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 |
|---|---|---|---|---|---|---|
A set \( \mathcal{T} \) of distinct positive integers has the property that for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 3003 is the largest element of \( \ma... | 30 | 0 | 8,192 | -1 | 8,192 | |
A factory received a task to process 6000 pieces of part P and 2000 pieces of part Q. The factory has 214 workers. Each worker spends the same amount of time processing 5 pieces of part P as they do processing 3 pieces of part Q. The workers are divided into two groups to work simultaneously on different parts. In orde... | 137 | 0 | 8,192 | -1 | 8,192 | |
Joy has $30$ thin rods, one each of every integer length from $1 \text{ cm}$ through $30 \text{ cm}$. She places the rods with lengths $3 \text{ cm}$, $7 \text{ cm}$, and $15 \text{cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How ma... | 17 | 1. **Identify the range for the fourth rod**: To form a quadrilateral, the sum of the lengths of any three sides must be greater than the length of the fourth side. This is known as the triangle inequality theorem. We apply this to the three rods of lengths $3 \text{ cm}$, $7 \text{ cm}$, and $15 \text{ cm}$.
2. **Cal... | 0.5625 | 4,832.25 | 4,503.888889 | 5,254.428571 |
A department store offers two promotions. Promotion A says, "Buy one pair of shoes, get the second pair for half the price." Promotion B says, "Buy one pair of shoes, get $\$10$ off the second pair." Jane wants to buy two pairs of shoes that cost $\$30$ each. She can only use one of the promotions, A or B. Jane decides... | 5 | 1 | 1,408.375 | 1,408.375 | -1 | |
While hiking in a valley, a hiker first walks 15 miles north, then 8 miles east, then 9 miles south, and finally 2 miles east. How far is the hiker from the starting point after completing these movements? | 2\sqrt{34} | 0.3125 | 620.9375 | 587.2 | 636.272727 | |
A man buys a house for $10,000 and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays $325 a year taxes and realizes $5\frac{1}{2}\%$ on his investment. The monthly rent (in dollars) is: | 83.33 | 1. **Understanding the problem**: A man buys a house for $10,000 and aims to realize a $5\frac{1}{2}\%$ return on his investment. This means he wants to earn $5.5\%$ of $10,000$ annually:
\[
0.055 \times 10,000 = 550 \text{ dollars per year}
\]
2. **Accounting for taxes**: He also pays $325$ in taxes each yea... | 0.5625 | 5,368.25 | 4,987 | 5,858.428571 |
Martin is playing a game. His goal is to place tokens on an 8 by 8 chessboard in such a way that there is at most one token per square, and each column and each row contains at most 4 tokens.
a) How many tokens can Martin place, at most?
b) If, in addition to the previous constraints, each of the two main diagonals c... | 32 | 0.1875 | 8,045.375 | 7,410 | 8,192 | |
What is the number of degrees in the smaller angle formed by the hour and minute hands of a clock at 8:15? Express your answer as a decimal to the nearest tenth.
[asy]
size(200);
draw(Circle((0,0),5),linewidth(1.2));
pair[] mins;
for(int i = 0; i < 60; ++i){
mins[i] = 4.5*dir(-6*i + 90);
dot(mins[i]);
}
for(int i = 1... | 157.5 | 1 | 2,672 | 2,672 | -1 | |
Find the minimum value of
\[ x^3 + 9x + \frac{81}{x^4} \]
for \( x > 0 \). | 21 | 0 | 8,192 | -1 | 8,192 | |
The base of the pyramid is a right triangle with a hypotenuse equal to 6 and an acute angle of $15^{\circ}$. All lateral edges are inclined to the plane of the base at an angle of $45^{\circ}$. Find the volume of the pyramid. | 4.5 | 0 | 6,569.4375 | -1 | 6,569.4375 | |
Let triangle $ABC$ be equilateral with each side measuring 18 inches. Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC$. If $OQ$ is 6 inches, then $AQ$, in inches, is:
A) $6$ inches
B) $12$ inches
C) $15$ inches
D) $18$ inches
E) $21$ inches | 18 | 0 | 7,460.9375 | -1 | 7,460.9375 | |
The integers $1,2,4,5,6,9,10,11,13$ are to be placed in the circles and squares below with one number in each shape. Each integer must be used exactly once and the integer in each circle must be equal to the sum of the integers in the two neighbouring squares. If the integer $x$ is placed in the leftmost square and the... | 20 | From the given information, if $a$ and $b$ are in two consecutive squares, then $a+b$ goes in the circle between them. Since all of the numbers that we can use are positive, then $a+b$ is larger than both $a$ and $b$. This means that the largest integer in the list, which is 13, cannot be either $x$ or $y$ (and in fact... | 0 | 8,192 | -1 | 8,192 |
Given a four-digit number $\overline{ABCD}$ such that $\overline{ABCD} + \overline{AB} \times \overline{CD}$ is a multiple of 1111, what is the minimum value of $\overline{ABCD}$? | 1729 | 0 | 8,192 | -1 | 8,192 | |
In the arithmetic sequence $\{a\_n\}$, it is given that $a\_3 + a\_4 = 12$ and $S\_7 = 49$.
