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 |
|---|---|---|---|---|---|---|
Let $ABC$ be triangle such that $|AB| = 5$ , $|BC| = 9$ and $|AC| = 8$ . The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$ . Let $Z$ be the intersection of lines $XY$ and $AC$ . What is $|AZ|$ ? $
\textbf{a)}\ \sqrt{104}
\qquad\te... | 10 | 0 | 8,018.375 | -1 | 8,018.375 | |
Let $\clubsuit$ and $\heartsuit$ be whole numbers such that $\clubsuit \times \heartsuit = 48$ and $\clubsuit$ is even, find the largest possible value of $\clubsuit + \heartsuit$. | 26 | 0.375 | 3,820.6875 | 3,222 | 4,179.9 | |
Consider all four-digit numbers (including leading zeros) from $0000$ to $9999$. A number is considered balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. Calculate the total number of such balanced four-digit numbers. | 670 | 0.75 | 6,278.9375 | 5,641.25 | 8,192 | |
Find the units digit of $13 \cdot 41$. | 3 | 0.9375 | 2,422.125 | 2,037.466667 | 8,192 | |
In a circle with radius 7, two intersecting chords $PQ$ and $RS$ are given such that $PQ=10$ and $RS$ bisects $PQ$ at point $T$. It is also given that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Determine the sine of the central angle subtended by the minor arc $PR$ and express it as a fraction $\... | 15 | 0 | 7,103.8125 | -1 | 7,103.8125 | |
Given that the math scores of a certain high school approximately follow a normal distribution N(100, 100), calculate the percentage of students scoring between 80 and 120 points. | 95.44\% | 0.1875 | 4,361.4375 | 1,777 | 4,957.846154 | |
A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\bigtriangleup ACE$. Express your answer in simplest radical form. | \frac{9\sqrt{3}}{4} | 0 | 4,269.5625 | -1 | 4,269.5625 | |
The population of a city increases annually by $1 / 50$ of the current number of inhabitants. In how many years will the population triple? | 55 | 0 | 6,016.5 | -1 | 6,016.5 | |
The area of this region formed by six congruent squares is 294 square centimeters. What is the perimeter of the region, in centimeters?
[asy]
draw((0,0)--(-10,0)--(-10,10)--(0,10)--cycle);
draw((0,10)--(0,20)--(-30,20)--(-30,10)--cycle);
draw((-10,10)--(-10,20));
draw((-20,10)--(-20,20));
draw((-20,20)--(-20,30)--(-40... | 98 | 0 | 7,004.1875 | -1 | 7,004.1875 | |
In $\triangle ABC$, it is known that $\cos A=\frac{4}{5}$ and $\tan (A-B)=-\frac{1}{2}$. Find the value of $\tan C$. | \frac{11}{2} | 0.9375 | 3,520.875 | 3,209.466667 | 8,192 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\overrightarrow{a} \cdot (\overrightarrow{a} - 2\overrightarrow{b}) = 3$, and $|\overrightarrow{a}| = 1$, $\overrightarrow{b} = (1,1)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{4} | 0.4375 | 2,149.1875 | 2,083.714286 | 2,200.111111 | |
$(1)$ Calculate: $|2-\sqrt{3}|+(\sqrt{2}+1)^0+3\tan30°+(-1)^{2023}-(\frac{1}{2})^{-1}$.
$(2)$ Simplify first and then find the value: $(\frac{x-1}{x}-\frac{x-2}{x+1})÷\frac{2x^2-x}{x^2+2x+1}$, where $x$ satisfies $x^{2}-2x-2=0$. | \frac{1}{2} | 0.9375 | 3,770.875 | 3,476.133333 | 8,192 | |
A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$ ?
<details><summary>Clarifications</summary>
- $S$ is the ``set of 10 distinct integers'' from the first sentence.
</details>
*Ray Li* | 511 | 0.25 | 7,542.6875 | 5,880 | 8,096.916667 | |
$n$ mushroom gatherers went to the forest and brought back a total of 338 mushrooms (it is possible that some of the gatherers did not bring back any mushrooms). A boy named Petya, upon learning this, stated, "Some two of them must have brought an identical number of mushrooms!" For what smallest $n$ will Petya certain... | 27 | 0.375 | 7,140.5625 | 5,471.833333 | 8,141.8 | |
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$. | 53 |
We begin with the sequence \(\{a_n\}\) of integers defined such that \(a_1 = 1\), \(a_2 = 2\), and for \(n \geq 3\), \(a_{n+1}\) is the largest prime divisor of the sum \(S_n = a_1 + a_2 + \ldots + a_n\). We are tasked with finding the value of \(a_{100}\).
