problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
John draws a regular five pointed star in the sand, and at each of the 5 outward-pointing points and 5 inward-pointing points he places one of ten different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent? | 362880 | 0.75 | 5,002.5 | 3,939.333333 | 8,192 | |
If \( x^{4} + ax^{2} + bx + c = 0 \) has roots 1, 2, and 3 (one root is repeated), find \( a + c \).
(17th Annual American High School Mathematics Examination, 1966) | -61 | 0.5 | 6,240.5625 | 4,289.125 | 8,192 | |
If $\lceil{\sqrt{x}}\rceil=12$, how many possible integer values of $x$ are there? | 23 | 0.1875 | 5,248.1875 | 4,427.666667 | 5,437.538462 | |
Compute
$$
\int_{1}^{\sqrt{3}} x^{2 x^{2}+1}+\ln \left(x^{2 x^{2 x^{2}+1}}\right) d x. \text{ }$$ | 13 | 0.125 | 7,399.9375 | 5,383.5 | 7,688 | |
Given a function $f(x)$ whose domain is $D$, if for any $x_1, x_2 \in D$, it holds that $f(x_1) \leq f(x_2)$ whenever $x_1 < x_2$, then the function $f(x)$ is called a non-decreasing function on $D$. Suppose $f(x)$ is a non-decreasing function on the interval $[0,1]$ and satisfies the following three conditions:
1. $f(... | \frac{11}{4} | 0.125 | 8,136.5 | 7,748 | 8,192 | |
Given the complex number $z$ that satisfies $z= \frac {1-i}{i+1}$, find the value of $|1+z|$. | \sqrt{2} | 1 | 2,451.0625 | 2,451.0625 | -1 | |
The coordinates of $A, B$ and $C$ are $(5,5), (2,1)$ and $(0,k)$ respectively.
The value of $k$ that makes $\overline{AC}+\overline{BC}$ as small as possible is: | 2\frac{1}{7} | 1. **Identify the problem**: We need to find the value of $k$ such that the sum of the distances $\overline{AC} + \overline{BC}$ is minimized, where $A = (5,5)$, $B = (2,1)$, and $C = (0,k)$.
2. **Geometric interpretation**: Consider the reflection of point $A$ across the y-axis to get point $A'$ with coordinates $(-5... | 0 | 5,282.1875 | -1 | 5,282.1875 |
Find the number that becomes a perfect square either by adding 5 or by subtracting 11. | 20 | 0.25 | 6,954.5625 | 4,858.25 | 7,653.333333 | |
Given the points $(7, -9)$ and $(1, 7)$ as the endpoints of a diameter of a circle, calculate the sum of the coordinates of the center of the circle, and also determine the radius of the circle. | \sqrt{73} | 0.9375 | 1,710.4375 | 1,732.933333 | 1,373 | |
Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$, and $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$, and let $M$ be the point on line $BK$ f... | 72 | Using the diagram above, we can solve this problem by using mass points. By angle bisector theorem: \[\frac{BL}{CB}=\frac{AL}{CA}\implies\frac{BL}{300}=\frac{AL}{450}\implies 3BL=2AL\] So, we can weight $A$ as $2$ and $B$ as $3$ and $L$ as $5$. Since $K$ is the midpoint of $A$ and $C$, the weight of $A$ is equal to the... | 0.3125 | 7,984 | 7,526.4 | 8,192 |
Jia draws five lines in a plane without any three lines intersecting at one point. For each pair of intersecting lines, Jia gets a candy. If there is a set of parallel lines, Jia also gets a candy. For example, in a diagram with seven intersection points and one set of parallel lines, Jia receives 8 candies. How many c... | 11 | 0 | 8,192 | -1 | 8,192 | |
Consider a trapezoidal field where it's planted uniformly with wheat. The trapezoid has the following measurements: side $AB$ is 150 m, base $AD$ (the longest side) is 300 m, and the other base $BC$ is 150 m. The angle at $A$ is $75^\circ$, and the angle at $B$ is $105^\circ$. At harvest time, all the wheat is collecte... | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2510$. Let $N$ be the largest of the sums $a+b$, $b+c$, $c+d$, and $d+e$. What is the smallest possible value of $N$? | 1255 | 0 | 7,809.3125 | -1 | 7,809.3125 | |
Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$ ? | 3267 | 0 | 8,067.4375 | -1 | 8,067.4375 | |
In a right square prism \( P-ABCD \) with side edges and base edges both equal to 4, determine the total length of all the curves formed on its surface by points that are 3 units away from vertex \( P \). | 6\pi | 0.125 | 7,023.9375 | 6,414 | 7,111.071429 | |
In the diagram, triangles $ABC$ and $CBD$ are isosceles. The perimeter of $\triangle CBD$ is $19,$ the perimeter of $\triangle ABC$ is $20,$ and the length of $BD$ is $7.$ What is the length of $AB?$ [asy]
size(7cm);
defaultpen(fontsize(11));
pair b = (0, 0);
pair d = 7 * dir(-30);
pair a = 8 * dir(-140);
pair c = 6 ... | 8 | 0.25 | 6,927.6875 | 3,518.25 | 8,064.166667 | |
Let points $A = (0,0)$, $B = (1,2)$, $C = (3,3)$, and $D = (4,0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$, where these fractions are in lowest terms. What is $p + q + r + s$? | 58 | 1. **Identify Coordinates and Setup**: We are given the coordinates of the quadrilateral $ABCD$ as $A = (0,0)$, $B = (1,2)$, $C = (3,3)$, and $D = (4,0)$. We need to find the point where a line through $A$ that divides the area of $ABCD$ into two equal parts intersects $\overline{CD}$.
