prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can this be done? | 1152 |
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base $10$ with no leading zeros. A $6$-digit palindrome $n$ is chosen uniformly at random. Find the probability that \(\frac{n}{11}\) is also a palindrome. The original answer is in the form of \(\frac{k}{m}\), where... | 41 |
Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether? | 3932 |
In $\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is the integer length of $BC$? | 61 |
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline... | 21 |
A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. Find how many numbers in the set $\{ 2010,2011,2012,\cdots,2019 \}$ are nice. | 1 |
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are \(R_1 = 100\) inches, \(R_2 = 60\) inches, and \(R_3 = 80\) inches, respectively. The ball always remains in contact with the track and does not slip. If the answer is \(x\pi\), w... | 238 |
For certain real numbers \(a\), \(b\), and \(c\), the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] Find the value of \(f(1)\). The original answer is in the form \(m - n\), please give the value of ... | 7007 |
Three numbers $a, b, c$, none zero, form an arithmetic progression. Increasing $a$ by $1$ or increasing $c$ by $2$ results in a geometric progression. Find the value of $b$. | 12 |
Square $ABCD$ has side length $s$, and a circle centered at $E$ has radius $r$. Both $r$ and $s$ are rational numbers. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF = \sqrt{9 + 5\sq... | 14 |
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. Find the sum of the... | 23 |
Given $A = \{1, 2, 3, 5, 8, 13, 21, 34, 55\}$, find how many of the numbers between $3$ and $89$ cannot be written as the sum of two elements of set $A$. | 51 |
In \(\triangle ABC\) shown in the figure, \(AB=7\), \(BC=8\), \(CA=9\), and \(\overline{AH}\) is an altitude. Points \(D\) and \(E\) lie on sides \(\overline{AC}\) and \(\overline{AB}\), respectively, so that \(\overline{BD}\) and \(\overline{CE}\) are angle bisectors, intersecting \(\overline{AH}\) at \(Q\) and \(P\),... | 28 |
Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $\frac{a}{b}$, where $a$ and $b$ are relatively pr... | 151 |
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$, and $AB$, respectively, and $CE=CD$ and $BF=BD$, then if $\measuredangle EDF$ equals $x^\circ$, what is the value of $x$? | 50 |
The solutions of the equation $z^4 + 4z^3i - 6z^2 - 4zi - i = 0$ are the vertices of a convex polygon in the complex plane. What is the area of the polygon? The original answer is in the form $2^{x/y}$, where $x$ and $y$ are integers, and $y$ cannot be zero. Please give the value of $x + y$. | 9 |
The sides of a right triangle are $a$ and $b$, and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, find the ratio of $r$ to $s$. The original answer is in the format of k:m, please give the value of k + m. | 10 |
If 64 is divided into three parts proportional to 2, 4, and 6, find the smallest part and express your answer in the form \frac{x}{y}, where \frac{x}{y} is in simplest terms. Please provide the value of x + y. | 35 |
A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $150,000,000\text{ km}$ from the Sun. What is the closest value for the acceleration of the Earth in this orbit? Provide your answer as a number without including the unit, and multiply the original ans... | 6 |
For $x^2 + 2x + 5$ to be a factor of $x^4 + px^2 + q$, the original answer can be expressed as two integers $m$ and $n$ for the values of $p$ and $q$, respectively. Please find the value of $m + n$. | 31 |
Edward is trying to spell the word "CAT". He has an equal chance of spelling the word in any order of letters (e.g., TAC or TCA). What is the probability that he spells "CAT" incorrectly? Provide your answer as a simplified fraction in the form \(\frac{k}{m}\), and then calculate the value of k + m. | 11 |
The area of a circle inscribed in a regular hexagon is \(100\pi\). What is the area of the hexagon? The original answer is in the form of \(k\sqrt{m}\), please provide the value of k + m. | 203 |
Mary is about to pay for five items at the grocery store. The prices of the items are $7.99, $4.99, $2.99, $1.99, and $0.99. Mary will pay with a twenty-dollar bill. What percentage of the $20.00 will she receive in change? Provide your answer as an integer percentage. | 5 |
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? Provide your answer as an integer. | 100 |
The graphs of $y = -|x - a| + b$ and $y = |x - c| + d$ intersect at points $(2,5)$ and $(8,3)$. Find the value of $a + c$. | 10 |
How many unique $3$-letter sequences with no spaces can be made using the letters in "AUGUSTIN LOUIS CAUCHY", which contains $19$ letters? For example, "GAA" is one acceptable sequence, but "GGA" is not an acceptable sequence because there is only one G available. The original ordering of the letters does not have to b... | 1486 |