### Step-by-Step Process
1. **Calculate Initial Sums and Pr... | 0.0625 | 8,192 | 8,192 | 8,192 |
Each of $a_1,$ $a_2,$ $\dots,$ $a_{100}$ is equal to $1$ or $-1.$ Find the minimum positive value of
\[\sum_{1 \le i < j \le 100} a_i a_j.\] | 22 | 0.9375 | 6,308.0625 | 6,182.466667 | 8,192 | |
If the line $l_1: x + ay + 6 = 0$ is parallel to the line $l_2: (a-2)x + 3y + 2a = 0$, calculate the distance between lines $l_1$ and $l_2$. | \frac{8\sqrt{2}}{3} | 0 | 5,598.75 | -1 | 5,598.75 | |
Compute \[\frac{(N-1)!(N)}{(N+1)!}.\] | \frac{1}{N+1} | 1 | 1,456.3125 | 1,456.3125 | -1 | |
There are 5 blue chips and 3 yellow chips in a bag. One chip is drawn from the bag. That chip is placed back into the bag. A second chip is then drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction. | \frac{15}{32} | 1 | 1,677.625 | 1,677.625 | -1 | |
Determine the degree of the polynomial resulting from $(5x^6 - 4x^5 + x^2 - 18)(2x^{12} + 6x^9 - 11x^6 + 10) - (x^3 + 4)^6$ when this expression is expanded and simplified. | 18 | 0.875 | 2,742.1875 | 2,232.285714 | 6,311.5 | |
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 65 islanders gathered for a meeting. Each of them made the following statement in turn: "Among the statements made earlier, the true ones are exactly 20 less than the false ones." How many knights were present at this meeting? | 23 | 0 | 8,127.3125 | -1 | 8,127.3125 | |
$ABCD$ is a rectangle whose area is 12 square units. How many square units are contained in the area of trapezoid $EFBA$?
[asy]
size(4cm,4cm);
for(int i=0; i < 4; ++i){
for(int k=0; k < 5; ++k){
draw((0,i)--(4,i));
draw((k,0)--(k,3));
} }
draw((0,0)--(1,3));
draw((3,3)--(4,0));
label("$A$",(0,0),SW);
label("$B$",(... | 9 | 0.6875 | 5,529.375 | 4,319.090909 | 8,192 | |
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? | 3 | 1. **Calculate the percentage of students scoring 100 points**:
Given that $10\%$ scored $70$ points, $35\%$ scored $80$ points, and $30\%$ scored $90$ points, the percentage of students scoring $100$ points is:
\[
100\% - (10\% + 35\% + 30\%) = 100\% - 75\% = 25\%
\]
2. **Determine the median score**:
... | 0.875 | 4,078.6875 | 3,917.642857 | 5,206 |
Compute the number of ways to color 3 cells in a $3 \times 3$ grid so that no two colored cells share an edge. | 22 | If the middle square is colored, then two of the four corner squares must be colored, and there are $\binom{4}{2}=6$ ways to do this. If the middle square is not colored, then after coloring one of the 8 other squares, there are always 6 ways to place the other two squares. However, the number of possibilities is overc... | 0 | 8,192 | -1 | 8,192 |
Let $z=a+bi$ be the complex number with $\vert z \vert = 5$ and $b > 0$ such that the distance between $(1+2i)z^3$ and $z^5$ is maximized, and let $z^4 = c+di$. Find $c+d$. | 125 | Let's consider the maximization constraint first: we want to maximize the value of $|z^5 - (1+2i)z^3|$ Simplifying, we have
$|z^3| * |z^2 - (1+2i)|$
$=|z|^3 * |z^2 - (1+2i)|$
$=125|z^2 - (1+2i)|$
Thus we only need to maximize the value of $|z^2 - (1+2i)|$.