2. **Calculate the Area of Quadr... | 0.875 | 4,610.5625 | 4,423.142857 | 5,922.5 |
In how many ways can we enter numbers from the set $\{1,2,3,4\}$ into a $4 \times 4$ array so that all of the following conditions hold? (a) Each row contains all four numbers. (b) Each column contains all four numbers. (c) Each "quadrant" contains all four numbers. (The quadrants are the four corner $2 \times 2$ squar... | 288 | Call a filled $4 \times 4$ array satisfying the given conditions cool. There are 4 ! possibilities for the first row; WLOG, let it be 1234. Since each quadrant has to contain all four numbers, we have exactly four possibilities for the second row, namely: (i) 3412 (ii) 3421 (iii) 4312 (iv) 4321 I claim that the number ... | 0.125 | 7,946.5625 | 6,228.5 | 8,192 |
A hollow glass sphere with uniform wall thickness and an outer diameter of $16 \mathrm{~cm}$ floats in water in such a way that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness, given that the specific gravity of the glass is $s = 2.523$? | 0.8 | 0.125 | 7,820.125 | 5,217 | 8,192 | |
Given that Jennifer plans to build a fence around her garden in the shape of a rectangle, with $24$ fence posts, and evenly distributing the remaining along the edges, with $6$ yards between each post, and with the longer side of the garden, including corners, having three times as many posts as the shorter side, calcu... | 855 | 0 | 6,703.375 | -1 | 6,703.375 | |
Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2010} \right| = \frac{1}{2010}.\] | \frac{2008}{1005} | 0.25 | 7,791.5 | 6,798.75 | 8,122.416667 | |
Each square in the following hexomino has side length 1. Find the minimum area of any rectangle that contains the entire hexomino. | \frac{21}{2} | If a rectangle contains the entire hexomino, it must also contain its convex hull, which is an origin-symmetric hexagon. It is fairly clear that the smallest rectangle that contains such a hexagon must share one set of parallel sides with the hexagon. There are three such rectangles, and checking them all, we find that... | 0 | 5,742 | -1 | 5,742 |
In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.
\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200\]What is the median of the numbers in this list? | 142 | To find the median of the list, we first need to determine the total number of elements in the list. Each integer $n$ from $1$ to $200$ appears $n$ times. Therefore, the total number of elements, $N$, is the sum of the first $200$ positive integers:
\[
N = 1 + 2 + 3 + \ldots + 200 = \frac{200 \times (200 + 1)}{2} = \f... | 0.875 | 4,857.0625 | 4,710.785714 | 5,881 |
In $\triangle ABC$, $AB > AC$, $\angle BAC = 45^\circ$. Point $E$ is the intersection of the external angle bisector of $\angle BAC$ with the circumcircle of $\triangle ABC$. Point $F$ is on $AB$ such that $EF \perp AB$. Given $AF = 1$ and $BF = 5$, find the area of $\triangle ABC$. | 6 \sqrt{2} | 0.0625 | 7,985.3125 | 5,756 | 8,133.933333 | |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.5 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
... | \text{C} | 0 | 6,975.9375 | -1 | 6,975.9375 | |
Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, where $a_n > 0$. If $S_6 - 2S_3 = 5$, then the minimum value of $S_9 - S_6$ is ______. | 20 | 0.3125 | 7,782.8125 | 6,882.6 | 8,192 | |
Evaluate the expression \[(5^{1001} + 6^{1002})^2 - (5^{1001} - 6^{1002})^2\] and express it in the form \(k \cdot 30^{1001}\) for some integer \(k\). | 24 | 0.625 | 2,476.5 | 2,123 | 3,065.666667 | |
Juan rolls a fair twelve-sided die marked with the numbers 1 through 12. Then Amal rolls a fair ten-sided die. What is the probability that the product of the two rolls is a multiple of 3? | \frac{8}{15} | 0.9375 | 2,825.9375 | 2,468.2 | 8,192 | |
Given that $(1+\sin t)(1+\cos t)=5/4$ and
$(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},$
where $k, m,$ and $n_{}$ are positive integers with $m_{}$ and $n_{}$ relatively prime, find $k+m+n.$ | 27 | We want $1+\sin t \cos t-\sin t-\cos t$. However, note that we only need to find $\sin t+\cos t$.