(I) Find the general term formula for the sequence $\{a\_n\}$.
(II) Let $[x]$ denote the greatest integer not exceeding $x$, for example, $[0.9] = 0$ and $[2.6] = 2$. Define a new sequence $\{b\_n\}$ where $b\_n = [\log_{10} a\_... | 5445 | 0.75 | 5,470.0625 | 5,101.583333 | 6,575.5 | |
There are three kinds of saltwater solutions: A, B, and C, with concentrations of 5%, 8%, and 9% respectively, and their weights are 60 grams, 60 grams, and 47 grams. Now, we want to prepare 100 grams of 7% saltwater solution. What is the maximum and minimum amount of solution A that can be used? | 35 | 0.8125 | 4,065.375 | 3,911.615385 | 4,731.666667 | |
Given a biased coin with probabilities of $\frac{3}{4}$ for heads and $\frac{1}{4}$ for tails, determine the difference between the probability of winning Game A, which involves 4 coin tosses and at least three heads, and the probability of winning Game B, which involves 5 coin tosses with the first two tosses and the ... | \frac{89}{256} | 0.375 | 5,861.875 | 4,440 | 6,715 | |
A hemisphere is placed on a sphere of radius \(100 \text{ cm}\). The second hemisphere is placed on the first one, and the third hemisphere is placed on the second one (as shown below). Find the maximum height of the tower (in cm). | 300 | 0 | 7,690.3125 | -1 | 7,690.3125 | |
Let $U$ be a positive integer whose only digits are 0s and 1s. If $Y = U \div 18$ and $Y$ is an integer, what is the smallest possible value of $Y$? | 61728395 | 0.125 | 7,895 | 5,816 | 8,192 | |
A semicircle has diameter $XY$ . A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$ , $U$ on the semicircle, and $V$ on $XY$ . What is the area of $STUV$ ? | 36 | 0.25 | 7,681.5 | 6,150 | 8,192 | |
Find the number of positive divisors of 2002. | 16 | 1 | 1,587.0625 | 1,587.0625 | -1 | |
Let $O$ be the origin. $y = c$ intersects the curve $y = 2x - 3x^3$ at $P$ and $Q$ in the first quadrant and cuts the y-axis at $R$ . Find $c$ so that the region $OPR$ bounded by the y-axis, the line $y = c$ and the curve has the same area as the region between $P$ and $Q$ under the curve and above... | 4/9 | 0.0625 | 8,192 | 8,192 | 8,192 | |
How many 7-digit positive integers are made up of the digits 0 and 1 only, and are divisible by 6? | 11 | 0.4375 | 6,279.6875 | 5,034.428571 | 7,248.222222 | |
Find the number of subsets $S$ of $\{1,2, \ldots 6\}$ satisfying the following conditions: - $S$ is non-empty. - No subset of $S$ has the property that the sum of its elements is 10. | 34 | We do casework based on the largest element of $S$. Call a set $n$-free if none of its subsets have elements summing to $n$. Case 1: The largest element of $S$ is 6. Then $4 \notin S$. If $5 \notin S$, then we wish to find all 4-free subsets of $\{1,2,3\}$ (note that $1+2+3=6<10$). We just cannot include both 1,3, so w... | 0 | 8,182 | -1 | 8,182 |
In $\triangle ABC$ , medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$ , $PE=1.5$ , $PD=2$ , and $DE=2.5$ . What is the area of $AEDC?$
[asy]
unitsize(75);
pathpen = black; pointpen=black;
pair A = MP("A", D((0,0)), dir(200));
pair B = MP("B", D((2,0)), dir(-20));
pair C = MP("C", D((1/2,1)), dir... | 13.5 | 0 | 7,996.875 | -1 | 7,996.875 | |
If $\mathbf{A}^{-1} = \begin{pmatrix} -4 & 1 \\ 0 & 2 \end{pmatrix},$ then find the inverse of $\mathbf{A}^2.$ | \begin{pmatrix}16 & -2 \\ 0 & 4 \end{pmatrix} | 0.75 | 4,656.3125 | 3,477.75 | 8,192 | |
How many integers from 1 to 16500
a) are not divisible by 5;
b) are not divisible by either 5 or 3;
c) are not divisible by either 5, 3, or 11? | 8000 | 1 | 3,473.875 | 3,473.875 | -1 | |
Let $p, q, r, s, t, u$ be positive real numbers such that $p+q+r+s+t+u = 11$. Find the minimum value of
\[
\frac{1}{p} + \frac{9}{q} + \frac{25}{r} + \frac{49}{s} + \frac{81}{t} + \frac{121}{u}.