To maximize this value, we must have that $z^2$ is in the opp... | 0.3125 | 7,673.9375 | 6,534.2 | 8,192 |
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000... | 100 | 1. **Define Variables:**
Let $a$ be the number of sets of twins, $b$ be the number of sets of triplets, and $c$ be the number of sets of quadruplets.
2. **Set Up Equations:**
From the problem, we have the following relationships:
- There are four times as many sets of triplets as sets of quadruplets: $b = 4c$... | 1 | 2,377.75 | 2,377.75 | -1 |
On a 10-ring target, the probabilities of hitting scores 10, 9, 8, 7, and 6 are $\frac{1}{5}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8},$ and $\frac{1}{10}$ respectively. The probability of hitting any other score (from 5 to 1) is $\frac{1}{12}$. $A$ pays $B$ the score amount in forints for any hit that is at least 6, and ... | 96 | 0 | 8,192 | -1 | 8,192 | |
A class has 54 students, and there are 4 tickets for the Shanghai World Expo to be distributed among the students using a systematic sampling method. If it is known that students with numbers 3, 29, and 42 have already been selected, then the student number of the fourth selected student is ▲. | 16 | 0.125 | 7,151.8125 | 4,764.5 | 7,492.857143 | |
The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | 18 | 0.9375 | 2,724.3125 | 2,359.8 | 8,192 | |
The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$ . | -24 | 0.125 | 7,971.0625 | 7,111 | 8,093.928571 | |
The $10\times15$ rectangle $EFGH$ is cut into two congruent pentagons, which are repositioned to form a square. Determine the length $z$ of one side of the pentagons that aligns with one side of the square.
A) $5\sqrt{2}$
B) $5\sqrt{3}$
C) $10\sqrt{2}$
D) $10\sqrt{3}$ | 5\sqrt{3} | 0 | 7,213.25 | -1 | 7,213.25 | |
Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$. | 1003 |
To solve the problem of finding the smallest integer \( n \) such that each subset of \(\{1, 2, \ldots, 2004\}\) with \( n \) elements has two distinct elements \( a \) and \( b \) for which \( a^2 - b^2 \) is a multiple of \( 2004 \), we start by analyzing the structure of the number \( 2004 \).
Firstly, factorize \... | 0.0625 | 5,731.5 | 4,794 | 5,794 |
Given a set of $n$ positive integers in which the difference between any two elements is either divisible by 5 or divisible by 25, find the maximum value of $n$. | 25 | 0.25 | 8,146.75 | 8,011 | 8,192 | |
Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$ , respectively, and meet perpendicularly at $T$ . $Q$ is on $AT$ , $S$ is on $BT$ , and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$ . Determine the radius of the circle. | 29 | 0.5625 | 6,250.25 | 4,740 | 8,192 | |
Given an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity $e = \dfrac{\sqrt{2}}{2}$, the length of the minor axis is $2$.
$(1)$ Find the equation of the ellipse;
$(2)$ Point $A$ is a moving point on the ellipse (not ... | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Mackenzie bought 142 feet of fencing with which to enclose her rectangular garden. If the numbers of feet in the lengths of the garden's sides are natural numbers, what is the maximum number of square feet that can be enclosed by the fencing? | 1260 | 1 | 2,009.3125 | 2,009.3125 | -1 | |
In trapezoid \(ABCD\), \(\overrightarrow{AB} = 2 \overrightarrow{DC}\), \(|\overrightarrow{BC}| = 6\). Point \(P\) is a point in the plane of trapezoid \(ABCD\) and satisfies \(\overrightarrow{AP} + \overrightarrow{BP} + 4 \overrightarrow{DP} = 0\). Additionally, \(\overrightarrow{DA} \cdot \overrightarrow{CB} = |\over... | \frac{4 \sqrt{2}}{3} | 0 | 7,825.1875 | -1 | 7,825.1875 | |
From $A$ to $B$ it is 999 km. Along the road, there are kilometer markers with distances written to $A$ and to $B$:
$0|999,1|998, \ldots, 999|0$.