Let $y = \sin t+\cos t \rightarrow y^2 = \sin^2 t + \cos^2 t + 2\sin t \cos t = 1 + 2\sin t \cos x$
From this we have $\sin t \cos t = \frac{y^2-1}{2}$ and $\sin t + \cos t = y$
Substituting, we have $2y^2+4y-3=0 \right... | 0.8125 | 4,993.25 | 4,607 | 6,667 |
Place 5 balls numbered 1, 2, 3, 4, 5 into 5 boxes also numbered 1, 2, 3, 4, 5.
(1) How many ways are there to do this?
(2) If each box can hold at most one ball, how many ways are there?
(3) If exactly one box is to remain empty, how many ways are there?
(4) If each box contains one ball, and exactly one ball's... | 20 | 0.25 | 6,181.6875 | 5,310.75 | 6,472 | |
The volume of the parallelepiped generated by $\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix},$ $\begin{pmatrix} 1 \\ k \\ 2 \end{pmatrix},$ and $\begin{pmatrix} 1 \\ 2 \\ k \end{pmatrix}$ is 15. Find $k,$ where $k > 0.$ | \frac{9}{2} | 0.9375 | 4,027.375 | 3,916.466667 | 5,691 | |
Find the $2 \times 2$ matrix $\mathbf{M}$ such that $\mathbf{M} \begin{pmatrix} 3 \\ 0 \end{pmatrix} = \begin{pmatrix} 6 \\ 21 \end{pmatrix}$ and $\mathbf{M} \begin{pmatrix} -1 \\ 5 \end{pmatrix} = \begin{pmatrix} 3 \\ -17 \end{pmatrix}.$ | \begin{pmatrix} 2 & 1 \\ 7 & -2 \end{pmatrix} | 1 | 1,833.875 | 1,833.875 | -1 | |
Evaluate: \( \frac {\tan 150^{\circ} \cos (-210^{\circ}) \sin (-420^{\circ})}{\sin 1050^{\circ} \cos (-600^{\circ})} \). | -\sqrt{3} | 0.8125 | 5,245.25 | 4,939.461538 | 6,570.333333 | |
There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\em advisory meeting} (for a particular club) can be realized only when each participant is a member of t... | \binom{2003}{11} | 0 | 8,148.5625 | -1 | 8,148.5625 | |
In the table shown, the formula relating \(x\) and \(y\) is:
\[\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5\\ \hline y & 3 & 7 & 13 & 21 & 31\\ \hline\end{array}\] | y = x^2 + x + 1 | To find the correct formula relating $x$ and $y$, we will substitute the given values of $x$ into each formula choice and check if the resulting $y$ matches the values in the table.
#### Checking Choice (A) $y = 4x - 1$
1. For $x = 1$, $y = 4(1) - 1 = 3$
2. For $x = 2$, $y = 4(2) - 1 = 7$
3. For $x = 3$, $y = 4(3) - 1... | 0.9375 | 4,053.8125 | 3,777.933333 | 8,192 |
$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP < DP.$
[asy]
unitsize(0.6 inch);
draw(circle((0,0),1));
draw((-0.3,0.94)--(0.3,-0.94));
draw((-0.7,-0.7)--(0.7,-0.7));
label("$A$",(-0.3,0... | 2 | 1 | 1,437.9375 | 1,437.9375 | -1 | |
There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? | 40 | To solve this problem, we need to find the minimum number of people that must be seated such that any additional person must sit next to someone already seated. We aim to maximize the number of empty seats between seated people under this constraint.
