, Bella has 
points, and Chris has 
points. They have reached the final round, which works as follows:
- they are given a hint as to what the only question of the round will be about;
- then, each of them must bet some amount of their points---this bet must be a nonnegative integer (a player does not know any of the other players' bets, and this bet cannot be changed later on);
- then, they will be shown the question, where they will have
seconds to individually submit a response (a player does not know any of the other players' answers); - finally, once time runs out, their responses and bets are revealed---if a player's response is correct, then the number of points they bet will be added to their score, otherwise, it will be subtracted from their score, and whoever ends up having the most points wins the contest (if there is a tie for the win, then the winner is randomly decided).
seconds to individually submit a response (a player does not know any of the other players' answers);4
For a positive integer , let 
denote the product of the digits of , and let 
denote the sum of the digits of . Find the sum of all positive integers satisfying 
.
, let 
denote the product of the digits of , and let 
denote the sum of the digits of . Find the sum of all positive integers satisfying 
.5
Three congruent equilateral triangles 
, 
, and 
are stacked from left to right inside rectangle such that the bottom left vertex of is , the bottom side of lies on 
, the bottom left vertex of is the midpoint of a side of , the bottom left vertex of is the midpoint of a side of , and the other two vertices of lie on and 
, as shown below. Given that rectangle has area , find the area of any one of the triangles , , or .
![[asy]
unitsize(0.111111111111111111cm);
real s = sqrt(4044/sqrt(75));
real l = 5s/2;
real w = s * sqrt(3);
pair J,H,M,T,V1,V2,V3,V4,V5,V6,V7,V8,C1,C2,C3;
J = (0,w);
H = (l,w);
M = (l,0);
T = (0,0);
V1 = (s,0);
V2 = (s/2,s * sqrt(3)/2);
V3 = (V1+V2)/2;
V4 = (3 * s/4+s,s * sqrt(3)/4);
V5 = (3 * s/4+s/2,s * sqrt(3)/4+s * sqrt(3)/2);
V6 = (V4+V5)/2;
V7 = (l,s * sqrt(3)/4+s * sqrt(3)/4);
V8 = (l-s/2,w);
C1 = (T+V1+V2)/3;
C2 = (V3+V4+V5)/3;
C3 = (V6+V7+V8)/3;
draw(J--H--M--T--cycle);
draw(V1--V2--T);
draw(V3--V4--V5--cycle);
draw(V6--V7--V8--cycle);
label("$J$", J, NW);
label("$H$", H, NE);
label("$M$", M, SE);
label("$T$", T, SW);
label("$T_1$", C1);
label("$T_2$", C2);
label("$T_3$", C3);
[/asy]](//latex.artofproblemsolving.com/7/8/3/78387641d1b899533d361c8dee7b48f9b3c169d2.png)
, 
, and 
are stacked from left to right inside rectangle such that the bottom left vertex of is , the bottom side of lies on 
, the bottom left vertex of is the midpoint of a side of , the bottom left vertex of is the midpoint of a side of , and the other two vertices of lie on and 
, as shown below. Given that rectangle has area , find the area of any one of the triangles , , or .![[asy]
unitsize(0.111111111111111111cm);
real s = sqrt(4044/sqrt(75));
real l = 5s/2;
real w = s * sqrt(3);
pair J,H,M,T,V1,V2,V3,V4,V5,V6,V7,V8,C1,C2,C3;
J = (0,w);
H = (l,w);
M = (l,0);
T = (0,0);
V1 = (s,0);
V2 = (s/2,s * sqrt(3)/2);
V3 = (V1+V2)/2;
V4 = (3 * s/4+s,s * sqrt(3)/4);
V5 = (3 * s/4+s/2,s * sqrt(3)/4+s * sqrt(3)/2);
V6 = (V4+V5)/2;
V7 = (l,s * sqrt(3)/4+s * sqrt(3)/4);
V8 = (l-s/2,w);
C1 = (T+V1+V2)/3;
C2 = (V3+V4+V5)/3;
C3 = (V6+V7+V8)/3;
draw(J--H--M--T--cycle);
draw(V1--V2--T);
draw(V3--V4--V5--cycle);