\] | \frac{1296}{11} | 0.75 | 5,508.625 | 4,614.166667 | 8,192 | |
Alexio has 100 cards numbered 1-100, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3 or 5? Express your answer as a common fraction. | \dfrac{37}{50} | 0.875 | 4,855.75 | 4,379.142857 | 8,192 | |
Three points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle's center? | \frac{3}{16} | 0.1875 | 7,476.625 | 7,577.666667 | 7,453.307692 | |
For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. What is the sum of the 10 least values of $n$ such that $S_n$ is divisible by $3$? | 197 | 1. **Define the sets and express $S_n$:**
Let $A_{n, <} = \{(j, k) : 1 \leq j < k \leq n\}$ and $A_{n, >} = \{(j, k) : 1 \leq k < j \leq n\}$. Note that $A_{n, <}$ and $A_{n, >}$ are symmetric, and each product $jk$ where $j < k$ has a corresponding product $kj$ where $k < j$. Therefore, the sum of products over $A_... | 0.125 | 7,802.6875 | 6,680.5 | 7,963 |
Given that $α$ is an angle in the third quadrant and $\cos(α+π)=\frac{4}{5}$, find the value of $\tan 2α$. | \frac{24}{7} | 0.75 | 5,055 | 4,009.333333 | 8,192 | |
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they d... | $25\%$ | 1. **Assume Total Number of Students**: Let's assume there are 100 students at Typico High School for simplicity in calculation.
2. **Students Who Like Dancing**:
- Given that 60% of the students like dancing.
- Therefore, $60\% \times 100 = 60$ students like dancing.
3. **Students Who Dislike Dancing**:
- ... | 0 | 2,676.875 | -1 | 2,676.875 |
Triangle $A B C$ obeys $A B=2 A C$ and $\angle B A C=120^{\circ}$. Points $P$ and $Q$ lie on segment $B C$ such that $$\begin{aligned} A B^{2}+B C \cdot C P & =B C^{2} \\ 3 A C^{2}+2 B C \cdot C Q & =B C^{2} \end{aligned}$$ Find $\angle P A Q$ in degrees. | 40^{\circ} | We have $A B^{2}=B C(B C-C P)=B C \cdot B P$, so triangle $A B C$ is similar to triangle $P B A$. Also, $A B^{2}=B C(B C-2 C Q)+A C^{2}=(B C-C Q)^{2}-C Q^{2}+A C^{2}$, which rewrites as $A B^{2}+C Q^{2}=$ $B Q^{2}+A C^{2}$. We deduce that $Q$ is the foot of the altitude from $A$. Thus, $\angle P A Q=90^{\circ}-\angle Q... | 0 | 5,757.9375 | -1 | 5,757.9375 |
Draw a tangent line MN to the circle $(x-2)^2+(y-2)^2=1$ at point N, where N is the point of tangency. If $|MN|=|MO|$ (where O is the origin), then the minimum value of $|MN|$ is \_\_\_\_\_\_. | \frac{7\sqrt{2}}{8} | 0 | 7,820.5 | -1 | 7,820.5 | |
The function $f$ is defined on positive integers as follows:
\[f(n) = \left\{
\begin{array}{cl}
n + 15 & \text{if $n < 15$}, \\
f(n - 7) & \text{if $n \ge 15$}.