1. **Understanding the Pattern**:
- If we place a person in a se... | 0.5625 | 6,356.8125 | 5,287.111111 | 7,732.142857 |
After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds? | 24 | 1. **Convert Naomi's travel time to hours**: Naomi travels $6$ miles in $10$ minutes. To convert minutes to hours, we divide by $60$:
\[
10 \text{ minutes} = \frac{10}{60} \text{ hours} = \frac{1}{6} \text{ hours}
\]
2. **Calculate Naomi's average speed**: Using the formula for speed, which is $\text{speed} =... | 0 | 5,797.9375 | -1 | 5,797.9375 |
Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least amounts of money that Ricardo can have? | 8072 | 1. **Define Variables:**
Let $p$ be the number of pennies Ricardo has. Since the rest of the coins are nickels, he has $2020 - p$ nickels.
2. **Constraints:**
Ricardo has at least one penny and at least one nickel. Therefore, we have the constraints:
\[
p \geq 1 \quad \text{and} \quad 2020 - p \geq 1
\]... | 1 | 2,477.1875 | 2,477.1875 | -1 |
Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of $2$. In the interval $[-1,1)$, $f(x)$ is defined as follows:
$$
f(x)=\begin{cases}
x + a, & -1 \leqslant x < 0, \\
\left| \frac{2}{5} - x \right|, & 0 \leqslant x < 1,
\end{cases}
$$
where $a \in \mathbb{R}$. If $f\left( -\frac{5}{2} \right) = f\left( \... | -\frac{2}{5} | 0.875 | 3,033.5625 | 2,977.571429 | 3,425.5 | |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\] | \frac{89}{9} | 0 | 8,192 | -1 | 8,192 | |
In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, where $j$ and $k$ are positive integers, find the sum of all possible values for $k$. | 51 | 1 | 2,458.5 | 2,458.5 | -1 | |
Paul Erdős was one of the most prolific mathematicians of all time and was renowned for his many collaborations. The Erdős number of a mathematician is defined as follows. Erdős has an Erdős number of 0, a mathematician who has coauthored a paper with Erdős has an Erdős number of 1, a mathematician who has not coauthor... | 4.65 | We'll suppose that each mathematician collaborates with approximately 20 people (except for Erdős himself, of course). Furthermore, if a mathematician has Erdős number $k$, then we'd expect him to be the cause of approximately $\frac{1}{2^{k}}$ of his collaborators' Erdős numbers. This is because as we get to higher Er... | 0 | 5,551.8125 | -1 | 5,551.8125 |
Petrov writes down odd numbers: \(1, 3, 5, \ldots, 2013\), and Vasechkin writes down even numbers: \(2, 4, \ldots, 2012\). Each of them calculates the sum of all the digits of all their numbers and tells it to the star student Masha. Masha subtracts Vasechkin's result from Petrov's result. What is the outcome? | 1007 | 0 | 8,179.0625 | -1 | 8,179.0625 | |
Given real numbers \( x \) and \( y \) that satisfy \( x^{2} + y^{2} \leq 5 \), find the maximum and minimum values of the function \( f(x, y) = 3|x+y| + |4y+9| + |7y-3x-18| \). | 27 + 6\sqrt{5} | 0 | 8,166.1875 | -1 | 8,166.1875 | |
In the polar coordinate system, the equation of curve $C$ is $\rho^{2}\cos 2\theta=9$. Point $P(2 \sqrt {3}, \frac {\pi}{6})$ is given. Establish a Cartesian coordinate system with the pole $O$ as the origin and the positive half-axis of the $x$-axis as the polar axis.
$(1)$ Find the parametric equation of line $OP$ ... | \sqrt {2} | 0 | 5,436.0625 | -1 | 5,436.0625 | |
In the diagram, $AB = 13\text{ cm},$ $DC = 20\text{ cm},$ and $AD = 5\text{ cm}.$ What is the length of $AC,$ to the nearest tenth of a centimeter?