draw(V6--V7--V8--cycle);
label("$J$", J, NW);
label("$H$", H, NE);
label("$M$", M, SE);
label("$T$", T, SW);
label("$T_1$", C1);
label("$T_2$", C2);
label("$T_3$", C3);
[/asy]](http://latex.artofproblemsolving.com/7/8/3/78387641d1b899533d361c8dee7b48f9b3c169d2.png)
6
For positive real numbers and 
let 
denote the real number such that area of the (non-degenerate) triangle with side lengths 
and is maximized. Find
![\[ \sum_{n=2}^{100}f\left(\sqrt{\tbinom{n}{2}},\sqrt{\tbinom{n+1}{2}}\right). \]](//latex.artofproblemsolving.com/9/f/8/9f8a34fd2ff8ed77ca1e1c31eb28f853bcaa71c3.png)
and 
let 
denote the real number such that area of the (non-degenerate) triangle with side lengths 
and is maximized. Find![\[ \sum_{n=2}^{100}f\left(\sqrt{\tbinom{n}{2}},\sqrt{\tbinom{n+1}{2}}\right). \]](http://latex.artofproblemsolving.com/9/f/8/9f8a34fd2ff8ed77ca1e1c31eb28f853bcaa71c3.png)
7
Let 
be an irregular convex heptagon (i.e., its angles and side lengths are all distinct, with the angles all having measure less than 
) with area 
such that all of its side lengths are greater than 
, 
, and 
. Arcs with radius are drawn inside with their centers at each of the vertices and their endpoints on the sides, creating circular sectors. Find the area of the region inside but outside the sectors.
be an irregular convex heptagon (i.e., its angles and side lengths are all distinct, with the angles all having measure less than 
) with area 
such that all of its side lengths are greater than 
, 
, and 
. Arcs with radius are drawn inside with their centers at each of the vertices and their endpoints on the sides, creating circular sectors. Find the area of the region inside but outside the sectors.8
An ant is walking on a sidewalk and discovers sidewalk panels with leaves inscribed in them, as shown below. Find the number of ways in which the ant can traverse from point to point if it can only move
![[asy]
unitsize(1cm);
int r = 4;
int c = 5;
for (int i = 0; i < r; ++i) {
for (int j = 0; j < c; ++j) {
pair A = (j,i);
}
}
for (int i = 0; i < r; ++i) {
for (int j = 0; j < c; ++j) {
if (j != c-1) {
draw((j,i)--(j+1,i));
}
if (i != r-1) {
draw((j,i)--(j,i+1));
}
}
}
for (int i = 1; i < r+1; ++i) {
for (int j = 0; j < c-2; ++j) {
fill(arc((i,j),1,90,180)--cycle,deepgreen);
fill(arc((i-1,j+1),1,270,360)--cycle,deepgreen);
draw((i-1,j)--(i,j+1), heavygreen+linewidth(0.5));
draw((i-2/3,j+1/3)--(i-2/3,j+1/3+0.1), heavygreen);
draw((i-1/3,j+2/3)--(i-1/3,j+2/3+0.1), heavygreen);
draw((i-2/3,j+1/3)--(i-2/3+0.1,j+1/3), heavygreen);
draw((i-1/3,j+2/3)--(i-1/3+0.1,j+2/3), heavygreen);
draw(arc((i,j),1,90,180));
draw(arc((i-1,j+1),1,270,360));
}
}
draw((0,3)--(0,1), red+linewidth(1.5));
draw((0,3)--(0,1), red+linewidth(1.5));
draw(arc((1,1),1,90,180), red+linewidth(1.5));
draw((1,2)--(1,1)--(2,1), red+linewidth(1.5));
draw(arc((2,2),1,270,360), red+linewidth(1.5));
draw(arc((4,2),1,90,180), red+linewidth(1.5));
draw((4,3)--(4,0), red+linewidth(1.5));
dot((0,3));
dot((4,0));
label("$A$", (0,3), NW);
label("$B$", (4,0), SE);
[/asy]](//latex.artofproblemsolving.com/a/b/8/ab803b6227aec07f7263389dd709748ec81781c5.png)
sidewalk panels with leaves inscribed in them, as shown below. Find the number of ways in which the ant can traverse from point to point if it can only move
- up, down, or right (along the border of a sidewalk panel), or
- up-right (along one of two margin halves of a leaf)
![[asy]
unitsize(1cm);
int r = 4;
int c = 5;
