talzoomanzoo/olympiadbench · Datasets at Hugging Face

Question stringlengths 49 4.42k	answer listlengths 1 1
Let $T=20$. In $\triangle L E O, \sin \angle L E O=\frac{1}{T}$. If $L E=\frac{1}{n}$ for some positive real number $n$, then $E O=$ $n^{3}-4 n^{2}+5 n$. As $n$ ranges over the positive reals, compute the least possible value of $[L E O]$.	[ "$\\frac{1}{40}$" ]
Let $T=\frac{1}{40}$. Given that $x, y$, and $z$ are real numbers such that $x+y=5, x^{2}-y^{2}=\frac{1}{T}$, and $x-z=-7$, compute $x+z$	[ "20" ]
Let $T=20$. The product of all positive divisors of $2^{T}$ can be written in the form $2^{K}$. Compute $K$.	[ "210" ]
Let $T=210$. At the Westward House of Supper ("WHS"), a dinner special consists of an appetizer, an entrée, and dessert. There are 7 different appetizers and $K$ different entrées that a guest could order. There are 2 dessert choices, but ordering dessert is optional. Given that there are $T$ possible different orders that could be placed at the WHS, compute $K$.	[ "10" ]
Let $S=15$ and let $M=10$ . Sam and Marty each ride a bicycle at a constant speed. Sam's speed is $S \mathrm{~km} / \mathrm{hr}$ and Marty's speed is $M \mathrm{~km} / \mathrm{hr}$. Given that Sam and Marty are initially $100 \mathrm{~km}$ apart and they begin riding towards one another at the same time, along a straight path, compute the number of kilometers that Sam will have traveled when Sam and Marty meet.	[ "60" ]
Compute the $2011^{\text {th }}$ smallest positive integer $N$ that gains an extra digit when doubled.	[ "6455" ]
In triangle $A B C, C$ is a right angle and $M$ is on $\overline{A C}$. A circle with radius $r$ is centered at $M$, is tangent to $\overline{A B}$, and is tangent to $\overline{B C}$ at $C$. If $A C=5$ and $B C=12$, compute $r$.	[ "$\\frac{12}{5}$" ]
The product of the first five terms of a geometric progression is 32 . If the fourth term is 17 , compute the second term.	[ "$\\frac{4}{17}$" ]
Polygon $A_{1} A_{2} \ldots A_{n}$ is a regular $n$-gon. For some integer $k<n$, quadrilateral $A_{1} A_{2} A_{k} A_{k+1}$ is a rectangle of area 6 . If the area of $A_{1} A_{2} \ldots A_{n}$ is 60 , compute $n$.	[ "40" ]
A bag contains 20 lavender marbles, 12 emerald marbles, and some number of orange marbles. If the probability of drawing an orange marble in one try is $\frac{1}{y}$, compute the sum of all possible integer values of $y$.	[ "69" ]
Compute the number of ordered quadruples of integers $(a, b, c, d)$ satisfying the following system of equations: $$ \left\{\begin{array}{l} a b c=12,000 \\ b c d=24,000 \\ c d a=36,000 \end{array}\right. $$	[ "12" ]
Let $n$ be a positive integer such that $\frac{3+4+\cdots+3 n}{5+6+\cdots+5 n}=\frac{4}{11}$. Compute $\frac{2+3+\cdots+2 n}{4+5+\cdots+4 n}$.	[ "$\\frac{27}{106}$" ]
The quadratic polynomial $f(x)$ has a zero at $x=2$. The polynomial $f(f(x))$ has only one real zero, at $x=5$. Compute $f(0)$.	[ "$-\\frac{32}{9}$" ]
The Local Area Inspirational Math Exam comprises 15 questions. All answers are integers ranging from 000 to 999, inclusive. If the 15 answers form an arithmetic progression with the largest possible difference, compute the largest possible sum of those 15 answers.	[ "7530" ]
Circle $\omega_{1}$ has center $O$, which is on circle $\omega_{2}$. The circles intersect at points $A$ and $C$. Point $B$ lies on $\omega_{2}$ such that $B A=37, B O=17$, and $B C=7$. Compute the area of $\omega_{1}$.	[ "$548 \\pi$" ]
Compute the number of integers $n$ for which $2^{4}<8^{n}<16^{32}$.	[ "41" ]
Let $T=41$. Compute the number of positive integers $b$ such that the number $T$ has exactly two digits when written in base $b$.	[ "35" ]