\end{array}
\right.\]
Find the maximum value of the function. | 29 | 0.875 | 5,290.4375 | 4,875.928571 | 8,192 | |
What is the largest value of $n$ less than 100,000 for which the expression $10(n-3)^5 - n^2 + 20n - 30$ is a multiple of 7? | 99999 | 0 | 5,832.9375 | -1 | 5,832.9375 | |
Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying
the real and imaginary part of $z$ are both integers;
$|z|=\sqrt{p},$ and
there exists a triangle whose three side lengths are $p,$ the real part of $z^{3},$ and the imaginary part of $z^{3}.$ | 349 | Assume that $z=a+bi$. Then, \[z^3=(a^3-3ab^2)+(3a^2b-b^3)i\]Note that by the Triangle Inequality, \[|(a^3-3ab^2)-(3a^2b-b^3)|<p\implies |a^3+b^3-3ab^2-3a^2b|<a^2+b^2\]Thus, we know \[|a+b||a^2+b^2-4ab|<a^2+b^2\]Without loss of generality, assume $a>b$ (as otherwise, consider $i^3\overline z=b+ai$). If $|a/b|\geq 4$, th... | 0 | 7,891.0625 | -1 | 7,891.0625 |
The sides of triangle $PQR$ are in the ratio $3:4:5$. Segment $QS$ is the angle bisector drawn to the longest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the shorter subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express your answer as a common fraction. | \frac{45}{7} | 0.875 | 3,446.25 | 3,516.285714 | 2,956 | |
The ship decided to determine the depth of the ocean at its location. The signal sent by the echo sounder was received on the ship 8 seconds later. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean. | 6000 | 0 | 1,341.3125 | -1 | 1,341.3125 | |
Given that $\cos\left(\frac {\pi}{4} + \theta\right) = -\frac {3}{5}$, and $\frac {11\pi}{12} < \theta < \frac {5\pi}{4}$, find the value of $\frac {\sin{2\theta} + 2\sin^{2}{\theta}}{1 - \tan{\theta}}$. | \frac {28}{75} | 0.6875 | 5,593 | 4,411.636364 | 8,192 | |
Given sets $A=\{-1,1,2\}$ and $B=\{-2,1,2\}$, a number $k$ is randomly selected from set $A$ and a number $b$ is randomly selected from set $B$. The probability that the line $y=kx+b$ does not pass through the third quadrant is $\_\_\_\_\_\_$. | P = \frac{2}{9} | 0.1875 | 7,227.375 | 5,686.666667 | 7,582.923077 | |
There are three islands A, B, and C at sea. It is measured that the distance between islands A and B is 10n miles, $\angle BAC=60^\circ$, and $\angle ABC=75^\circ$. The distance between islands B and C is \_\_\_\_\_\_ n miles. | 5\sqrt{6} | 0 | 5,133.875 | -1 | 5,133.875 | |
Given that the complex number $z$ satisfies the equation $\frac{1-z}{1+z}={i}^{2018}+{i}^{2019}$ (where $i$ is the imaginary unit), find the value of $|2+z|$. | \frac{5\sqrt{2}}{2} | 0 | 3,541.625 | -1 | 3,541.625 | |
If $\sqrt[3]{3}$ is approximated as $1.442$, calculate the value of $\sqrt[3]{3}-3\sqrt[3]{3}-98\sqrt[3]{3}$. | -144.2 | 0.875 | 2,432 | 2,161.5 | 4,325.5 | |
The figure drawn is not to scale. Which of the five segments shown is the longest? [asy]
pair A = (-3,0), B=(0,2), C=(3,0), D=(0,-1);
draw(D(MP("A", A, W))--D(MP("B", B, N))--D(MP("C", C, E))--D(MP("D", D, S))--A);
draw(B--D);
MP("55^\circ", (0,-0.75), NW);
MP("55^\circ", (0,-0.75), NE);
MP("40^\circ", (0,1.5), SW);
MP... | CD | 0 | 8,047.0625 | -1 | 8,047.0625 | |
What is the measure of an orthogonal trihedral angle? What is the sum of the measures of polyhedral angles that share a common vertex, have no common interior points, and together cover the entire space? | 4\pi | 0.875 | 3,610.125 | 3,754.142857 | 2,602 | |
An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 (\angle D$) and $\angle BAC = t \pi$ in radians, then find $t$.