for (int i = 0; i < r; ++i) {
for (int j = 0; j < c; ++j) {
pair A = (j,i);
}
}
for (int i = 0; i < r; ++i) {
for (int j = 0; j < c; ++j) {
if (j != c-1) {
draw((j,i)--(j+1,i));
}
if (i != r-1) {
draw((j,i)--(j,i+1));
}
}
}
for (int i = 1; i < r+1; ++i) {
for (int j = 0; j < c-2; ++j) {
fill(arc((i,j),1,90,180)--cycle,deepgreen);
fill(arc((i-1,j+1),1,270,360)--cycle,deepgreen);
draw((i-1,j)--(i,j+1), heavygreen+linewidth(0.5));
draw((i-2/3,j+1/3)--(i-2/3,j+1/3+0.1), heavygreen);
draw((i-1/3,j+2/3)--(i-1/3,j+2/3+0.1), heavygreen);
draw((i-2/3,j+1/3)--(i-2/3+0.1,j+1/3), heavygreen);
draw((i-1/3,j+2/3)--(i-1/3+0.1,j+2/3), heavygreen);
draw(arc((i,j),1,90,180));
draw(arc((i-1,j+1),1,270,360));
}
}
draw((0,3)--(0,1), red+linewidth(1.5));
draw((0,3)--(0,1), red+linewidth(1.5));
draw(arc((1,1),1,90,180), red+linewidth(1.5));
draw((1,2)--(1,1)--(2,1), red+linewidth(1.5));
draw(arc((2,2),1,270,360), red+linewidth(1.5));
draw(arc((4,2),1,90,180), red+linewidth(1.5));
draw((4,3)--(4,0), red+linewidth(1.5));
dot((0,3));
dot((4,0));
label("$A$", (0,3), NW);
label("$B$", (4,0), SE);
[/asy]](http://latex.artofproblemsolving.com/a/b/8/ab803b6227aec07f7263389dd709748ec81781c5.png)
9
In 
, 
, 
, and 
. Assume that an equilateral hexagon 
is able to be drawn inside so that is parallel to 
, 
is parallel to 
, 
is parallel to 
, lies on , 
lies on , and 
lies on . Find the area of hexagon .
, 
, 
, and 
. Assume that an equilateral hexagon 
is able to be drawn inside so that is parallel to 
, 
is parallel to 
, 
is parallel to 
, lies on , 
lies on , and 
lies on . Find the area of hexagon .10
Let 
denote the set of points 
in 2D space with integer coordinates such that 
and 
. That is,
![\[ \Lambda=\{ (x,y) \in \mathbb{Z}^2: 0\leq x\leq 4, \ 0\leq y\leq 2 \}. \]](//latex.artofproblemsolving.com/2/b/3/2b3043c1933da946e270ccf23d8260355fae196d.png)
Find the number of ways to connect points of with segments of length or 
such that the interior of any unit square with vertices in contains part of exactly one segment; an example is shown below (connections that differ by reflections are distinct).
![[asy]
unitsize(1cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((4,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((4,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((4,2));
draw((0,0)--(1,1));
draw((0,2)--(2,1));
draw((1,1)--(2,0));
draw((2,0)--(3,2));
draw((3,1)--(4,2));
draw((3,0)--(4,1));
[/asy]](//latex.artofproblemsolving.com/b/8/a/b8a9207e8be842bb146ed25114fe46eed0b6abb8.png)
denote the set of points 
in 2D space with integer coordinates such that 
and 
. That is,![\[ \Lambda=\{ (x,y) \in \mathbb{Z}^2: 0\leq x\leq 4, \ 0\leq y\leq 2 \}. \]](http://latex.artofproblemsolving.com/2/b/3/2b3043c1933da946e270ccf23d8260355fae196d.png)
Find the number of ways to connect points of with segments of length or 
such that the interior of any unit square with vertices in contains part of exactly one segment; an example is shown below (connections that differ by reflections are distinct).![[asy]
unitsize(1cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((4,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((4,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((4,2));
draw((0,0)--(1,1));
draw((0,2)--(2,1));
draw((1,1)--(2,0));
draw((2,0)--(3,2));
draw((3,1)--(4,2));
draw((3,0)--(4,1));
[/asy]](http://latex.artofproblemsolving.com/b/8/a/b8a9207e8be842bb146ed25114fe46eed0b6abb8.png)
