prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
In square \(ABCD\), points \(E\) and \(H\) lie on \(\overline{AB}\) and \(\overline{DA}\), respectively, so that \(AE=AH.\) Points \(F\) and \(G\) lie on \(\overline{BC}\) and \(\overline{CD}\), respectively, and points \(I\) and \(J\) lie on \(\overline{EH}\) so that \(\overline{FI} \perp \overline{EH}\) and \(\overli... | 14 |
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$? | 48 |
How many positive integers $n$ satisfy $\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$ (Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.) | 6 |
For how many positive integers $n \le 1000$ is $\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$ not divisible by $3$? | 22 |
What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$? | 201 |
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that \[ (2 + i)^n = a_n + b_ni \] for all integers $n\geq 0$, where $i = \sqrt{-1}$. Find the value of \[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,.\] The original answer is in the form \(\frac{k}{m}\). Please give the value of \(k + m\). | 23 |
What is the maximum value of \(\frac{(2^t-3t)t}{4^t}\) for real values of \(t?\) The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of k + m. | 13 |
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below. [asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.2... | 83 |
Square \(ABCD\) in the coordinate plane has vertices at the points \(A(1,1), B(-1,1), C(-1,-1),\) and \(D(1,-1).\) Consider the following four transformations: \(\quad\bullet\qquad\) \(L,\) a rotation of \(90^{\circ}\) counterclockwise around the origin; \(\quad\bullet\qquad\) \(R,\) a rotation of \(90^{\circ}\) clockw... | 39 |
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly \(7\). Jason always plays to optimize his chances of winnin... | 43 |
Let \( n \) be the least positive integer greater than 1000 for which \(\gcd(63, n+120) = 21\) and \(\gcd(n+63, 120) = 60.\) Find the sum of the digits of \( n \). | 18 |
How many positive integers $n$ satisfy $$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$ | 6 |
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counte... | 112 |
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly \(7\). Jason always plays to optimize his chances of winnin... | 43 |
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying \[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\] is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloo... | 929 |
For each real number \(a\) with \(0 \leq a \leq 1\), let numbers \(x\) and \(y\) be chosen independently at random from the intervals \([0, a]\) and \([0, 1]\), respectively, and let \(P(a)\) be the probability that \(\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1\). Find the maximum value of \(P(a)\). The original answer is in... | 4 |
At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? | 32 |
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? | 20 |
What is the area of the shaded figure shown below? [asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label(... | 6 |
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have an area of $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? | 20 |
What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube? | 6 |
The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is the value of $A$? | 9 |
When a student multiplied the number $66$ by the repeating decimal, $\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \underline{b}\cdots=\underline{1}.\overline{\underline{a} \ \underline{b}},$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66~ times$ $\underline{1}.\under... | 75 |
The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$. The image of $P$ after these two transformations is at $(-6,3)$. What is the value of $b - a$? | 7 |
The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is the value of $A?$ | 9 |
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. Find the number of distinct integers that can be written as the sum of two, not necessarily different, special fractions. | 11 |
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at \(4\) miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to \(2\) miles per hour. After reaching the tower, she immediately turns aro... | 25 |
The least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \cdot 6^k$, where $m$ and $k$ are integers and $6$ is not a divisor of $m$. What is the value of $m+k$? | 58 |
The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. Find the sum of the digits of the greatest prime number that is a divisor of $16{,}383$. | 10 |
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. If the answer is $x^\circ$, what is the degree measure of $\angle AFE$? | 170 |
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. Find how many distinct integers can be written as the sum of two, not necessarily different, special fractions. | 11 |
When a student multiplied the number $66$ by the repeating decimal, $$\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},$$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\underline{1}.\un... | 75 |
Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. Find the sum of the greatest number and the least number that appear in the second row from... | 367 |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines? Provide your answer as an integer. | 6 |
A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly? | 1 |
The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$? Provide your answer as a single integer. | 10 |
Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M, 32, 33, 34, 35, 36, 37, 38, 39,$ and $40.$ Find the value of $\frac{N}{M}.$ | 74 |
The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure of the vertex angle of this triangle, in degrees? | 150 |
Which expression is equivalent to $(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$ Provide your answer in the form $a^{b} - c^{b}$, and then find the value of $a + b + c$. | 133 |
When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. Find the probability that the sum of the numbers rolled is even. The original answer is in \frac{k}{m} format, please give the value of k + m. | 13 |
A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\log_{2}x, \log_{3}x,$ and $\log_{4}x.$ What is the integer value of $x$? | 576 |
Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. If the area of the circle passing through $A$, $O$, and $C$ is $x\pi$, what is the value of $x$? | 12 |
Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers? | 10 |
$A$ school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is ... | -135 |
Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number p... | 27 |
The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ Find the remainder when $N$ is divided by $5$. | 3 |
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perim... | 60 |
A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance $d$ the beam will travel along this path? If the answer is $a\sqrt{b}$, where $a$ and $... | 10 |
Triangle \(ABC\) has \(AB=13, BC=14\) and \(AC=15\). Let \(P\) be the point on \(\overline{AC}\) such that \(PC=10\). There are exactly two points \(D\) and \(E\) on line \(BP\) such that quadrilaterals \(ABCD\) and \(ABCE\) are trapezoids. The distance \(DE\) can be expressed in the form \(k\sqrt{m}\). Please find the... | 14 |
Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the... | 70 |
A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. Find the area of the region bounded by these $6$ reflected arcs. The original answer is in the form of k\sqrt{m} - n\pi, please provide the value of k + m + n. | 7 |
Consider two concentric circles of radius $17$ and $19.$ The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle? The original answer is in the form of k\sqrt{m}, please give the value of k + m. | 14 |
Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? Provide your answer in the form of \(\frac{k}{m}\) where the fraction is in simplest form. Please give the value of \(k + m\). | 123 |
Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the value of $y$ if the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$ is $1 : y$? | 14 |
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then re... | 368 |
For \(n\) a positive integer, let \(f(n)\) be the quotient obtained when the sum of all positive divisors of \(n\) is divided by \(n.\) For example, \[f(14)=(1+2+7+14)\div 14=\frac{12}{7}\] Find the value of \(f(768)-f(384).\) The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 193 |
The angle bisector of the acute angle formed at the origin by the graphs of the lines \(y = x\) and \(y=3x\) has equation \(y=kx.\) The original answer is in the form \(\frac{k+\sqrt{m}}{n}\). Please find the value of \(k + m + n\). | 8 |
How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy $1-3\sin\theta+5\cos3\theta = 0?$ | 6 |
Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls? Express your answer in the form \(\frac{k}{m}\) and calculate the value of \(k + m\). | 21 |
A square with side length \(3\) is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length \(2\) has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle in the form of a fraction \(\frac{... | 85 |
Let $c = \frac{2\pi}{11}.$ What is the value of $\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?$ | 1 |
All the roots of the polynomial \(z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16\) are positive integers, possibly repeated. The original answer for the value of \(B\) can be expressed as k + m, where k and m are integers. Please find the value of k + m. | -88 |
Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$ Expres... | 29 |
How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \begin{align*} x^2+3y&=9 \\ (|x|+|y|-4)^2 &= 1 \end{align*} Find the number of solutions. | 5 |
Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? Provide your answer as a simplified fraction in the form \frac{k}{m}. Please give the value of k + m. | 123 |
In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon? Express your answer in the form $k\sqrt{m}$, and find the value of $k + m$. | 15 |
Values for $A,B,C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the c... | 90 |
The real number $x$ satisfies the equation $x+\frac{1}{x} = \sqrt{5}$. What is the value of $x^{11}-7x^{7}+x^3$? | 0 |