Let $T=35$. Triangle $A B C$ has a right angle at $C$, and $A B=40$. If $A C-B C=T-1$, compute $[A B C]$, the area of $\triangle A B C$.	[ "111" ]
Let $x$ be a positive real number such that $\log _{\sqrt{2}} x=20$. Compute $\log _{2} \sqrt{x}$.	[ "5" ]
Let $T=5$. Hannah flips two fair coins, while Otto flips $T$ fair coins. Let $p$ be the probability that the number of heads showing on Hannah's coins is greater than the number of heads showing on Otto's coins. If $p=q / r$, where $q$ and $r$ are relatively prime positive integers, compute $q+r$.	[ "17" ]
Let $T=17$. In ARMLovia, the unit of currency is the edwah. Janet's wallet contains bills in denominations of 20 and 80 edwahs. If the bills are worth an average of $2 T$ edwahs each, compute the smallest possible value of the bills in Janet's wallet.	[ "1020" ]
Spheres centered at points $P, Q, R$ are externally tangent to each other, and are tangent to plane $\mathcal{M}$ at points $P^{\prime}, Q^{\prime}, R^{\prime}$, respectively. All three spheres are on the same side of the plane. If $P^{\prime} Q^{\prime}=Q^{\prime} R^{\prime}=12$ and $P^{\prime} R^{\prime}=6$, compute the area of $\triangle P Q R$.	[ "$18 \\sqrt{6}$" ]
Let $f(x)=x^{1}+x^{2}+x^{4}+x^{8}+x^{16}+x^{32}+\cdots$. Compute the coefficient of $x^{10}$ in $f(f(x))$.	[ "40" ]
Compute $\left\lfloor 100000(1.002)^{10}\right\rfloor$.	[ "102018" ]
If $1, x, y$ is a geometric sequence and $x, y, 3$ is an arithmetic sequence, compute the maximum value of $x+y$.	[ "$\\frac{15}{4}$" ]
Define the sequence of positive integers $\left\{a_{n}\right\}$ as follows: $$ \left\{\begin{array}{l} a_{1}=1 \\ \text { for } n \geq 2, a_{n} \text { is the smallest possible positive value of } n-a_{k}^{2}, \text { for } 1 \leq k<n . \end{array}\right. $$ For example, $a_{2}=2-1^{2}=1$, and $a_{3}=3-1^{2}=2$. Compute $a_{1}+a_{2}+\cdots+a_{50}$.	[ "253" ]
Compute the base $b$ for which $253_{b} \cdot 341_{b}=\underline{7} \underline{4} \underline{X} \underline{Y} \underline{Z}_{b}$, for some base- $b$ digits $X, Y, Z$.	[ "20" ]
Some portions of the line $y=4 x$ lie below the curve $y=10 \pi \sin ^{2} x$, and other portions lie above the curve. Compute the sum of the lengths of all the segments of the graph of $y=4 x$ that lie in the first quadrant, below the graph of $y=10 \pi \sin ^{2} x$.	[ "$\\frac{5 \\pi}{4} \\sqrt{17}$" ]
In equilateral hexagon $A B C D E F, \mathrm{~m} \angle A=2 \mathrm{~m} \angle C=2 \mathrm{~m} \angle E=5 \mathrm{~m} \angle D=10 \mathrm{~m} \angle B=10 \mathrm{~m} \angle F$, and diagonal $B E=3$. Compute $[A B C D E F]$, that is, the area of $A B C D E F$.	[ "$\\frac{9}{2}$" ]
The taxicab distance between points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$ is defined as $d(A, B)=$ $\left\|x_{A}-x_{B}\right\|+\left\|y_{A}-y_{B}\right\|$. Given some $s>0$ and points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$, define the taxicab ellipse with foci $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$ to be the set of points $\{Q \mid d(A, Q)+d(B, Q)=s\}$. Compute the area enclosed by the taxicab ellipse with foci $(0,5)$ and $(12,0)$, passing through $(1,-1)$.	[ "96" ]
The function $f$ satisfies the relation $f(n)=f(n-1) f(n-2)$ for all integers $n$, and $f(n)>0$ for all positive integers $n$. If $f(1)=\frac{f(2)}{512}$ and $\frac{1}{f(1)}=2 f(2)$, compute $f(f(4))$.	[ "4096" ]
