problem stringlengths 10 4.31k | answer stringlengths 0 159 | correct_count float64 1 9 |
|---|---|---|
Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? | 26 | 3 |
Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$ | \frac{\sqrt{2}}{2} | 9 |
Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$. | \frac{5}{3} | 1 |
The graph, $G$ of $y=\log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. What is the equation for $G'$? | 10^{-x} | 7 |
A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased? | 7 | 1 |
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
[asy]
unitsize(2mm); defaultpen(... | \frac{1}{5} | 9 |
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 | 1 |
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ | 360 | 1 |
A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$? | 8 | 6 |
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values... | 60 | 8 |
A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube? | 90 | 1 |
In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?
$\begin{array}{cccc} & A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}$ | 1 | 6 |
If $a$ and $b$ are digits for which
$\begin{array}{ccc}& 2 & a\ \times & b & 3\ \hline & 6 & 9\ 9 & 2 & \ \hline 9 & 8 & 9\end{array}$
then $a+b =$ | 7 | 9 |
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | 37 | 8 |
Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$? | \frac{25}{32} | 1 |
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the... | 16 | 7 |
Letters $A, B, C,$ and $D$ represent four different digits selected from $0, 1, 2, \ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$? | 17 | 7 |
A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? | \frac{5}{6} | 3 |
Medians $AD$ and $CE$ of $\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$.
Let the area of $\triangle MNE$ be $k$ times the area of $\triangle ABC$. Then $k$ equals: | \frac{1}{6} | 2 |
How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive? | 58 | 9 |
How many positive factors of 36 are also multiples of 4? | 3 | 9 |
The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:
1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't first or last.
What is the average of the first and l... | 6.5 | 6 |
The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? | 90 | 2 |
Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? | 12 | 6 |
If $q_1(x)$ and $r_1$ are the quotient and remainder, respectively, when the polynomial $x^8$ is divided by $x + \frac{1}{2}$, and if $q_2(x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x + \frac{1}{2}$, then $r_2$ equals | -\frac{1}{16} | 2 |
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 $i = \sqrt{-1}$. What is
\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\] | \frac{7}{16} | 5 |
An $11 \times 11 \times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point? | 331 | 2 |
A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies hal... | 2 | 8 |
A palindrome between $1000$ and $10000$ is chosen at random. What is the probability that it is divisible by $7$? | \frac{1}{5} | 8 |
What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers? | 996 | 1 |
In $\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$, then the area of $\triangle ABC$ is | 4K | 7 |
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? | 143 | 8 |
Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \cdot a - a \cdot b$? | -15 | 3 |
Given $0 \le x_0 < 1$, let
\[x_n = \begin{cases} 2x_{n-1} & \text{ if } 2x_{n-1} < 1 \\ 2x_{n-1} - 1 & \text{ if } 2x_{n-1} \ge 1 \end{cases}\]for all integers $n > 0$. For how many $x_0$ is it true that $x_0 = x_5$? | 31 | 2 |
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? | 0.4 | 9 |
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and $d$ wer... | 3 | 9 |
Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?
$\textbf{(I)}\ x+y < a+b$
$\textbf{(II)}\ x-y < a-b$
$\textbf{(III)}\ xy < ab$
$\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}$ | 1 | 7 |
The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=12$, then the radius of the larger circle is | 18 | 8 |
How many positive integer factors of $2020$ have more than $3$ factors? | 7 | 5 |
Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser? | 10 | 9 |
What is the greatest number of consecutive integers whose sum is $45?$ | 90 | 9 |
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? | 22 | 1 |
The fraction $\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to: | a^{-2}+b^{-2} | 1 |
What is the value of $\frac{11! - 10!}{9!}$? | 100 | 9 |
Let $A, M$, and $C$ be digits with $(100A+10M+C)(A+M+C) = 2005$. What is $A$? | 4 | 9 |
Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. | -1/9 | 7 |
The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is | 28 | 9 |
How many integers $n \geq 2$ are there such that whenever $z_1, z_2, \dots, z_n$ are complex numbers such that
\[|z_1| = |z_2| = \dots = |z_n| = 1 \text{ and } z_1 + z_2 + \dots + z_n = 0,\]
then the numbers $z_1, z_2, \dots, z_n$ are equally spaced on the unit circle in the complex plane? | 2 | 1 |
If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk? | \frac{x(x+2)(x+5)}{(x+1)(x+3)} | 8 |
The set $\{3,6,9,10\}$ is augmented by a fifth element $n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $n$? | 26 | 7 |
Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the sta... | 47 | 1 |
The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner? | 14 | 4 |
In a magic triangle, each of the six whole numbers $10-15$ is placed in one of the circles so that the sum, $S$, of the three numbers on each side of the triangle is the same. The largest possible value for $S$ is
[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(... | 39 | 5 |
Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
[asy]
unitsize(6mm);
defaultpen(linewidth(.8pt));
path p=(1,1)--(-2,1)--(-2,2)--(1,2);
draw(p);
draw... | 3 | 4 |
A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of water, in cubic feet? | $48\pi - 36 \sqrt {3}$ | 8 |
A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx +2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$? | \frac{50}{99} | 1 |
Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
(1) It is prime.
(2) It is even.
(3) It is divisible by 7.
