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 |
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A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 15. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RS$, respectively. Determine the perimeter of triangle $MNO$. | 32.5 | 0 | 6,789.8125 | -1 | 6,789.8125 | |
The operation $\nabla$ is defined by $a \nabla b=4 a+b$. What is the value of $(5 \nabla 2) \nabla 2$? | 90 | Using the definition, $(5 \nabla 2) \nabla 2=(4 \times 5+2) \nabla 2=22 \nabla 2=4 \times 22+2=90$. | 1 | 388.5 | 388.5 | -1 |
Find the value of $f(2017)$ for a function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(x) \cdot f(x+2) = 13$ given that $f(3) = 4$. | \frac{13}{4} | 0.9375 | 4,243.6875 | 3,980.466667 | 8,192 | |
Compute $\arccos(\sin 3)$, where all functions are in radians. | 3 - \frac{\pi}{2} | 0.8125 | 5,363.8125 | 4,748.692308 | 8,029.333333 | |
For how many pairs of consecutive integers in $\{1000,1001,1002,\ldots,2000\}$ is no carrying required when the two integers are added? | 156 | Consider what carrying means: If carrying is needed to add two numbers with digits $abcd$ and $efgh$, then $h+d\ge 10$ or $c+g\ge 10$ or $b+f\ge 10$. 6. Consider $c \in \{0, 1, 2, 3, 4\}$. $1abc + 1ab(c+1)$ has no carry if $a, b \in \{0, 1, 2, 3, 4\}$. This gives $5^3=125$ possible solutions.
With $c \in \{5, 6, 7, 8\... | 0 | 8,192 | -1 | 8,192 |
Given an isosceles triangle \(ABC\) with \(AB = AC\) and \(\angle ABC = 53^\circ\), find the measure of \(\angle BAM\). Point \(K\) is such that \(C\) is the midpoint of segment \(AK\). Point \(M\) is chosen such that:
- \(B\) and \(M\) are on the same side of line \(AC\);
- \(KM = AB\);
- \(\angle MAK\) is the maximu... | 44 | 0 | 8,192 | -1 | 8,192 | |
Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$. | 40 | 0 | 8,192 | -1 | 8,192 | |
Let $P$ be a point not on line $XZ$ and $Q$ a point on line $XZ$ such that $PQ \perp XZ.$ Meanwhile, $R$ is a point on line $PZ$ such that $SR \perp PZ.$ If $SR = 5,$ $PQ = 6,$ and $XZ = 7,$ then what is the length of $PZ?$ | 8.4 | 0 | 8,192 | -1 | 8,192 | |
(1901 + 1902 + 1903 + \cdots + 1993) - (101 + 102 + 103 + \cdots + 193) = | 167400 | 1. **Identify the pattern in the sequences**:
The problem involves two sums of arithmetic sequences. The first sequence starts at 1901 and ends at 1993, and the second sequence starts at 101 and ends at 193.
2. **Recognize the relationship between the sequences**:
Each term in the first sequence can be expresse... | 0.0625 | 2,307.0625 | 815 | 2,406.533333 |
Vertex $E$ of equilateral $\triangle{ABE}$ is in the interior of unit square $ABCD$. Let $R$ be the region consisting of all points inside $ABCD$ and outside $\triangle{ABE}$ whose distance from $AD$ is between $\frac{1}{3}$ and $\frac{2}{3}$. What is the area of $R$? | \frac{3-\sqrt{3}}{9} | 1. **Identify the region $R$:** The region $R$ consists of all points inside the unit square $ABCD$ but outside the equilateral triangle $\triangle{ABE}$, and whose distance from side $AD$ is between $\frac{1}{3}$ and $\frac{2}{3}$.
2. **Calculate the area of the middle third of the square:** The middle third of the s... | 0 | 7,948.3125 | -1 | 7,948.3125 |
In the trapezoid \(ABCD\), the bases are given as \(AD = 4\) and \(BC = 1\), and the angles at \(A\) and \(D\) are \(\arctan 2\) and \(\arctan 3\) respectively.
Find the radius of the circle inscribed in triangle \(CBE\), where \(E\) is the intersection point of the diagonals of the trapezoid. | \frac{18}{25 + 2 \sqrt{130} + \sqrt{445}} | 0 | 8,192 | -1 | 8,192 | |
The positive integers $A, B, C$, and $D$ form an arithmetic and geometric sequence as follows: $A, B, C$ form an arithmetic sequence, while $B, C, D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3}$, what is the smallest possible value of $A + B + C + D$? | 76 | 0 | 8,192 | -1 | 8,192 | |
Find all numbers in the range of
\[f(x) = \arctan x + \arctan \frac{1 - x}{1 + x},\]expressed in radians. Enter all the numbers, separated by commas. | -\frac{3 \pi}{4}, \frac{\pi}{4} | 0 | 7,741.125 | -1 | 7,741.125 | |
Riquinho distributed $R \$ 1000.00$ among his friends: Antônio, Bernardo, and Carlos in the following manner: he successively gave 1 real to Antônio, 2 reais to Bernardo, 3 reais to Carlos, 4 reais to Antônio, 5 reais to Bernardo, and so on. How much did Bernardo receive? | 345 | 0.0625 | 7,113.8125 | 5,561 | 7,217.333333 | |
Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \[
\{-8, -6, -4, -1, 3, 5, 7, 10\}.