Frank Narf accidentally read a degree $n$ polynomial with integer coefficients backwards. That is, he read $a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ as $a_{0} x^{n}+\ldots+a_{n-1} x+a_{n}$. Luckily, the reversed polynomial had the same zeros as the original polynomial. All the reversed polynomial's zeros were real, and also integers. If $1 \leq n \leq 7$, compute the number of such polynomials such that $\operatorname{GCD}\left(a_{0}, a_{1}, \ldots, a_{n}\right)=1$.	[ "70" ]
Given a regular 16-gon, extend three of its sides to form a triangle none of whose vertices lie on the 16-gon itself. Compute the number of noncongruent triangles that can be formed in this manner.	[ "11" ]
Two square tiles of area 9 are placed with one directly on top of the other. The top tile is then rotated about its center by an acute angle $\theta$. If the area of the overlapping region is 8 , compute $\sin \theta+\cos \theta$.	[ "$\\frac{5}{4}$" ]
Suppose that neither of the three-digit numbers $M=\underline{4} \underline{A} \underline{6}$ and $N=\underline{1} \underline{B} \underline{7}$ is divisible by 9 , but the product $M \cdot N$ is divisible by 9 . Compute the largest possible value of $A+B$.	[ "12" ]
Let $T=12$. Each interior angle of a regular $T$-gon has measure $d^{\circ}$. Compute $d$.	[ "150" ]
Suppose that $r$ and $s$ are the two roots of the equation $F_{k} x^{2}+F_{k+1} x+F_{k+2}=0$, where $F_{n}$ denotes the $n^{\text {th }}$ Fibonacci number. Compute the value of $(r+1)(s+1)$.	[ "2" ]
Let $T=2$. Compute the product of $-T-i$ and $i-T$, where $i=\sqrt{-1}$.	[ "5" ]
Let $T=5$. Compute the number of positive divisors of the number $20^{4} \cdot 11^{T}$ that are perfect cubes.	[ "12" ]
Let $T=72 \sqrt{2}$, and let $K=\left(\frac{T}{12}\right)^{2}$. In the sequence $0.5,1,-1.5,2,2.5,-3, \ldots$, every third term is negative, and the absolute values of the terms form an arithmetic sequence. Compute the sum of the first $K$ terms of this sequence.	[ "414" ]
Let $A$ be the sum of the digits of the number you will receive from position 7 , and let $B$ be the sum of the digits of the number you will receive from position 9 . Let $(x, y)$ be a point randomly selected from the interior of the triangle whose consecutive vertices are $(1,1),(B, 7)$ and $(17,1)$. Compute the probability that $x>A-1$.	[ "$\\frac{79}{128}$" ]
Let $T=9.5$. If $\log _{2} x^{T}-\log _{4} x=\log _{8} x^{k}$ is an identity for all $x>0$, compute the value of $k$.	[ "27" ]
Let $T=16$. An isosceles trapezoid has an area of $T+1$, a height of 2 , and the shorter base is 3 units shorter than the longer base. Compute the sum of the length of the shorter base and the length of one of the congruent sides.	[ "9.5" ]
Let $T=10$. Susan flips a fair coin $T$ times. Leo has an unfair coin such that the probability of flipping heads is $\frac{1}{3}$. Leo gets to flip his coin the least number of times so that Leo's expected number of heads will exceed Susan's expected number of heads. Compute the number of times Leo gets to flip his coin.	[ "16" ]
Let $T=1$. Dennis and Edward each take 48 minutes to mow a lawn, and Shawn takes 24 minutes to mow a lawn. Working together, how many lawns can Dennis, Edward, and Shawn mow in $2 \cdot T$ hours? (For the purposes of this problem, you may assume that after they complete mowing a lawn, they immediately start mowing the next lawn.)	[ "10" ]
Let T be a rational number. Compute $\sin ^{2} \frac{T \pi}{2}+\sin ^{2} \frac{(5-T) \pi}{2}$.	[ "1" ]
Let $T=11$. Compute the value of $x$ that satisfies $\sqrt{20+\sqrt{T+x}}=5$.	[ "14" ]
The sum of the interior angles of an $n$-gon equals the sum of the interior angles of a pentagon plus the sum of the interior angles of an octagon. Compute $n$.	[ "11" ]