(4) One of its dig... | 8 | 5 |
The manager of a company planned to distribute a $50 bonus to each employee from the company fund, but the fund contained $5 less than what was needed. Instead the manager gave each employee a $45 bonus and kept the remaining $95 in the company fund. The amount of money in the company fund before any bonuses were pai... | 995 | 9 |
$6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = $ | $6^7$ | 1 |
Rectangle $ABCD$ has $AB=4$ and $BC=3$. Segment $EF$ is constructed through $B$ so that $EF$ is perpendicular to $DB$, and $A$ and $C$ lie on $DE$ and $DF$, respectively. What is $EF$? | \frac{125}{12} | 1 |
If $f(x)=3x+2$ for all real $x$, then the statement:
"$|f(x)+4|<a$ whenever $|x+2|<b$ and $a>0$ and $b>0$"
is true when | $b \le a/3$ | 3 |
In a collection of red, blue, and green marbles, there are $25\%$ more red marbles than blue marbles, and there are $60\%$ more green marbles than red marbles. Suppose that there are $r$ red marbles. What is the total number of marbles in the collection? | 3.4r | 9 |
If an integer of two digits is $k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by | 11-k | 4 |
If $F(n+1)=\frac{2F(n)+1}{2}$ for $n=1,2,\cdots$ and $F(1)=2$, then $F(101)$ equals: | 52 | 9 |
A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? | \frac{1}{3} | 9 |
Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth? | $1.15 | 4 |
A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$ | 4 | 4 |
In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument? | 154 | 9 |
The number of points common to the graphs of
$(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$ is: | 4 | 9 |
If $\log_6 x=2.5$, the value of $x$ is: | 36\sqrt{6} | 8 |
Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of $\triangle CEF$ is $200$ square inches. Then the number of inches in $BE$ is: | 12 | 2 |
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squ... | \sqrt{2}+\frac{1}{2} | 1 |
The product of $\sqrt[3]{4}$ and $\sqrt[4]{8}$ equals | 2\sqrt[12]{32} | 2 |
Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? | \frac{3}{8} | 9 |
Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$? | 18 | 2 |
Three times Dick's age plus Tom's age equals twice Harry's age.
Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age.
Their respective ages are relatively prime to each other. The sum of the squares of their ages is | 42 | 3 |
How many unordered pairs of edges of a given cube determine a plane? | 42 | 2 |
If $1998$ is written as a product of two positive integers whose difference is as small as possible, then the difference is | 17 | 6 |
If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is | \frac{x}{x-1} | 4 |
What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?
[asy]
unitsize(12);
draw((0,0)--(10,0));
draw((-1.5,1.5)--(-1.5,2.5));
draw((-1,2)--(-2,2));
draw((1,1)--(3,1)--(3,3)--(1,3)--cycle);
draw((1,4)--(3,4)... | 1047 | 8 |
Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\]
Then $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals: | f(n) | 5 |
If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: | $3x^2+4$ | 4 |
A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is: | 0 | 8 |
Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
[asy]
unitsize(8mm); defaultpen(linewidth(.8pt));
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((0,3)--(0,4)--(1,4)--(1,3)--cycle);
draw((1,3)--(1,4)--(2,... | 3 | 3 |
Three generous friends, each with some money, redistribute the money as followed:
Amy gives enough money to Jan and Toy to double each amount has.
Jan then gives enough to Amy and Toy to double their amounts.
Finally, Toy gives enough to Amy and Jan to double their amounts.
If Toy had 36 dollars at the beginning and 3... | 252 | 5 |
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 the beam will travel along this path? | 10\sqrt{2} | 7 |
Placing no more than one X in each small square, what is the greatest number of X's that can be put on the grid shown without getting three X's in a row vertically, horizontally, or diagonally?
[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) { draw((0,b)--(3,b)); } [/asy] | 6 | 5 |
A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? | $2\sqrt{5}$ | 1 |
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units? | 0 | 1 |
In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? | 2\frac{2}{3} | 8 |
Given a finite sequence $S=(a_1,a_2,\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence
$\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$
of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and le... | \sqrt{2}-1 | 3 |
If $y+4 = (x-2)^2$ and $x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$? | 15 | 4 |
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 | 8 |
Let $P$ equal the product of 3,659,893,456,789,325,678 and 342,973,489,379,256. The number of digits in $P$ is: | 34 | 8 |
If $\frac{\frac{x}{4}}{2}=\frac{4}{\frac{x}{2}}$, then $x=$ | \pm 8 | 1 |
Points $P$ and $Q$ lie in a plane with $PQ=8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P$, $Q$, and $R$ is a right triangle with area $12$ square units? | 8 | 7 |
Let $x$ and $y$ be two-digit positive integers with mean $60$. What is the maximum value of the ratio $\frac{x}{y}$? | \frac{33}{7} | 8 |
A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring ... | 173 | 1 |
Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere? | \( \frac{2\sqrt{3}-3}{2} \) | 3 |
Three-digit powers of $2$ and $5$ are used in this "cross-number" puzzle. What is the only possible digit for the outlined square?
\[\begin{array}{lcl} \textbf{ACROSS} & & \textbf{DOWN} \\ \textbf{2}.~ 2^m & & \textbf{1}.~ 5^n \end{array}\] | 6 | 5 |
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