[asy]
import graph;
unitsize(2 cm);
pair O, A, B, C, D;
O = (0,0);
A = dir(90);
B... | \frac{5}{11} | 0.4375 | 5,920.4375 | 4,093.571429 | 7,341.333333 | |
How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 8? | 14 | 0 | 6,337 | -1 | 6,337 | |
Consider \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that
\[ A + A^{\times} = I_{2020} \]
\[ A \cdot A^{\times} = I_{2020} \]
where \(A^{\times}\) is the adjugate matrix of \(A\), i.e., the matrix whose elements are \(a_{ij} = (-1)^{i+j} d_{ji}\), where \(d_{ji}\) is the determinant obtained from \(A\), eliminating t... | 673 | 0 | 6,258.1875 | -1 | 6,258.1875 | |
Two bullets are placed in two consecutive chambers of a 6-chamber pistol. The cylinder is then spun. The pistol is fired but the first shot is a blank. Let \( p \) denote the probability that the second shot is also a blank if the cylinder is spun after the first shot and let \( q \) denote the probability that the sec... | 89 | 0.0625 | 7,828.75 | 3,819 | 8,096.066667 | |
3 red marbles, 4 blue marbles, and 5 green marbles are distributed to 12 students. Each student gets one and only one marble. In how many ways can the marbles be distributed so that Jamy and Jaren get the same color and Jason gets a green marble? | 3150 | 0.0625 | 7,785.3125 | 5,094 | 7,964.733333 | |
Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$ . For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer? | 22 | 0.25 | 7,822.5625 | 6,714.25 | 8,192 | |
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John'... | 1001 | 0 | 8,192 | -1 | 8,192 | |
Charlie is making a necklace with yellow beads and green beads. She has already used 4 green beads and 0 yellow beads. How many yellow beads will she have to add so that $ rac{4}{5}$ of the total number of beads are yellow? | 16 | If $ rac{4}{5}$ of the beads are yellow, then $ rac{1}{5}$ are green. Since there are 4 green beads, the total number of beads must be $4 imes 5=20$. Thus, Charlie needs to add $20-4=16$ yellow beads. | 1 | 1,452.75 | 1,452.75 | -1 |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | \sqrt{244} | 0 | 3,395.875 | -1 | 3,395.875 | |
Determine the value of the following expression, simplified as a fraction: $$1+\cfrac{3}{2+\cfrac{5}{6}}$$ | \frac{35}{17} | 1 | 1,572.25 | 1,572.25 | -1 | |
Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there? | 112 | 0.375 | 6,988.375 | 5,119.166667 | 8,109.9 | |
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$ | 2 | We start by defining the function $R(n)$ as the sum of the remainders when $n$ is divided by each integer from $2$ to $10$. We are interested in finding two-digit integers $n$ such that $R(n) = R(n+1)$.
#### Step 1: Define the Change in Remainder Function
Let $\Delta(n, k) = \text{Rem}(n+1, k) - \text{Rem}(n, k)$, whe... | 0 | 8,109.9375 | -1 | 8,109.9375 |
Given complex numbers $u$ and $v$ such that $|u+v| = 3$ and $|u^2 + v^2| = 10,$ determine the largest possible value of $|u^3 + v^3|.$ | 31.5 | 0 | 8,091.875 | -1 | 8,091.875 | |
Determine all triplets of real numbers $(x, y, z)$ satisfying the system of equations $x^{2} y+y^{2} z =1040$, $x^{2} z+z^{2} y =260$, $(x-y)(y-z)(z-x) =-540$. | (16,4,1),(1,16,4) | Call the three equations $(1),(2),(3) \cdot(1) /(2)$ gives $y=4 z .(3)+(1)-(2)$ gives $\left(y^{2}-z^{2}\right) x=15 z^{2} x=240$ so $z^{2} x=16$. Therefore $z(x+2 z)^{2}=x^{2} z+z^{2} y+4 z^{2} x=\frac{81}{5}$, $z(x-2 z)^{2}=x^{2} z+z^{2} y-4 z^{2} x=\frac{49}{5}$ so $\left|\frac{x+2 z}{x-2 z}\right|=\frac{9}{7}$. Thu... | 0 | 7,356.25 | -1 | 7,356.25 |
Given the parabola $y=x^2$ and the moving line $y=(2t-1)x-c$ have common points $(x_1, y_1)$, $(x_2, y_2)$, and $x_1^2+x_2^2=t^2+2t-3$.