[asy]
draw((0,0)--(5,12)--(21,12)--(5,0)--cycle,black+linewidth(1));
draw((5,12)--(5,0),black+linewidth(1));
draw((0,0)--(21,12),black+linewidth(1));
draw((5,0)--(5,0.5)--... | 24.2 | 0.5 | 6,838.0625 | 5,484.125 | 8,192 | |
Given the sets $A=\{x|x^{2}-px-2=0\}$ and $B=\{x|x^{2}+qx+r=0\}$, if $A\cup B=\{-2,1,5\}$ and $A\cap B=\{-2\}$, find the value of $p+q+r$. | -14 | 0.625 | 4,961.1875 | 4,042.2 | 6,492.833333 | |
Find the largest natural number \( n \) for which the product of the numbers \( n, n+1, n+2, \ldots, n+20 \) is divisible by the square of one of these numbers. | 20 | 0 | 8,192 | -1 | 8,192 | |
The probability of an event occurring in each of 900 independent trials is 0.5. Find a positive number $\varepsilon$ such that with a probability of 0.77, the absolute deviation of the event frequency from its probability of 0.5 does not exceed $\varepsilon$. | 0.02 | 0.125 | 7,499.875 | 5,561 | 7,776.857143 | |
Evaluate $\lfloor 3.998 \rfloor + \lceil 7.002 \rceil$. | 11 | 0.875 | 1,723.875 | 1,766.857143 | 1,423 | |
A point in space $(x,y,z)$ is randomly selected so that $-1\le x \le 1$,$-1\le y \le 1$,$-1\le z \le 1$. What is the probability that $x^2+y^2+z^2\le 1$? | \frac{\pi}{6} | 1 | 1,553.4375 | 1,553.4375 | -1 | |
If $a$ , $b$ are integers and $s=a^3+b^3-60ab(a+b)\geq 2012$ , find the least possible value of $s$. | 2015 |
Given the inequality \( s = a^3 + b^3 - 60ab(a + b) \geq 2012 \), we are tasked with finding the least possible value of \( s \).
To solve this, we start by rewriting \( s \) in terms of a simplified expression:
\[
s = a^3 + b^3 - 60ab(a + b).
\]
Notice that we can factor and simplify the expression using the identi... | 0 | 8,192 | -1 | 8,192 |
How many times do the graphs of the line $5x + 8y = 10$ and the circle $x^2 + y^2 = 1$ intersect? | 0 | 1 | 3,334.6875 | 3,334.6875 | -1 | |
How many different prime numbers are factors of $N$ if
$\log_2 ( \log_3 ( \log_5 (\log_ 7 N))) = 11?$ | 1 | 1. **Rewriting the given logarithmic equation:**
Start with the equation:
\[
\log_2 ( \log_3 ( \log_5 (\log_ 7 N))) = 11
\]
Rewriting this equation in terms of exponents, we start from the innermost logarithm and work our way outwards:
\[
\log_3 ( \log_5 (\log_ 7 N)) = 2^{11}
\]
2. **Solving ... | 0.875 | 3,966.3125 | 3,362.642857 | 8,192 |
Three positive integers have an arithmetic mean of 26 and median of 27. If the median is 5 less than the largest number, what is the smallest number? | 19 | 1 | 1,289.5625 | 1,289.5625 | -1 | |
Two sectors of a circle of radius $12$ overlap as shown, with $P$ and $R$ as the centers of the respective circles. Determine the area of the shaded region.
[asy]
draw((0,0)--(10.3923,-6)--(20.7846,0)--(10.3923,6)--cycle,black+linewidth(1));
filldraw((10.3923,6)..(12,0)..(10.3923,-6)--cycle,gray,black+linewidth(1));
f... | 48\pi-72\sqrt{3} | 0.625 | 6,358 | 5,257.6 | 8,192 | |
For rational numbers $x$, $y$, $a$, $t$, if $|x-a|+|y-a|=t$, then $x$ and $y$ are said to have a "beautiful association number" of $t$ with respect to $a$. For example, $|2-1|+|3-1|=3$, then the "beautiful association number" of $2$ and $3$ with respect to $1$ is $3$. <br/> $(1)$ The "beautiful association number" of $... | 2001000 | 0.0625 | 8,045.375 | 8,192 | 8,035.6 | |
Let $a$, $b$, $c$ be positive numbers, and $a+b+9c^2=1$. The maximum value of $\sqrt{a} + \sqrt{b} + \sqrt{3}c$ is \_\_\_\_\_\_. | \frac{\sqrt{21}}{3} | 0 | 7,609.625 | -1 | 7,609.625 | |
At the beginning of the school year, Lisa's goal was to earn an $A$ on at least $80\%$ of her $50$ quizzes for the year. She earned an $A$ on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an $A$? | 2 | 1. **Determine the total number of quizzes Lisa needs to score an A on to meet her goal**: Lisa's goal is to earn an A on at least 80% of her 50 quizzes. Therefore, the total number of quizzes she needs to score an A on is:
\[
0.80 \times 50 = 40
\]
quizzes.