Let $T=20$. In $\triangle L E O, \sin \angle L E O=\frac{1}{T}$. If $L E=\frac{1}{n}$ for some positive real number $n$, then $E O=$ $n^{3}-4 n^{2}+5 n$. As $n$ ranges over the positive reals, compute the least possible value of $[L E O]$.

[ "$\\frac{1}{40}$" ]

Let $T=\frac{1}{40}$. Given that $x, y$, and $z$ are real numbers such that $x+y=5, x^{2}-y^{2}=\frac{1}{T}$, and $x-z=-7$, compute $x+z$

[ "20" ]

Let $T=20$. The product of all positive divisors of $2^{T}$ can be written in the form $2^{K}$. Compute $K$.

[ "210" ]

Let $T=210$. At the Westward House of Supper ("WHS"), a dinner special consists of an appetizer, an entrée, and dessert. There are 7 different appetizers and $K$ different entrées that a guest could order. There are 2 dessert choices, but ordering dessert is optional. Given that there are $T$ possible different orders that could be placed at the WHS, compute $K$.

[ "10" ]

Let $S=15$ and let $M=10$ . Sam and Marty each ride a bicycle at a constant speed. Sam's speed is $S \mathrm{~km} / \mathrm{hr}$ and Marty's speed is $M \mathrm{~km} / \mathrm{hr}$. Given that Sam and Marty are initially $100 \mathrm{~km}$ apart and they begin riding towards one another at the same time, along a straight path, compute the number of kilometers that Sam will have traveled when Sam and Marty meet.

[ "60" ]

Compute the $2011^{\text {th }}$ smallest positive integer $N$ that gains an extra digit when doubled.

[ "6455" ]

In triangle $A B C, C$ is a right angle and $M$ is on $\overline{A C}$. A circle with radius $r$ is centered at $M$, is tangent to $\overline{A B}$, and is tangent to $\overline{B C}$ at $C$. If $A C=5$ and $B C=12$, compute $r$.

[ "$\\frac{12}{5}$" ]

The product of the first five terms of a geometric progression is 32 . If the fourth term is 17 , compute the second term.

[ "$\\frac{4}{17}$" ]

Polygon $A_{1} A_{2} \ldots A_{n}$ is a regular $n$-gon. For some integer $k<n$, quadrilateral $A_{1} A_{2} A_{k} A_{k+1}$ is a rectangle of area 6 . If the area of $A_{1} A_{2} \ldots A_{n}$ is 60 , compute $n$.

[ "40" ]

A bag contains 20 lavender marbles, 12 emerald marbles, and some number of orange marbles. If the probability of drawing an orange marble in one try is $\frac{1}{y}$, compute the sum of all possible integer values of $y$.

[ "69" ]

Compute the number of ordered quadruples of integers $(a, b, c, d)$ satisfying the following system of equations: $$ \left\{\begin{array}{l} a b c=12,000 \\ b c d=24,000 \\ c d a=36,000 \end{array}\right. $$

[ "12" ]

Let $n$ be a positive integer such that $\frac{3+4+\cdots+3 n}{5+6+\cdots+5 n}=\frac{4}{11}$. Compute $\frac{2+3+\cdots+2 n}{4+5+\cdots+4 n}$.

[ "$\\frac{27}{106}$" ]

The quadratic polynomial $f(x)$ has a zero at $x=2$. The polynomial $f(f(x))$ has only one real zero, at $x=5$. Compute $f(0)$.

[ "$-\\frac{32}{9}$" ]

The Local Area Inspirational Math Exam comprises 15 questions. All answers are integers ranging from 000 to 999, inclusive. If the 15 answers form an arithmetic progression with the largest possible difference, compute the largest possible sum of those 15 answers.

[ "7530" ]

Circle $\omega_{1}$ has center $O$, which is on circle $\omega_{2}$. The circles intersect at points $A$ and $C$. Point $B$ lies on $\omega_{2}$ such that $B A=37, B O=17$, and $B C=7$. Compute the area of $\omega_{1}$.

[ "$548 \\pi$" ]

Compute the number of integers $n$ for which $2^{4}<8^{n}<16^{32}$.

[ "41" ]

Let $T=41$. Compute the number of positive integers $b$ such that the number $T$ has exactly two digits when written in base $b$.