How many of these markers have only two different digits? | 40 | 0 | 8,069.125 | -1 | 8,069.125 | |
An isosceles right triangle $A B C$ has area 1. Points $D, E, F$ are chosen on $B C, C A, A B$ respectively such that $D E F$ is also an isosceles right triangle. Find the smallest possible area of $D E F$. | \frac{1}{5} | Without loss of generality, suppose that $A B$ is the hypotenuse. If $F$ is the right angle, then $F$ must be the midpoint of $A B$. To prove this, let $X$ and $Y$ be the feet from $F$ to $B C$ and $A C$. Since $\angle X F Y=\angle D F E=90^{\circ}$, we have $\angle X F D=\angle Y F E$ so $$X F=D F \cos \angle X F D=E ... | 0 | 8,192 | -1 | 8,192 |
A bus traveling a 100 km route is equipped with a computer that forecasts the remaining time to arrival at the final destination. This time is calculated based on the assumption that the average speed of the bus on the remaining part of the route will be the same as it was on the part already traveled. Forty minutes af... | 85 | 0 | 8,192 | -1 | 8,192 | |
Let $\mathbf{a} = \begin{pmatrix} 3 \\ p \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 1 \\ q \end{pmatrix}$ be vectors that are equal in magnitude, and orthogonal. Enter the ordered pair $(p,q).$ | \left( -\frac{31}{12}, \frac{41}{12} \right) | 1 | 2,414.8125 | 2,414.8125 | -1 | |
The function $f(x)$ satisfies $f(2+x)=f(2-x)$ for all real numbers $x$. If the equation $f(x)=0$ has exactly four distinct real roots, then the sum of these roots is | 8 | 1. **Identify Symmetry in Function**: Given the function $f(x)$ satisfies $f(2+x) = f(2-x)$ for all real numbers $x$. This implies that $f(x)$ is symmetric about $x = 2$.
2. **Roots of the Function**: We know that $f(x) = 0$ has exactly four distinct real roots. Let's denote these roots as $r_1, r_2, r_3,$ and $r_4$.
... | 1 | 2,155.8125 | 2,155.8125 | -1 |
Solve for $x$: $0.05x - 0.09(25 - x) = 5.4$. | 54.6428571 | 0 | 6,406.875 | -1 | 6,406.875 | |
Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$ 4abc = (a + 3) (b + 3) (c + 3). $$ Find the numerical value of $a + b + c$ . | 16 | 0.1875 | 7,927.75 | 6,782.666667 | 8,192 | |
A diagonal from a vertex of a polygon forms a triangle with the two adjacent sides, so the number of triangles formed by the polygon is at least the number of sides of the polygon. However, the number of triangles cannot exceed the number of ways to choose 2 sides from the polygon, which is given by the combination for... | 2023 | 0 | 7,845.75 | -1 | 7,845.75 | |
A parking lot in Flower Town is a square with $7 \times 7$ cells, each of which can accommodate a car. The parking lot is enclosed by a fence, and one of the corner cells has an open side (this is the gate). Cars move along paths that are one cell wide. Neznaika was asked to park as many cars as possible in such a way ... | 28 | 0 | 7,611.6875 | -1 | 7,611.6875 | |
During a math competition organized in a certain city, the scores of all participating students approximately follow a normal distribution $N(60, 100)$. It is known that there are 13 students who scored 90 or above.
(1) Calculate the total number of students who participated in the competition.
(2) If it is planned to ... | 80 | 0.6875 | 5,158.4375 | 3,779.545455 | 8,192 | |
Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $9$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ | 913 | A positive integer $n$ has exactly two 1s in its binary representation exactly when $n = 2^j + 2^k$ for $j \neq k$ nonnegative integers. Thus, the set $S$ is equal to the set $\{n \in \mathbb{Z} \mid n = 2^j + 2^k \,\mathrm{ and }\, 0 \leq j < k \leq 39\}$. (The second condition ensures simultaneously that $j \neq k$ a... | 0.1875 | 6,659.9375 | 4,774.666667 | 7,095 |
What is the $y$-intercept of the line $y = x + 4$ after it is translated down 6 units? | -2 | The line with equation $y = x + 4$ has a $y$-intercept of 4. When the line is translated 6 units downwards, all points on the line are translated 6 units down. This moves the $y$-intercept from 4 to $4 - 6 = -2$. | 1 | 1,343.5625 | 1,343.5625 | -1 |
Let $p$, $q$, $r$, and $s$ be the roots of $x^4 - 24x^3 + 50x^2 - 26x + 7 = 0$. Compute \[(p+q)^2 + (q+r)^2 + (r+s)^2 + (s+p)^2 + (p+r)^2 + (q+s)^2.\] | 1052 | 0.0625 | 5,321.6875 | 2,671 | 5,498.4 | |
Given that the number of rabbits in a farm increases such that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$, and the populations in the years $2001$, $2002$, and $2004$ were $50$, $80$, and $170$, respectively, determine the population in $20... | 120 | 0 | 8,192 | -1 | 8,192 | |
Knowing that the system
\[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\]
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution. | 83 |
To solve for \( x^5 + y^5 + z^5 \) given the system of equations:
1. \( x + y + z = 3 \)
2. \( x^3 + y^3 + z^3 = 15 \)
3. \( x^4 + y^4 + z^4 = 35 \)
and the condition:
\[ x^2 + y^2 + z^2 < 10, \]
we will utilize symmetric polynomials and Newton's identities.