2. **Calculate the number of quizzes she has alr... | 1 | 1,535.4375 | 1,535.4375 | -1 |
Find the maximum value of
\[\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 + \cos \theta & 1 & 1 \end{vmatrix},\]as $\theta$ ranges over all angles. | \frac{1}{2} | 0.75 | 5,532.9375 | 4,646.583333 | 8,192 | |
Given that there are 5 cards of the same size and shape, each marked with the numbers 1, 2, 3, 4, and 5 respectively. If two cards are drawn at random, the probability that the larger number on these two cards is 3 is ______. | \dfrac {1}{5} | 1 | 2,893.75 | 2,893.75 | -1 | |
If $f(x) = 3x + 1$, what is $f(3)$? | 10 | 1 | 935.375 | 935.375 | -1 | |
Given that Jessie moves from 0 to 24 in six steps, and travels four steps to reach point x, then one more step to reach point z, and finally one last step to point y, calculate the value of y. | 24 | 0.625 | 495.875 | 504.3 | 481.833333 | |
Three distinct integers, $x$, $y$, and $z$, are randomly chosen from the set $\{1, 2, 3, \dots, 12\}$. What is the probability that $xyz - x - y - z$ is even? | \frac{1}{11} | 0 | 7,525.25 | -1 | 7,525.25 | |
For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$ . | 125 | 0.8125 | 5,447.8125 | 4,915.153846 | 7,756 | |
Five rays $\overrightarrow{OA}$ , $\overrightarrow{OB}$ , $\overrightarrow{OC}$ , $\overrightarrow{OD}$ , and $\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that $\angle EOD = 2\angle COB$ , $\angle COB = 2\angle BOA$ , while $\angle DOC = 3\angle BOA$ . ... | 90 | 0.8125 | 4,471.9375 | 3,779 | 7,474.666667 | |
Let $(a_1,a_2,a_3,\ldots,a_{12})$ be a permutation of $(1,2,3,\ldots,12)$ for which
$a_1>a_2>a_3>a_4>a_5>a_6 \mathrm{\ and \ } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}.$
An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations. | 462 | Clearly, $a_6=1$. Now, consider selecting $5$ of the remaining $11$ values. Sort these values in descending order, and sort the other $6$ values in ascending order. Now, let the $5$ selected values be $a_1$ through $a_5$, and let the remaining $6$ be $a_7$ through ${a_{12}}$. It is now clear that there is a bijection b... | 0.375 | 6,340.0625 | 4,365 | 7,525.1 |
A parabola has vertex $V = (0,0)$ and focus $F = (0,1).$ Let $P$ be a point in the first quadrant, lying on the parabola, so that $PF = 101.$ Find $P.$ | (20,100) | 1 | 2,272.9375 | 2,272.9375 | -1 | |
Compute the smallest positive integer that does not appear in any problem statement on any round at HMMT November 2023. | 22 | The number 22 does not appear on any round. On the other hand, the numbers 1 through 21 appear as follows. \begin{tabular}{c|c|c} Number & Round & Problem \\ \hline 1 & Guts & 21 \\ 2 & Guts & 13 \\ 3 & Guts & 17 \\ 4 & Guts & 13 \\ 5 & Guts & 14 \\ 6 & Guts & 2 \\ 7 & Guts & 10 \\ 8 & Guts & 13 \\ 9 & Guts & 28 \\ 10 ... | 0 | 4,476 | -1 | 4,476 |
A person has 13 pieces of a gold chain containing 80 links. Separating one link costs 1 cent, and attaching a new one - 2 cents.
What is the minimum amount needed to form a closed chain from these pieces?
Remember, larger and smaller links must alternate. | 30 | 0 | 7,657.75 | -1 | 7,657.75 | |
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden? | 6 | 1. **Interpreting the Graph**: The problem states that Suzanna rides her bike at a constant rate, and the graph shows that every 5 minutes, her distance increases by 1 mile.
2. **Total Riding Time**: Suzanna rides for a total of half an hour. Half an hour is equivalent to 30 minutes.
3. **Calculating Distance per 5 M... | 0.375 | 6,953.6875 | 4,889.833333 | 8,192 |
If $x$, $y$, $z$ are positive numbers satisfying
\[x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.\]
Find all the possible values of $x+y+z$. | 3 |
We are given that \(x\), \(y\), and \(z\) are positive numbers satisfying the system of equations:
\[
x + \frac{y}{z} = 2,
\]
\[
y + \frac{z}{x} = 2,
\]
\[
z + \frac{x}{y} = 2.
\]
Our goal is to find all possible values of \(x + y + z\).