[ "35" ]

Let $T=35$. Triangle $A B C$ has a right angle at $C$, and $A B=40$. If $A C-B C=T-1$, compute $[A B C]$, the area of $\triangle A B C$.

[ "111" ]

Let $x$ be a positive real number such that $\log _{\sqrt{2}} x=20$. Compute $\log _{2} \sqrt{x}$.

[ "5" ]

Let $T=5$. Hannah flips two fair coins, while Otto flips $T$ fair coins. Let $p$ be the probability that the number of heads showing on Hannah's coins is greater than the number of heads showing on Otto's coins. If $p=q / r$, where $q$ and $r$ are relatively prime positive integers, compute $q+r$.

[ "17" ]

Let $T=17$. In ARMLovia, the unit of currency is the edwah. Janet's wallet contains bills in denominations of 20 and 80 edwahs. If the bills are worth an average of $2 T$ edwahs each, compute the smallest possible value of the bills in Janet's wallet.

[ "1020" ]

Spheres centered at points $P, Q, R$ are externally tangent to each other, and are tangent to plane $\mathcal{M}$ at points $P^{\prime}, Q^{\prime}, R^{\prime}$, respectively. All three spheres are on the same side of the plane. If $P^{\prime} Q^{\prime}=Q^{\prime} R^{\prime}=12$ and $P^{\prime} R^{\prime}=6$, compute the area of $\triangle P Q R$.

[ "$18 \\sqrt{6}$" ]

Let $f(x)=x^{1}+x^{2}+x^{4}+x^{8}+x^{16}+x^{32}+\cdots$. Compute the coefficient of $x^{10}$ in $f(f(x))$.

[ "40" ]

Compute $\left\lfloor 100000(1.002)^{10}\right\rfloor$.

[ "102018" ]

If $1, x, y$ is a geometric sequence and $x, y, 3$ is an arithmetic sequence, compute the maximum value of $x+y$.

[ "$\\frac{15}{4}$" ]

Define the sequence of positive integers $\left\{a_{n}\right\}$ as follows: $$ \left\{\begin{array}{l} a_{1}=1 \\ \text { for } n \geq 2, a_{n} \text { is the smallest possible positive value of } n-a_{k}^{2}, \text { for } 1 \leq k<n . \end{array}\right. $$ For example, $a_{2}=2-1^{2}=1$, and $a_{3}=3-1^{2}=2$. Compute $a_{1}+a_{2}+\cdots+a_{50}$.

[ "253" ]

Compute the base $b$ for which $253_{b} \cdot 341_{b}=\underline{7} \underline{4} \underline{X} \underline{Y} \underline{Z}_{b}$, for some base- $b$ digits $X, Y, Z$.

[ "20" ]

Some portions of the line $y=4 x$ lie below the curve $y=10 \pi \sin ^{2} x$, and other portions lie above the curve. Compute the sum of the lengths of all the segments of the graph of $y=4 x$ that lie in the first quadrant, below the graph of $y=10 \pi \sin ^{2} x$.

[ "$\\frac{5 \\pi}{4} \\sqrt{17}$" ]

In equilateral hexagon $A B C D E F, \mathrm{~m} \angle A=2 \mathrm{~m} \angle C=2 \mathrm{~m} \angle E=5 \mathrm{~m} \angle D=10 \mathrm{~m} \angle B=10 \mathrm{~m} \angle F$, and diagonal $B E=3$. Compute $[A B C D E F]$, that is, the area of $A B C D E F$.

[ "$\\frac{9}{2}$" ]

The taxicab distance between points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$ is defined as $d(A, B)=$ $\left|x_{A}-x_{B}\right|+\left|y_{A}-y_{B}\right|$. Given some $s>0$ and points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$, define the taxicab ellipse with foci $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$ to be the set of points $\{Q \mid d(A, Q)+d(B, Q)=s\}$. Compute the area enclosed by the taxicab ellipse with foci $(0,5)$ and $(12,0)$, passing through $(1,-1)$.

[ "96" ]

The function $f$ satisfies the relation $f(n)=f(n-1) f(n-2)$ for all integers $n$, and $f(n)>0$ for all positive integers $n$. If $f(1)=\frac{f(2)}{512}$ and $\frac{1}{f(1)}=2 f(2)$, compute $f(f(4))$.