### Step 1: Establish Variables and Polynomials
Let:
-... | 0.5625 | 5,883 | 4,454.111111 | 7,720.142857 |
How many pairs of values \( p, q \in \mathbf{N} \), not exceeding 100, exist for which the equation
\[ x^{5} + p x + q = 0 \]
has solutions in rational numbers? | 133 | 0.1875 | 7,774.0625 | 5,963 | 8,192 | |
The equilateral triangle has sides of \(2x\) and \(x+15\) as shown. Find the perimeter of the triangle in terms of \(x\). | 90 | 0.5625 | 2,755 | 1,355.777778 | 4,554 | |
You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$ , the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$ .) At any ste... | 695 | 0.125 | 8,138 | 7,760 | 8,192 | |
The probability of snow for each of the next three days is $\frac{2}{3}$. What is the probability that it will snow at least once during those three days? Express your answer as a common fraction. | \dfrac{26}{27} | 1 | 1,794.8125 | 1,794.8125 | -1 | |
There are two lathes processing parts of the same model. The yield rate of the first lathe is $15\%$, and the yield rate of the second lathe is $10\%$. Assuming that the yield rates of the two lathes do not affect each other, the probability of both lathes producing excellent parts simultaneously is ______; if the proc... | 13\% | 0.0625 | 1,532.375 | 1,322 | 1,546.4 | |
How many ways can we put 4 math books and 6 English books on a shelf if all the math books must stay together, but the English books must be split into two groups of 3 each, with each group staying together? | 5184 | 0.1875 | 7,877.875 | 7,118.666667 | 8,053.076923 | |
If the system of equations
\begin{align*}
6x-4y&=a,\\
6y-9x &=b.
\end{align*}has a solution $(x, y)$ where $x$ and $y$ are both nonzero, find $\frac{a}{b},$ assuming $b$ is nonzero. | -\frac{2}{3} | 1 | 2,587.625 | 2,587.625 | -1 | |
Find the number of positive integer divisors of 12 ! that leave a remainder of 1 when divided by 3. | 66 | First we factor $12!=2^{10} 3^{5} 5^{2} 7^{1} 11^{1}$, and note that $2,5,11 \equiv-1(\bmod 3)$ while $7 \equiv 1$ $(\bmod 3)$. The desired divisors are precisely $2^{a} 5^{b} 7^{c} 11^{d}$ with $0 \leq a \leq 10,0 \leq b \leq 2,0 \leq c \leq 1,0 \leq d \leq 1$, and $a+b+d$ even. But then for any choice of $a, b$, exac... | 0.0625 | 7,572.25 | 4,754 | 7,760.133333 |
A student, Liam, wants to earn a total of 30 homework points. For earning the first four homework points, he has to do one homework assignment each; for the next four points, he has to do two homework assignments each; and so on, such that for every subsequent set of four points, the number of assignments he needs to d... | 128 | 0 | 7,701.625 | -1 | 7,701.625 | |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $D$ and $E$ are the upper and right vertices of the ellipse $C$, and $S_{\triangle DEF_2} = \frac{\sqrt{3}}{2}$, eccentricity $e = \frac{1}{2}$.