### Step 1: Analyze the equations.
Each equation can be rewritten as:
1. \(x ... | 0.125 | 7,786.25 | 4,946 | 8,192 |
Simplify $(9 \times 10^8) \div (3 \times 10^3)$. (Your answer should be a single number, with no products or quotients.) | 300,\!000 | 0 | 1,880.375 | -1 | 1,880.375 | |
Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 15, what is the smallest value of $n$ such that $f(n) > 15$? | n = 255 | 0.375 | 7,635.4375 | 6,707.833333 | 8,192 | |
Let $x_{1}=y_{1}=x_{2}=y_{2}=1$, then for $n \geq 3$ let $x_{n}=x_{n-1} y_{n-2}+x_{n-2} y_{n-1}$ and $y_{n}=y_{n-1} y_{n-2}- x_{n-1} x_{n-2}$. What are the last two digits of $\left|x_{2012}\right|$ ? | 84 | Let $z_{n}=y_{n}+x_{n} i$. Then the recursion implies that: $$\begin{aligned} & z_{1}=z_{2}=1+i \\ & z_{n}=z_{n-1} z_{n-2} \end{aligned}$$ This implies that $$z_{n}=\left(z_{1}\right)^{F_{n}}$$ where $F_{n}$ is the $n^{\text {th }}$ Fibonacci number $\left(F_{1}=F_{2}=1\right)$. So, $z_{2012}=(1+i)^{F_{2012}}$. Notice ... | 0 | 8,171.0625 | -1 | 8,171.0625 |
A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly $.500$. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater th... | 164 | Let $n$ be the number of matches won, so that $\frac{n}{2n}=\frac{1}{2}$, and $\frac{n+3}{2n+4}>\frac{503}{1000}$.
Cross multiplying, $1000n+3000>1006n+2012$, so $n<\frac{988}{6}=164 \dfrac {4}{6}=164 \dfrac{2}{3}$. Thus, the answer is $\boxed{164}$.
The problems on this page are copyrighted by the Mathematical Assoc... | 0.0625 | 7,657.3125 | 4,552 | 7,864.333333 |
For how many integer values of $n$ between 1 and 990 inclusive does the decimal representation of $\frac{n}{1000}$ terminate? | 990 | 0.75 | 6,363.375 | 5,753.833333 | 8,192 | |
A regular hexagon with side length 1 has an arbitrary interior point that is reflected over the midpoints of its six sides. Calculate the area of the hexagon formed in this way. | \frac{9\sqrt{3}}{2} | 0 | 8,192 | -1 | 8,192 | |
Let $a$ and $b$ be real numbers such that
\[\frac{a}{b} + \frac{a}{b^2} + \frac{a}{b^3} + \dots = 4.\]Find
\[\frac{a}{a + b} + \frac{a}{(a + b)^2} + \frac{a}{(a + b)^3} + \dotsb.\] | \frac{4}{5} | 1 | 3,229.875 | 3,229.875 | -1 | |
Find constants $A,$ $B,$ and $C$ so that
\[\frac{4x}{(x - 5)(x - 3)^2} = \frac{A}{x - 5} + \frac{B}{x - 3} + \frac{C}{(x - 3)^2}.\]Enter the ordered triple $(A,B,C).$ | (5,-5,-6) | 0.9375 | 2,853.0625 | 2,906.4 | 2,053 | |
Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 5$. Find $r$. | \frac{\sqrt{15}}{4} | 0 | 6,806.5 | -1 | 6,806.5 | |
Given that the terminal side of angle $\alpha$ passes through the point $(3a, 4a)$ ($a < 0$), then $\sin\alpha=$ ______, $\tan(\pi-2\alpha)=$ ______. | \frac{24}{7} | 0.9375 | 3,190.6875 | 2,868.066667 | 8,030 | |
A rectangle is inscribed in a triangle if its vertices all lie on the boundary of the triangle. Given a triangle \( T \), let \( d \) be the shortest diagonal for any rectangle inscribed in \( T \). Find the maximum value of \( \frac{d^2}{\text{area } T} \) for all triangles \( T \). | \frac{4\sqrt{3}}{7} | 0 | 8,049.9375 | -1 | 8,049.9375 | |
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$ | f(x) = x |
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function satisfying the following conditions:
1. \( f(x+1) = f(x) + 1 \).
2. \( f(x^2) = f(x)^2 \).
We aim to find all such functions.
### Step 1: Analyze the First Condition
The first condition \( f(x+1) = f(x) + 1 \) implies that \( f \) is a function that increases lin... | 0.0625 | 7,933.4375 | 5,102 | 8,122.2 |
Given an arithmetic sequence, the sum of the first four terms is 26, the sum of the last four terms is 110, and the sum of all terms in the sequence is 187. Determine the total number of terms in the sequence. | 11 | 0.8125 | 4,549.25 | 3,708.615385 | 8,192 | |
A sequence of 2020 natural numbers is written in a row. Each of them, starting from the third number, is divisible by the previous one and by the sum of the two preceding ones.