[ "4096" ]

Frank Narf accidentally read a degree $n$ polynomial with integer coefficients backwards. That is, he read $a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ as $a_{0} x^{n}+\ldots+a_{n-1} x+a_{n}$. Luckily, the reversed polynomial had the same zeros as the original polynomial. All the reversed polynomial's zeros were real, and also integers. If $1 \leq n \leq 7$, compute the number of such polynomials such that $\operatorname{GCD}\left(a_{0}, a_{1}, \ldots, a_{n}\right)=1$.

[ "70" ]

Given a regular 16-gon, extend three of its sides to form a triangle none of whose vertices lie on the 16-gon itself. Compute the number of noncongruent triangles that can be formed in this manner.

[ "11" ]

Two square tiles of area 9 are placed with one directly on top of the other. The top tile is then rotated about its center by an acute angle $\theta$. If the area of the overlapping region is 8 , compute $\sin \theta+\cos \theta$.

[ "$\\frac{5}{4}$" ]

Suppose that neither of the three-digit numbers $M=\underline{4} \underline{A} \underline{6}$ and $N=\underline{1} \underline{B} \underline{7}$ is divisible by 9 , but the product $M \cdot N$ is divisible by 9 . Compute the largest possible value of $A+B$.

[ "12" ]

Let $T=12$. Each interior angle of a regular $T$-gon has measure $d^{\circ}$. Compute $d$.

[ "150" ]

Suppose that $r$ and $s$ are the two roots of the equation $F_{k} x^{2}+F_{k+1} x+F_{k+2}=0$, where $F_{n}$ denotes the $n^{\text {th }}$ Fibonacci number. Compute the value of $(r+1)(s+1)$.

[ "2" ]

Let $T=2$. Compute the product of $-T-i$ and $i-T$, where $i=\sqrt{-1}$.

[ "5" ]

Let $T=5$. Compute the number of positive divisors of the number $20^{4} \cdot 11^{T}$ that are perfect cubes.

[ "12" ]

Let $T=72 \sqrt{2}$, and let $K=\left(\frac{T}{12}\right)^{2}$. In the sequence $0.5,1,-1.5,2,2.5,-3, \ldots$, every third term is negative, and the absolute values of the terms form an arithmetic sequence. Compute the sum of the first $K$ terms of this sequence.

[ "414" ]

Let $A$ be the sum of the digits of the number you will receive from position 7 , and let $B$ be the sum of the digits of the number you will receive from position 9 . Let $(x, y)$ be a point randomly selected from the interior of the triangle whose consecutive vertices are $(1,1),(B, 7)$ and $(17,1)$. Compute the probability that $x>A-1$.

[ "$\\frac{79}{128}$" ]

Let $T=9.5$. If $\log _{2} x^{T}-\log _{4} x=\log _{8} x^{k}$ is an identity for all $x>0$, compute the value of $k$.

[ "27" ]

Let $T=16$. An isosceles trapezoid has an area of $T+1$, a height of 2 , and the shorter base is 3 units shorter than the longer base. Compute the sum of the length of the shorter base and the length of one of the congruent sides.

[ "9.5" ]

Let $T=10$. Susan flips a fair coin $T$ times. Leo has an unfair coin such that the probability of flipping heads is $\frac{1}{3}$. Leo gets to flip his coin the least number of times so that Leo's expected number of heads will exceed Susan's expected number of heads. Compute the number of times Leo gets to flip his coin.

[ "16" ]

Let $T=1$. Dennis and Edward each take 48 minutes to mow a lawn, and Shawn takes 24 minutes to mow a lawn. Working together, how many lawns can Dennis, Edward, and Shawn mow in $2 \cdot T$ hours? (For the purposes of this problem, you may assume that after they complete mowing a lawn, they immediately start mowing the next lawn.)

[ "10" ]

Let T be a rational number. Compute $\sin ^{2} \frac{T \pi}{2}+\sin ^{2} \frac{(5-T) \pi}{2}$.

[ "1" ]

Let $T=11$. Compute the value of $x$ that satisfies $\sqrt{20+\sqrt{T+x}}=5$.

[ "14" ]

The sum of the interior angles of an $n$-gon equals the sum of the interior angles of a pentagon plus the sum of the interior angles of an octagon. Compute $n$.

[ "11" ]