(1) Find the standard equation of ellipse $C$;... | \frac{3}{2} | 0.3125 | 8,135.875 | 8,012.4 | 8,192 | |
Suppose two distinct competitors of the HMMT 2021 November contest are chosen uniformly at random. Let $p$ be the probability that they can be labelled $A$ and $B$ so that $A$ 's score on the General round is strictly greater than $B$ 's, and $B$ 's score on the theme round is strictly greater than $A$ 's. Estimate $P=... | 2443 | If competitors' scores on the General and Theme rounds were completely uncorrelated, we would expect the answer to be approximately $\frac{1}{2}$. If they were maximally correlated, we would expect the answer to be exactly 0 . It turns out that guessing $\frac{1}{4} \rightarrow 2500$ achieves almost full points $-17 / ... | 0 | 7,589.125 | -1 | 7,589.125 |
A person rolls a die twice, obtaining the numbers $m$ and $n$, which are used as the coefficients of a quadratic equation $x^2 + mx + n = 0$. The probability that the equation has real roots is ______. | \dfrac{19}{36} | 0.875 | 5,096.5 | 4,654.285714 | 8,192 | |
Simplify $\frac{{1+\cos{20}°}}{{2\sin{20}°}}-\sin{10°}\left(\frac{1}{{\tan{5°}}}-\tan{5°}\right)=\_\_\_\_\_\_$. | \frac{\sqrt{3}}{2} | 0 | 7,761.4375 | -1 | 7,761.4375 | |
In a class that includes Petya and Vanya, there are 31 students. In how many ways can a football team be selected from the class? | 2 \binom{29}{10} + \binom{29}{9} | 0 | 2,087.5625 | -1 | 2,087.5625 | |
How many integers (positive and negative) are divisors of 20? | 12 | 1 | 2,079.0625 | 2,079.0625 | -1 | |
Since December 2022, various regions in the country have been issuing multiple rounds of consumption vouchers in different forms to boost consumption recovery. Let the amount of issued consumption vouchers be denoted as $x$ (in hundreds of million yuan) and the consumption driven be denoted as $y$ (in hundreds of milli... | 35.25 | 0.4375 | 4,772.9375 | 4,287.571429 | 5,150.444444 | |
Each integer 1 through 9 is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer? | 0 | 1. **Understanding the Problem**: Each integer from $1$ to $9$ is placed on a slip of paper and put into a hat. Jack and Jill each draw a slip (with replacement), and we want to find the most likely units digit of the sum of the two integers they draw.
2. **Setting Up the Problem**: Let $J$ be the integer Jack draws a... | 0.25 | 8,106.5 | 7,850 | 8,192 |
What is the largest integer that must divide the product of any $5$ consecutive integers? | 24 | 0 | 6,596.0625 | -1 | 6,596.0625 | |
For any natural values \( m \) and \( n \) (with \( n > 1 \)), a function is defined as \( f\left(\frac{m}{n}\right)=\frac{\sqrt[n]{3^{m}}}{\sqrt[n]{3^{m}}+3} \). Compute the sum
\[ f\left(\frac{1}{2020}\right)+f\left(\frac{2}{2020}\right)+f\left(\frac{3}{2020}\right)+\cdots+f\left(\frac{4039}{2020}\right)+f\left(\fra... | 2020 | 0 | 8,192 | -1 | 8,192 | |
Maria needs to build a circular fence around a garden. Based on city regulations, the garden's diameter needs to be close to 30 meters, with an allowable error of up to $10\%$. After building, the fence turned out to have a diameter of 33 meters. Calculate the area she thought she was enclosing and the actual area encl... | 21\% | 0.875 | 2,290.875 | 2,220.642857 | 2,782.5 | |
Given the ellipse $\dfrac{{x}^{2}}{16}+ \dfrac{{y}^{2}}{9}=1$, with left and right foci $F_1$ and $F_2$ respectively, and a point $P$ on the ellipse, if $P$, $F_1$, and $F_2$ are the three vertices of a right triangle, calculate the distance from point $P$ to the $x$-axis. | \dfrac{9}{4} | 0.75 | 5,982.25 | 5,245.666667 | 8,192 | |
Let $p(x)$ be defined on $2 \le x \le 10$ such that $$p(x) = \begin{cases} x + 1 &\quad \lfloor x \rfloor\text{ is prime} \\ p(y) + (x + 1 - \lfloor x \rfloor) &\quad \text{otherwise} \end{cases}$$ where $y$ is the greatest prime factor of $\lfloor x\rfloor.$ Express the range of $p$ in interval notation. | [3,7] \cup [8,9) | 0.0625 | 7,877.5625 | 6,233 | 7,987.2 | |
In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C$ are given by: $\begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.