The coordinates of \(A\), \(B\), and \(C\) are \((5,5)\), \((2,1)\), and \((0,k)\) respectively. Find the value of \(k\) that makes \(\overline{AC} + \overline{BC}\) as small as possible. The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\). | 22 |
Point $P$ is $9$ units from the center of a circle with a radius of $15$. How many different chords of the circle contain $P$ and have integer lengths? | 12 |
When \(p = \sum_{k=1}^{6} k \ln{k}\), the number \(e^p\) is an integer. What is the largest power of \(2\) that is a factor of \(e^p\)? If the answer is \(2^x\), what is the value of \(x\)? | 16 |
Let \(ABCD\) be a trapezoid with \(AB \parallel CD\). Let \(a = AB\) and \(b = CD\). For \(MN \parallel AB\) such that \(M\) lies on \(AD\), \(N\) lies on \(BC\), and the trapezoids \(ABNM\) and \(MNCD\) have the same area, find the length of \(MN\). The original answer is in the form \(\sqrt{\frac{ka^2 + mb^2}{n}}\). ... | 4 |
If $R_n=\frac{1}{2}(a^n+b^n)$ where $a=3+2\sqrt{2}$, $b=3-2\sqrt{2}$, and $n=0,1,2, \ldots$, then $R_{12345}$ is an integer. Find the units digit of $R_{12345}$. | 9 |
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}. How many such polynomials satisfy $P(-1) = -9$? | 220 |
A uniform disk ($I = \frac{1}{2} MR^2$) of mass 8.0 kg can rotate without friction on a fixed axis. A string is wrapped around its circumference and is attached to a 6.0 kg mass. The string does not slip. What is the value of the tension in the cord while the mass is falling, in newtons? Please provide your answer as a... | 24 |
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 6... | 13 |
The vertices of triangle \(PQR\) have coordinates as follows: \(P(0,a)\), \(Q(b,0)\), \(R(c,d)\), where \(a\), \(b\), \(c\), and \(d\) are positive. The origin and point \(R\) lie on opposite sides of \(PQ\). Find the expression for the area of triangle \(PQR\). The original answer is in the form \(\frac{kac + mbd - na... | 5 |
For each real number \(x\), let \(f(x)\) be the minimum of the numbers \(4x+1\), \(x+2\), and \(-2x+4\). Find the maximum value of \(f(x)\). The original answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\). | 11 |
George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, find how many minutes after starting did they pass each other the second time. If the answer is x\frac{1}{2... | 9 |
What is the greatest integer not exceeding the number \(\left( 1 + \frac{\sqrt{2} + \sqrt{3} + \sqrt{4}}{\sqrt{2} + \sqrt{3} + \sqrt{6} + \sqrt{8} + 4}\right)^{10}\)? | 32 |
Vertex \(E\) of equilateral \(\triangle{ABE}\) is in the interior of unit square \(ABCD\). Let \(R\) be the region consisting of all points inside \(ABCD\) and outside \(\triangle{ABE}\) whose distance from \(\overline{AD}\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). Find the area of \(R\). The original answer is ... | 53 |
In the adjoining figure, \( \triangle CDE \) is an equilateral triangle and \( \square ABCD \) and \( \square DEFG \) are squares. If the measure of \( \angle GDA \) is \( x^\circ \), what is the value of \( x \)? | 120 |
If the angles $a, b, c, d,$ and $e$ are known, what is the expression for the angle $u$ in terms of $a$, $b$, $c$, $d$, and $e$? The original answer is in the form of m*a + n*c + p*d. Please provide the value of m + n + p. | 3 |
The Tigers beat the Sharks $2$ out of the first $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$? | 37 |
How many ordered pairs $(a, b)$ such that $a$ is a real positive number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017} = \log_b (a^{2017})$? | 597 |
Three balls are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\f... | 55 |
What is the greatest integer $n$ such that $$n \leq 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{2014}}?$$ | 88 |
Pablo buys popsicles for his friends. The store sells single popsicles for $1$ each, 3-popsicle boxes for $2$, and 5-popsicle boxes for $3$. What is the greatest number of popsicles that Pablo can buy with $8$? | 13 |
How many $5$-digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecutive digits whose sum is $10$? | 1845 |
The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111$, $R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=\frac{R_{24}}{R_4}$ is an integer whose base-ten representation is a sequence containing only ones and zeros. Find the number of ze... | 15 |
Juan rolls a fair regular octahedral die marked with the numbers $1$ through $8$. Then Amal rolls a fair six-sided die. Find the probability that the product of the two rolls is a multiple of $3$. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 3 |
Kayla rolls four fair 6-sided dice. Find the probability that at least one of the numbers Kayla rolls is greater than 4 and at least two of the numbers she rolls are greater than 2. Express your answer as a fraction in simplest form, \(\frac{k}{m}\). Please provide the value of k + m. | 142 |