(1) Find the range of the real number $t$;
(2) When does $t$ take the minimum value of $c$, and what is the minimum value of $c$? | \frac{11-6\sqrt{2}}{4} | 0 | 6,015.375 | -1 | 6,015.375 | |
Let $n$ be a positive integer. Let there be $P_{n}$ ways for Pretty Penny to make exactly $n$ dollars out of quarters, dimes, nickels, and pennies. Also, let there be $B_{n}$ ways for Beautiful Bill to make exactly $n$ dollars out of one dollar bills, quarters, dimes, and nickels. As $n$ goes to infinity, the sequence ... | 20 | Let $d_{x}$ be the number ways to make exactly $x$ cents using only dimes and nickels. It is easy to see that when $x$ is a multiple of 5 , $$d_{x}=\left\lfloor\frac{x}{10}\right\rfloor+1$$ Now, let $c_{x}$ be the number of ways to make exactly $x$ cents using only quarters, dimes and nickels. Again, it is easy to see ... | 0 | 8,192 | -1 | 8,192 |
Given an arithmetic sequence $\{a_n\}$, the sum of its first $n$ terms is $S_n$. It is known that $S_8 \leq 6$ and $S_{11} \geq 27$. Determine the minimum value of $S_{19}$. | 133 | 0.75 | 6,464.9375 | 6,117.916667 | 7,506 | |
A trapezoid is given where the ratio of the lengths of its bases is 2. If the trapezoid is rotated $360^\circ$ around the longer base, the resulting solid is $\Phi_1$. If the trapezoid is rotated $360^\circ$ around the shorter base, the resulting solid is $\Phi_2$. Find the ratio of the volumes of the solids $\Phi_1$ a... | 5/4 | 0.0625 | 7,709.6875 | 6,911 | 7,762.933333 | |
Let $t_n = \frac{n(n+1)}{2}$ be the $n$th triangular number. Find
\[\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2002}}\] | \frac {4004}{2003} | The problem statement and the solution provided seem unrelated. The problem statement asks for the sum of the reciprocals of the first 2002 triangular numbers, while the solution discusses logarithms and bases, which is irrelevant to the problem. Let's solve the original problem regarding triangular numbers.
1. **Unde... | 1 | 3,573.3125 | 3,573.3125 | -1 |
A natural number undergoes the following operation: the rightmost digit of its decimal representation is discarded, and then the number obtained after discarding is added to twice the discarded digit. For example, $157 \mapsto 15 + 2 \times 7 = 29$, $5 \mapsto 0 + 2 \times 5 = 10$. A natural number is called ‘good’ if ... | 19 | 0.5625 | 6,706.1875 | 5,550.555556 | 8,192 | |
Given \(0 \le x_0 < 1\), let
\[x_n = \left\{ \begin{array}{ll}
3x_{n-1} & \text{if } 3x_{n-1} < 1 \\
3x_{n-1} - 1 & \text{if } 1 \le 3x_{n-1} < 2 \\
3x_{n-1} - 2 & \text{if } 3x_{n-1} \ge 2
\end{array}\right.\]
for all integers \(n > 0\), determine the number of values of \(x_0\) for which \(x_0 = x_6\). | 729 | 0.0625 | 8,183.0625 | 8,049 | 8,192 | |
The U.S. produces about 5.5 million tons of apples each year. Of the total, $20\%$ is mixed with other products, with $50\%$ of the remainder used for apple juice and the other $50\%$ sold fresh. How many million tons of apples are used for apple juice? Express your answer as a decimal to the nearest tenth. | 2.2 | 1 | 1,228.3125 | 1,228.3125 | -1 | |
In different historical periods, the conversion between "jin" and "liang" was different. The idiom "ban jin ba liang" comes from the 16-based system. For convenience, we assume that in ancient times, 16 liang equaled 1 jin, with each jin being equivalent to 600 grams in today's terms. Currently, 10 liang equals 1 jin, ... | 2800 | 0.0625 | 5,474.4375 | 2,341 | 5,683.333333 | |
Given the user satisfaction ratings: 1 person with 10 points, 1 person with 9 points, 2 people with 8 points, 4 people with 7 points, 1 person with 5 points, and 1 person with 4 points, calculate the average, mode, median, and 85% percentile of the user satisfaction rating for this product. | 8.5 | 0.0625 | 1,065.8125 | 3,334 | 914.6 | |
Let $ABC$ be an isosceles triangle with $AC=BC$ , let $M$ be the midpoint of its side $AC$ , and let $Z$ be the line through $C$ perpendicular to $AB$ . The circle through the points $B$ , $C$ , and $M$ intersects the line $Z$ at the points $C$ and $Q$ . Find the radius of the circumcircle of the triangle $ABC$ in term... | \[ R = \frac{2}{3}m \] | Let length of side $CB = x$ and length of $QM = a$ . We shall first prove that $QM = QB$ .
Let $O$ be the circumcenter of $\triangle ACB$ which must lie on line $Z$ as $Z$ is a perpendicular bisector of isosceles $\triangle ACB$ .
So, we have $\angle ACO = \angle BCO = \angle C/2$ .