A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of different groups that could be selected. What i... | 95 |
Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$, and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$. What is the area of the circle that passes through vertices $A$, $B$, and $C?$ If the area is $x\pi$, what is the value of $x$? | 26 |
Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\overline{z}$. The polynomial $$P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$$ has four complex roots: $z_1$, $z_2$, $z_3$, ... | 208 |
In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$. $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200$ Find the median of the numbers in this list. | 142 |
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89,$ and $5$ are all uphill integers, but $32, 1240,$ and $466$ are not. Find how many uphill integers are divisible by $15$. | 6 |
In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$. $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \cdots, 200, 200, \cdots , 200$ What is the median of the numbers in this list? | 142 |
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. If the expected number of balls that occupy their original positions after these two successive transpositio... | 22 |
An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if th... | 191 |
Suppose $a$, $b$, $c$ are positive integers such that \[a+b+c=23\] and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^2+b^2+c^2$? | 438 |
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD},BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, whe... | 194 |
Let \(ABCD\) be an isosceles trapezoid having parallel bases \(\overline{AB}\) and \(\overline{CD}\) with \(AB>CD.\) Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles whose areas are \(2, 3, 4,\) and \(5\) starting with the triangle with base \(\overline{CD}\) and movin... | 4 |
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meter... | 17 |
Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. The equation of line $m$ is i... | -1 |
For how many ordered pairs \((b,c)\) of positive integers does neither \(x^2+bx+c=0\) nor \(x^2+cx+b=0\) have two distinct real solutions? | 6 |
A bug starts at a vertex of a grid made of equilateral triangles of side length \(1\). At each step, the bug moves in one of the \(6\) possible directions along the grid lines randomly and independently with equal probability. What is the probability that after \(5\) moves the bug never will have been more than \(1\) u... | 121 |
A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? Express your answer in the form \(\frac{k}{m}\), where k and m are integers with no common factors other than 1. Find the value of k +... | 21 |
A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section, the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not wan... | 84 |
Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon... | 147 |
Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. Find the value of $\frac{p}{q}$. | 16 |
Set $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence $$u_{k+1} = 2u_k - 2u_k^2.$$ This sequence tends to a limit; call it $L$. Find the least value of $k$ such that $$|u_k-L| \le \frac{1}{2^{1000}}.$$ | 10 |
The area of the region bounded by the graph of $x^2+y^2 = 3|x-y| + 3|x+y|$ is $m+n\pi$, where $m$ and $n$ are integers. What is the value of $m + n$? | 54 |
Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac{3}{4}$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\f... | 17 |
Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. Find the value of \[f(2) + f(3) + f(4) + f(5)+ f(6).\] | 8 |
Let $x$ be the least real number greater than $1$ such that $\sin(x)= \sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer? | 13 |
Suppose that on a parabola with vertex \(V\) and a focus \(F\) there exists a point \(A\) such that \(AF=20\) and \(AV=21\). Find the sum of all possible values of the length \(FV\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 43 |
Let \(Q(z)\) and \(R(z)\) be the unique polynomials such that\[z^{2021}+1=(z^2+z+1)Q(z)+R(z)\]and the degree of \(R\) is less than \(2.\) The original answer is in the form az^1 + b, please find the value of a + b. | -1 |
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. Find the number of values of $n \le 50$ for which $f_{50}(n) = 12$. | 10 |
Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is n... | 55 |
A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Suppose that $C'D = \frac{1}{3}$. What is the perimeter... | 2 |
The five solutions to the equation\[(z-1)(z^2+2z+4)(z^2+4z+6)=0\] may be written in the form $x_k+y_ki$ for $1\le k\le 5,$ where $x_k$ and $y_k$ are real. Let $\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\mathcal E$ ca... | 7 |
Let $ABCD$ be an isosceles trapezoid with $\overline{BC}\parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD$ in the form $m... | 38 |
For real numbers $x$, let $$P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)$$ where $i = \sqrt{-1}$. Find the number of values of $x$ with $0\leq x<2\pi$ for which $$P(x)=0.$$ | 0 |
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