Three cubes with volumes $1$, $8$, and $27$ are glued together at their faces. Find the smallest possible surface area of the resulting configuration. | 72 |
Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum \begin{align*} f\left(\frac{1}{2019}\right)-f\left(\frac{2}{2019}\right)+f\left(\frac{3}{2019}\right)-&f\left(\frac{4}{2019}\right)+\cdots\\ &\,+f\left(\frac{2017}{2019}\right) - f\left(\frac{2018}{2019}\right)? \end{align*} | 0 |
The polynomial $x^3 - 2004x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible? The original answer is in the form of k*n, please find the value of k. | 250500 |
Four rectangular paper strips of length $10$ and width $1$ are placed flat on a table, overlapping perpendicularly as shown. How much area of the table is covered? | 36 |
From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$, a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, find the... | 2 |
Beth adds the consecutive positive integers $a$, $b$, $c$, $d$, and $e$, and finds that the sum is a perfect square. She then adds $b$, $c$, and $d$ and finds that this sum is a perfect cube. What is the smallest possible value of $e$? | 1127 |
Letters $A, B, C,$ and $D$ represent four different digits from 0, 1, 2, 3, ..., 9. If $\frac{A+B}{C+D}$ is an integer that is as large as possible, what is the value of $A+B$? | 17 |
A child is sliding out of control with velocity \(v_{c}\) across a frozen lake. He runs head-on into another child, initially at rest, with 3 times the mass of the first child, who holds on so that the two now slide together. The original answer is in the form \(\frac{k \cdot v_{c}}{m}\). Please find the value of \(k +... | 5 |
How many integers $x$ satisfy the equation: \[ (x^2 - x - 1)^{x + 2} = 1 \] | 4 |
What is the largest possible value of $8x^2 + 9xy + 18y^2 + 2x + 3y$ such that $4x^2 + 9y^2 = 8$, where $x, y$ are real numbers? | 26 |
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 $DE$ is in the form of k\sqrt{m}. Please find the value of k + m. | 14 |
Represent the hypotenuse of a right triangle by $c$ and the area by $A$. What is the altitude on the hypotenuse? The original answer is in the form of $\frac{mA}{nc}$, where $m$ and $n$ are integers. Please give the value of $m + n$. | 3 |
What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping? The answer should be expressed in the form \(\frac{kM g}{m\mu_s}\), where \(k\) and \(m\) are integers. Please find the value of \(k + m\). | 2 |
Find a rational number between $\sqrt{2}$ and $\sqrt{3}$. If the answer is $x$, what is the value of $10x$? | 15 |
A square with an area of \(2\) is inscribed within a larger square with an area of \(3\), forming four congruent triangles. Determine the ratio of the shorter leg to the longer leg in the shaded right triangle. The original answer is in the form \(k - \sqrt{m}\). Please provide the value of \(k + m\). | 5 |
How many positive integers less than $1000$ are there such that they cannot be written as the sum of $2$ or more successive positive integers? Fill in the blank with the correct number. | 10 |
If the points $(1, y_1)$ and $(-1, y_2)$ lie on the graph of $y = ax^2 + bx + c$, and $y_1 - y_2 = -6$, the original answer is in the form of $k$. Please find the value of $k$. | -3 |
The fraction \(\frac{2(\sqrt{2} + \sqrt{6})}{3\sqrt{2 + \sqrt{3}}}\) is equal to what value? The answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please calculate the value of \(k + m\). | 7 |
Bertha has $6$ daughters and no sons. Some of her daughters have $6$ daughters, and the rest have none. Bertha has a total of $30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no children? Fill in the blank with the correct number. | 26 |
Let \(ABCD\) be an isosceles trapezoid with parallel bases \(\overline{AB}\) and \(\overline{CD}\), where \(AB > CD\). Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles with areas \(2\), \(3\), \(4\), and \(5\), starting with the triangle with base \(\overline{CD}\) and... | 4 |
The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $0.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}$. Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that t... | 7 |
Let $K$ be the point of intersection of $AB$ and the line touching the circumcircle of $\triangle ABC$ at $C$ where $m(\widehat {A}) > m(\widehat {B})$. Let $L$ be a point on $[BC]$ such that $m(\widehat{ALB})=m(\widehat{CAK})$, $5|LC|=4|BL|$, and $|KC|=12$. Find the length of $|AK|$ in centimeters, denoted as x. Pleas... | 8 |
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? | 8 |
A man with mass \(m\) jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance \(H\), at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant \(k\), ... | 9 |
The triangle $ABC$ has vertices such that $AB = 3$, $BC = 4$, and $AC = 5$. The inscribed circle is tangent to $AB$ at $C'$, $BC$ at $A'$, and $AC$ at $B'$. The ratio between the area of the triangles $A'B'C'$ and $ABC$ is given in the form \frac{k}{m}. Please find the value of k + m. | 6 |