(1) If the horizontal ... | \sqrt{3} | 0.8125 | 4,104.0625 | 3,895.384615 | 5,008.333333 | |
A circle \( K \) goes through the vertices \( A \) and \( C \) of triangle \( ABC \). The center of circle \( K \) lies on the circumcircle of triangle \( ABC \). Circle \( K \) intersects side \( AB \) at point \( M \). Find the angle \( BAC \) if \( AM : AB = 2 : 7 \) and \( \angle B = \arcsin \frac{4}{5} \). | 45 | 0 | 8,192 | -1 | 8,192 | |
The nine delegates to the Economic Cooperation Conference include $2$ officials from Mexico, $3$ officials from Canada, and $4$ officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of... | 139 | Like in the solution above, there are $84$ ways to pick $3$ delegates. We can use casework to find the probability that there aren't exactly $2$ sleepers from a county, then subtract from $1$.
If no country has at least $2$ delegates sleeping, then every country must have $1$ delegate sleeping. There are $2*3*4=24$ way... | 0.9375 | 3,582.8125 | 3,275.533333 | 8,192 |
Point \((x,y)\) is randomly picked from the rectangular region with vertices at \((0,0), (3014,0), (3014,3015)\), and \((0,3015)\). What is the probability that \(x > 8y\)? Express your answer as a common fraction. | \frac{7535}{120600} | 0 | 8,165.5625 | -1 | 8,165.5625 | |
Compute the determinant of the matrix:
\[
\begin{vmatrix} 2 & 4 & -2 \\ 0 & 3 & -1 \\ 5 & -1 & 2 \end{vmatrix}.
\] | 20 | 0.5 | 6,332.5625 | 4,473.125 | 8,192 | |
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i =... | 506 |
To solve the given problem, we first need to understand the requirements for a \(\pm 1\)-sequence. We are looking for the largest integer \( C \) such that, for any sequence of numbers \( a_1, a_2, \ldots, a_{2022} \) where each \( a_i \) is either \( +1 \) or \( -1 \), there exists a subsequence satisfying certain co... | 0 | 8,192 | -1 | 8,192 |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $a_1 = -3$, $a_{k+1} = \frac{3}{2}$, and $S_k = -12$, then calculate the value of $k$. | 13 | 1 | 2,723.3125 | 2,723.3125 | -1 | |
A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? | 9 | 1. **Identify the formula for the sum of the first $N$ natural numbers**: The sum of the first $N$ natural numbers is given by the formula:
\[
S = 1 + 2 + 3 + \cdots + N = \frac{N(N+1)}{2}
\]
This formula can be derived by pairing terms from the beginning and end of the sequence, each pair summing to $N+1$,... | 1 | 1,708.75 | 1,708.75 | -1 |
Given the function $f(x) = e^x \cos x - x$.
(I) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$;
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$. | -\frac{\pi}{2} | 1 | 4,020 | 4,020 | -1 | |
How many ordered four-tuples of integers $(a,b,c,d)\,$ with $0 < a < b < c < d < 500\,$ satisfy $a + d = b + c\,$ and $bc - ad = 93\,$? | 870 | Let $b = a + m$ and $c = a + m + n$. From $a + d = b + c$, $d = b + c - a = a + 2m + n$.
Substituting $b = a + m$, $c = a + m + n$, and $d = b + c - a = a + 2m + n$ into $bc - ad = 93$, \[bc - ad = (a + m)(a + m + n) - a(a + 2m + n) = m(m + n). = 93 = 3(31)\] Hence, $(m,n) = (1,92)$ or $(3,28)$.
For $(m,n) = (1,92)$,... | 0.75 | 6,485.375 | 6,184.5 | 7,388 |
Find $\tan G$ in the right triangle shown below.