Now $MQBC$ is a cyclic quadrilateral... | 0 | 5,897.1875 | -1 | 5,897.1875 |
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
[asy]
import three; size(2inch);
currentprojection=orthographic(4,2,2);
draw((0,0,0)--(0,0,3),dashed);
draw((0,0,0)--(0,4,0),dashed);
draw((0,0,0)--(5,0,0),dashed);
draw((5,4,3... | 10 | 0.125 | 8,126.875 | 7,671 | 8,192 | |
The ratio of the land area to the ocean area on the Earth's surface is 29:71. If three-quarters of the land is in the northern hemisphere, then calculate the ratio of the ocean area in the southern hemisphere to the ocean area in the northern hemisphere. | 171:113 | 0 | 6,520 | -1 | 6,520 | |
Given the points $(2, 3)$, $(10, 9)$, and $(6, m)$, where $m$ is an integer, determine the sum of all possible values of $m$ for which the area of the triangle formed by these points is a maximum. | 12 | 0.0625 | 7,824.8125 | 8,192 | 7,800.333333 | |
Joe has written 5 questions of different difficulties for a test with problems numbered 1 though 5. He wants to make sure that problem $i$ is harder than problem $j$ whenever $i-j \geq 3$. In how many ways can he order the problems for his test? | 25 | We will write $p_{i}>p_{j}$ for integers $i, j$ when the $i$ th problem is harder than the $j$ th problem. For the problem conditions to be true, we must have $p_{4}>p_{1}, p_{5}>p_{2}$, and $p_{5}>p_{1}$. Then, out of $5!=120$ total orderings, we see that in half of them satisfy $p_{4}>p_{1}$ and half satisfy $p_{5}>p... | 0 | 7,778.625 | -1 | 7,778.625 |
Let $F$ be the focus of the parabola $C: y^2=4x$, point $A$ lies on $C$, and point $B(3,0)$. If $|AF|=|BF|$, then calculate the distance of point $A$ from point $B$. | 2\sqrt{2} | 0.4375 | 4,006 | 3,587.142857 | 4,331.777778 | |
In the Chinese length measurement units, 1 meter = 3 chi, 1 zhang = 10 chi, and 1 kilometer = 2 li. How many zhang are in 1 li? | 150 | 0.625 | 546.875 | 549.1 | 543.166667 | |
A corporation plans to expand its sustainability team to include specialists in three areas: energy efficiency, waste management, and water conservation. The company needs 95 employees to specialize in energy efficiency, 80 in waste management, and 110 in water conservation. It is known that 30 employees will specializ... | 210 | 0.9375 | 2,894.5625 | 2,541.4 | 8,192 | |
There are 5 students on a team for a math competition. The math competition has 5 subject tests. Each student on the team must choose 2 distinct tests, and each test must be taken by exactly two people. In how many ways can this be done? | 2040 | We can model the situation as a bipartite graph on 10 vertices, with 5 nodes representing the students and the other 5 representing the tests. We now simply want to count the number of bipartite graphs on these two sets such that there are two edges incident on each vertex. Notice that in such a graph, we can start at ... | 0 | 7,883.75 | -1 | 7,883.75 |
A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? | \frac{5}{54} | To solve this problem, we need to minimize the white surface area on the larger $3$-inch cube constructed from $27$ smaller $1$-inch cubes. We have $21$ red cubes and $6$ white cubes.