What is the smallest possible value for the last number in the sequence? | 2019! | 0.1875 | 8,000 | 7,168 | 8,192 | |
If $\mathbf{v} \times \mathbf{w} = \begin{pmatrix} 5 \\ -2 \\ 4 \end{pmatrix},$ then find $(\mathbf{v} + \mathbf{w}) \times (\mathbf{v} + \mathbf{w}).$ | \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} | 1 | 1,826.0625 | 1,826.0625 | -1 | |
From an 8x8 chessboard, 10 squares were cut out. It is known that among the removed squares, there are both black and white squares. What is the maximum number of two-square rectangles (dominoes) that can still be guaranteed to be cut out from this board? | 23 | 0.125 | 7,645.1875 | 6,083.5 | 7,868.285714 | |
In triangle \( \triangle ABC \), \( |AB| = 13 \), \( |BC| = 14 \), \( |CA| = 15 \), an internal point \( P \) satisfies
\[
\overrightarrow{BP} \cdot \overrightarrow{CA} = 18 \text{ and } \overrightarrow{CP} \cdot \overrightarrow{BA} = 32.
\]
What is \( \overrightarrow{AP} \cdot \overrightarrow{BC} \)? | 14 | 1 | 3,250.75 | 3,250.75 | -1 | |
Let $p$ be a polynomial with integer coefficients such that $p(15)=6$ , $p(22)=1196$ , and $p(35)=26$ . Find an integer $n$ such that $p(n)=n+82$ . | 28 | 0.4375 | 7,084.625 | 5,660.857143 | 8,192 | |
Given a geometric sequence $\{a_n\}$ where each term is positive, and $a_1a_3=4$, $a_7a_9=25$, find $a_5$. | \sqrt{10} | 0.8125 | 5,913.5 | 5,387.692308 | 8,192 | |
Six semicircles are evenly arranged along the inside of a regular hexagon with a side length of 3 units. A circle is positioned in the center such that it is tangent to each of these semicircles. Find the radius of this central circle. | \frac{3 (\sqrt{3} - 1)}{2} | 0 | 7,272.1875 | -1 | 7,272.1875 | |
The three points $(3,-5)$, $(-a + 2, 3)$, and $(2a+3,2)$ lie on the same line. What is $a$? | \frac{-7}{23} | 0 | 2,634.125 | -1 | 2,634.125 | |
Calculate
\[T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!}\]
where the sum is taken over all 1994-tuples of the numbers $n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\}$ satisfying $n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots... | \frac{1}{1994!} | 0 | 8,192 | -1 | 8,192 | |
Given a sequence $\{a_n\}$ with $16$ terms, and $a_1=1, a_8=4$. Let the function related to $x$ be $f_n(x)=\frac{x^3}{3}-a_nx^2+(a_n^2-1)x$, where $n\in \mathbf{N}^*$. If $x=a_{n+1}$ ($1\leqslant n\leqslant 15$) is the extremum point of the function $f_n(x)$, and the slope of the tangent line at the point $(a_{16}, f_8... | 1176 | 0.1875 | 7,225 | 7,742.666667 | 7,105.538462 | |
Find $2^x$ if
\begin{align*}
2^x+3^y&=5,\\
2^{x+2}+3^{y+1} &=18.
\end{align*} | 3 | 0.9375 | 1,957.5 | 1,541.866667 | 8,192 | |
A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles? | 1002 | 0 | 7,477.5 | -1 | 7,477.5 | |
Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$ | 12 |
Let's denote Ana's age this year as $A$ and Bonita's age this year as $B$. Given that Ana and Bonita were born $n$ years apart, we have:
\[ A = B + n \]
From the problem, we know:
1. Last year, Ana was $5$ times as old as Bonita:
\[ A - 1 = 5(B - 1) \]
2. This year, Ana's age is the square of Bonita's age:
\[ A... | 1 | 2,268.25 | 2,268.25 | -1 |
Consider the permutation of $1, 2, \cdots, 20$ as $\left(a_{1} a_{2} \cdots a_{20}\right)$. Perform the following operation on this permutation: swap the positions of any two numbers. The goal is to transform this permutation into $(1, 2, \cdots, 20)$. Let $k_{a}$ denote the minimum number of operations needed to reach... | 19 | 0.3125 | 5,542.3125 | 4,446.2 | 6,040.545455 |
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