[asy]
pair H,F,G;
H = (0,0);
G = (15,0);
F = (0,8);
draw(F--G--H--F);
draw(rightanglemark(F,H,G,20));
label("$H$",H,SW);
label("$G$",G,SE);
label("$F$",F,N);
label("$17$",(F+G)/2,NE);
label("$15$",G/2,S);
[/asy] | \frac{8}{15} | 1 | 2,170.6875 | 2,170.6875 | -1 | |
In a tournament with 2017 participating teams, each round consists of three randomly chosen teams competing, with exactly one team surviving from each round. If only two teams remain, a one-on-one battle determines the winner. How many battles must take place to declare a champion? | 1008 | 0.5625 | 5,664.5 | 4,020.444444 | 7,778.285714 | |
Suppose $f(z)$ and $g(z)$ are polynomials in $z$, and the degree of $g(z)$ is less than the degree of $f(z)$. If the degree of $f(z)$ is two, what is the degree of $f(z)+g(z)$? | 2 | 1 | 2,155.4375 | 2,155.4375 | -1 | |
The local theater has one ticket window. In how many ways can six people line up to buy a ticket? | 720 | 1 | 1,334.625 | 1,334.625 | -1 | |
Given a sequence of 15 zeros and ones, determine the number of sequences where all the zeros are consecutive. | 121 | 0 | 8,192 | -1 | 8,192 | |
Let $x,$ $y,$ $z$ be real numbers such that $-1 < x,$ $y,$ $z < 1.$ Find the minimum value of
\[\frac{1}{(1 - x)(1 - y)(1 - z)} + \frac{1}{(1 + x)(1 + y)(1 + z)}.\] | 2 | 0.5 | 7,542.875 | 6,893.75 | 8,192 | |
In a right-angled triangle, the sum of the squares of the three side lengths is 1800. What is the length of the hypotenuse of this triangle? | 30 | 1 | 1,395.3125 | 1,395.3125 | -1 | |
For two real values of $n$, the equation $4x^2+nx+25=0$ has exactly one solution in $x$. What is the positive value of $n$? | 20 | 1 | 1,414 | 1,414 | -1 | |
Given the function $f(x)=\sin (\omega x-\varphi)$ $(\omega > 0,|\varphi| < \frac {\pi}{2})$ whose graph intersects with the x-axis at points that are a distance of $\frac {\pi}{2}$ apart, and it passes through the point $(0,- \frac {1}{2})$
$(1)$ Find the analytical expression of the function $f(x)$;
$(2)$ Let the ... | \frac {1}{2} | 1 | 4,598.125 | 4,598.125 | -1 | |
If the binomial coefficient of only the sixth term in the expansion of $(\sqrt{x} - \frac{2}{x^{2}})^{n}$ is the largest, then the constant term in the expansion is _______. | 180 | 0.875 | 4,296.3125 | 3,739.785714 | 8,192 | |
Given that the sum of the binomial coefficients in the expansion of $(5x- \frac{1}{\sqrt{x}})^n$ is 64, determine the constant term in its expansion. | 375 | 0.125 | 2,908.5625 | 2,453.5 | 2,973.571429 | |
Given that the function $f(x)= \frac{1}{2}(m-2)x^{2}+(n-8)x+1$ is monotonically decreasing in the interval $\left[ \frac{1}{2},2\right]$ where $m\geqslant 0$ and $n\geqslant 0$, determine the maximum value of $mn$. | 18 | 0.625 | 6,839.125 | 6,433.9 | 7,514.5 | |
A grocer stacks oranges in a pyramid-like stack whose rectangular base is $5$ oranges by $8$ oranges. Each orange above the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack? | 100 | 1. **Calculate the number of oranges in each layer**:
- The base layer (1st layer) is a rectangle of $5$ oranges by $8$ oranges. Thus, the number of oranges in the 1st layer is:
\[
5 \times 8 = 40
\]
- Each subsequent layer reduces in size by one orange in each dimension because each orange rests i... | 0.8125 | 4,942.3125 | 4,477.230769 | 6,957.666667 |
Given: $2x^2 - 4xy + 4y^2 + 6x + 9 = 0$, then $x + y =$ ? | -\frac{9}{2} | 1 | 4,392.8125 | 4,392.8125 | -1 | |
The third chick received as much porridge as the first two chicks combined. The fourth chick received as much porridge as the second and third chicks combined. The fifth chick received as much porridge as the third and fourth chicks combined. The sixth chick received as much porridge as the fourth and fifth chicks comb... | 40 | 0.8125 | 4,267.375 | 3,640.692308 | 6,983 | |
Calculate the sum of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}$ with an accuracy of $10^{-3}$. | 0.973 | 0.0625 | 7,524.375 | 5,058 | 7,688.8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.