1. **Understanding the structure of the cube:**
- The larger cube has dimensions $3 \times 3 \times 3$ inches, so it is composed of ... | 0.25 | 6,483.625 | 6,638.75 | 6,431.916667 |
In square \( A B C D \), \( P \) and \( Q \) are points on sides \( C D \) and \( B C \), respectively, such that \( \angle A P Q = 90^\circ \). If \( A P = 4 \) and \( P Q = 3 \), find the area of \( A B C D \). | \frac{256}{17} | 0.0625 | 8,139.4375 | 7,351 | 8,192 | |
Fold a 10m long rope in half 5 times, then cut it in the middle with scissors. How many segments is the rope cut into? | 33 | 0.125 | 470.25 | 589.5 | 453.214286 | |
Last year, Australian Suzy Walsham won the annual women's race up the 1576 steps of the Empire State Building in New York for a record fifth time. Her winning time was 11 minutes 57 seconds. Approximately how many steps did she climb per minute? | 130 | 0 | 392.375 | -1 | 392.375 | |
The function $f$ satisfies \[
f(x) + f(2x+y) + 5xy = f(3x - y) + 2x^2 + 1
\]for all real numbers $x,y$. Determine the value of $f(10)$. | -49 | 0.9375 | 3,774.75 | 3,480.266667 | 8,192 | |
Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)-... | 85 | 0.9375 | 2,167.1875 | 2,200.266667 | 1,671 | |
A square piece of paper has sides of length $120$. From each corner, a wedge is cut such that each of the two cuts for the wedge starts at a distance $10$ from the corner, and they meet on the diagonal at an angle of $45^{\circ}$. After the cuts, the paper is folded up along the lines joining the vertices of adjacent c... | 5\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
A teacher finds that when she offers candy to her class of 30 students, the mean number of pieces taken by each student is 5. If every student takes some candy, what is the greatest number of pieces one student could have taken? | 121 | 1 | 1,720.0625 | 1,720.0625 | -1 | |
In a checkered square with a side length of 2018, some cells are painted white and the rest are black. It is known that from this square, one can cut out a 10x10 square where all the cells are white, and a 10x10 square where all the cells are black. What is the smallest value for which it is guaranteed that one can cut... | 10 | 0.1875 | 8,019.375 | 7,568 | 8,123.538462 | |
The United States Postal Service charges an extra $\$0.11$ in postage if the length of an envelope, in inches, divided by its height, in inches, is less than $1.3$ or greater than $2.5.$ For how many of these four envelopes must the extra $\$0.11$ in postage be paid? \begin{tabular}[t]{ccc}
Envelope & Length in inches ... | 3 | 0.375 | 2,971.8125 | 1,851 | 3,644.3 | |
What is the sum of all the positive divisors of 91? | 112 | 1 | 1,565.4375 | 1,565.4375 | -1 | |
A square with an integer side length was cut into 2020 smaller squares. It is known that the areas of 2019 of these squares are 1, while the area of the 2020th square is not equal to 1. Find all possible values that the area of the 2020th square can take. Provide the smallest of these possible area values in the answer... | 112225 | 0.5625 | 6,280.8125 | 5,619.666667 | 7,130.857143 | |
Given that \( 169(157 - 77x)^2 + 100(201 - 100x)^2 = 26(77x - 157)(1000x - 2010) \), find the value of \( x \). | 31 | 0.1875 | 7,489.375 | 4,444.666667 | 8,192 | |
In $\triangle ABC$, it is known that the internal angle $A= \frac{\pi}{3}$, side $BC=2\sqrt{3}$. Let internal angle $B=x$, and the area be $y$.
(1) If $x=\frac{\pi}{4}$, find the length of side $AC$;
(2) Find the maximum value of $y$. | 3\sqrt{3} | 0.75 | 5,934.25 | 5,551.083333 | 7,083.75 | |
What is the greatest possible sum of the digits in the base-nine representation of a positive integer less than $3000$? | 24 | 0 | 8,192 | -1 | 8,192 | |
Through the midpoint $D$ of the base of an isosceles triangle, a line is drawn at an angle of $30^{\circ}$ to this base, on which the angle $\angle ACB$ intercepts a segment $EF$. It is known that $ED = 6$ and $FD = 4$. Find the height of the triangle drawn to the base. | 12 | 0.125 | 5,845.875 | 5,683 | 5,869.142857 | |
The number $6545$ can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers? | 162 | To solve the problem, we first need to find the prime factorization of the number $6545$.
1. **Prime Factorization of $6545$:**
\[
6545 = 5 \times 1309
\]
We need to check if $1309$ is prime or can be further factored. Testing divisibility by small primes:
\[
1309 \div 7 = 187 \quad (\text{since } 13... | 1 | 2,339.75 | 2,339.75 | -1 |
In the regular triangular pyramid S-ABC, M and N are the midpoints of edge SC and BC, respectively, and MN is perpendicular to AM. If the lateral edge SA equals $2\sqrt{3}$, then calculate the volume of the circumscribed sphere of the regular triangular pyramid S-ABC. | 36\pi | 0.5625 | 6,601.375 | 5,364.222222 | 8,192 | |
Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$ . | 45578899 | 0 | 8,187.125 | -1 | 8,187.125 | |
A natural number is written on the board. If its last digit (in the units place) is erased, the remaining non-zero number is divisible by 20. If the first digit is erased, the remaining number is divisible by 21. What is the smallest number that could be on the board if its second digit is not 0? | 1609 | 0 | 8,192 | -1 | 8,192 | |
A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads. | \frac{1}{2} | 0.1875 | 7,816.125 | 6,187.333333 | 8,192 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.