For how many integers $x$ does a triangle with side lengths $10$, $24$, and $x$ have all its angles acute? | 4 |
A very large number of small particles forms a spherical cloud. Initially, they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way. Find the time that passes until th... | 5427 |
If $m$ and $n$ are the roots of the quadratic equation $x^2 + mx + n = 0$, where $m \neq 0$ and $n \neq 0$, what is the value of the sum of the roots? | -1 |
How many primes \( p \) are there such that \( 5p(2^{p+1}-1) \) is a perfect square? Provide your answer as a single integer. | 1 |
In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. Find the greatest distance between a point of $A$ and a point of $B$. The original answer is in the form $a\sqrt{b}$, please give the value of $ab$. | 42 |
An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is \$20 or more? Express your answer in the form \(\frac{k}{m}\), where k and m are integers, and give the value of k + m. | 3 |
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 $ABCD$? Express yo... | 38 |
Liquid X does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$ cm thick. A rectangular box measuring $6$ cm by $3$ cm by $12$ cm is filled with liquid X. Its contents are poured onto a large body of water. The radius of the resulting circular film is in the for... | 2160 |
$K$ takes 30 minutes less time than $M$ to travel a distance of 30 miles. $K$ travels $\frac{1}{3}$ mile per hour faster than $M$. If $x$ is $K$'s rate of speed in miles per hour, find $K$'s time for the distance in hours. The original answer is in the form \(\frac{k}{x}\), please give the value of \(k\). | 30 |
Triangle $\triangle ABC$ in the figure has area $10$. Points $D$, $E$, and $F$, all distinct from $A$, $B$, and $C$, are on sides $AB$, $BC$, and $CA$ respectively, and $AD = 2$, $DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then what is that area? | 6 |
A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $105$ and $91$, respectively. Find the volume of the pyramid. | 784 |
For every positive integer $n$, the function $f$ is defined as follows: - $f(2n+1) = 2f(2n)$ - $f(2n) = f(2n-1) + 1$ - $f(1) = 0$ Determine the remainder when $f(2005)$ is divided by $5$. | 1 |
Let $U = 2 \times 2004^{2005}$, $V = 2004^{2005}$, $W = 2003 \times 2004^{2004}$, $X = 2 \times 2004^{2004}$, $Y = 2004^{2004}$, and $Z = 2004^{2003}$. Find the largest value among the expressions $U - V$, $V - W$, $W - X$, $X - Y$, and $Y - Z$. The answer should be given in the form of k - m, where k and m are placeho... | 43 |
A line passes through $A(1,1)$ and $B(100,1000)$. Find the number of other points with integer coordinates that are on the line and strictly between $A$ and $B$. | 8 |
In our number system, the base is ten. If the base were changed to four, you would count as follows: $1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, \ldots$ What would be the twentieth number in this system? | 110 |
Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her thir... | 31 |
Given the equation \(3x^2 - 4x + k = 0\) with real roots, find the value of \(k\) for which the product of the roots of the equation is a maximum. The original answer is in \(\frac{m}{n}\) format, where \(\frac{m}{n}\) is in its simplest form. Please give the value of \(m + n\). | 7 |
Find the number of positive integers less than $1000$ that are divisible by neither $5$ nor $7$. | 686 |
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order)... | 80 |
$AL$, $BM$, and $CN$ are the medians of $\triangle ABC$. $K$ is the intersection of the medians. If $C, K, L, M$ are concyclic and $AB = \sqrt{3}$, find the length of the median $CN$. The original answer is in the form \(\frac{k\sqrt{3}}{n}\), where k and n are integers. Please find the value of k + n. | 5 |
The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. What is the number of westbound vehicles present in a 100-mile section of h... | 200 |
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchange... | 103 |
What is the maximum value of the difference between the largest real root and the smallest real root of the equation system \[\begin{array}{rcl} ax^2 + bx+ c &=& 0 \\ bx^2 + cx+ a &=& 0 \\ cx^2 + ax+ b &=& 0 \end{array}\], where at least one of the reals $a,b,c$ is non-zero? | 0 |
For a rhombus with a side length of 5, the length of one of its diagonals is not larger than $6$, and the length of the other diagonal is not smaller than $6$. Determine the maximum value of the sum of the two diagonals. | 14 |
In a right triangle with sides \(a\) and \(b\), and hypotenuse \(c\), the altitude drawn on the hypotenuse is \(x\). Find the relationship between \(a\), \(b\), and \(x\) that satisfies \(\frac{k}{x^2} = \frac{m}{a^2} + \frac{n}{b^2}\). Please provide the sum of the coefficients \(k + m + n\). | 3 |
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