\] What is the minimum possible value of \[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\] | 18 | 0.8125 | 6,181.875 | 5,718 | 8,192 | |
Let $x$, $y$, and $z$ be real numbers greater than $1$, and let $z$ be the geometric mean of $x$ and $y$. The minimum value of $\frac{\log z}{4\log x} + \frac{\log z}{\log y}$ is \_\_\_\_\_\_. | \frac{9}{8} | 0.8125 | 4,817.75 | 4,423.538462 | 6,526 | |
Consider a sphere inscribed in a right cone with the base radius of 10 cm and height of 40 cm. The radius of the inscribed sphere can be expressed as $b\sqrt{d} - b$ cm. Determine the value of $b+d$. | 19.5 | 0 | 8,028.0625 | -1 | 8,028.0625 | |
Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle OIA=90^{\circ}$. Given that $AI=97$ and $BC=144$, compute the area of $\triangle ABC$. | 14040 | We present five different solutions and outline a sixth and seventh one. In what follows, let $a=BC$, $b=CA$, $c=AB$ as usual, and denote by $r$ and $R$ the inradius and circumradius. Let $s=\frac{1}{2}(a+b+c)$. In the first five solutions we will only prove that $\angle AIO=90^{\circ} \Longrightarrow b+c=2a$. Let us s... | 0 | 8,192 | -1 | 8,192 |
A point $P$ lies at the center of square $A B C D$. A sequence of points $\left\{P_{n}\right\}$ is determined by $P_{0}=P$, and given point $P_{i}$, point $P_{i+1}$ is obtained by reflecting $P_{i}$ over one of the four lines $A B, B C, C D, D A$, chosen uniformly at random and independently for each $i$. What is the p... | \frac{1225}{16384} | Solution 1. WLOG, $A B$ and $C D$ are horizontal line segments and $B C$ and $D A$ are vertical. Then observe that we can consider the reflections over vertical lines separately from those over horizontal lines, as each reflection over a vertical line moves $P_{i}$ horizontally to point $P_{i+1}$, and vice versa. Now c... | 0 | 8,192 | -1 | 8,192 |
At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle School are taking the co... | 28 | 1. **Identify the number of students in each class:**
- Mrs. Germain's class: 11 students
- Mr. Newton's class: 8 students
- Mrs. Young's class: 9 students
2. **Assumption of no overlap in students:**
Since the problem does not mention any students being in more than one class, we assume that all students ... | 1 | 1,084.5 | 1,084.5 | -1 |
The number of natural numbers from 1 to 1992 that are multiples of 3, but not multiples of 2 or 5, is
(Ninth "Jinyun Cup" Middle School Mathematics Invitational Competition, 1992) | 266 | 0.8125 | 5,549.8125 | 5,005.692308 | 7,907.666667 | |
Determine the number of ways a student can schedule four mathematics courses — algebra, geometry, number theory, and statistics — on an 8-period day, given that no two mathematics courses can be scheduled in consecutive periods. | 120 | 0.3125 | 7,356.6875 | 6,180.4 | 7,891.363636 | |
What numeral is in the 100th decimal place in the decimal representation of $\frac{6}{7}$? | 1 | 0.9375 | 2,439.0625 | 2,055.533333 | 8,192 | |
If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is | 18 | 1. **Identify the Midpoints**: Points E and G are the midpoints of sides AD and CD, respectively, in rectangle ABCD.
2. **Properties of Midpoints in a Rectangle**: Since E and G are midpoints, segment EG is parallel to sides AB and CD, and its length is half the length of AB (or CD). Similarly, since F is the midpoint... | 0.875 | 3,983.5 | 3,382.285714 | 8,192 |
The cost prices of three types of clothing, A, B, and C, are respectively 20 yuan, 30 yuan, and 40 yuan. Their selling prices are 24 yuan, m yuan, and 52 yuan, respectively. After calculation, the total profit from the three types of clothing is the same, and the sum of the sales volumes of clothing A and B is four tim... | 42 | 0.9375 | 2,451.625 | 2,180.6 | 6,517 | |
Explain how any unit fraction $\frac{1}{n}$ can be decomposed into other unit fractions. | \frac{1}{2n}+\frac{1}{3n}+\frac{1}{6n} | $\frac{1}{2n}+\frac{1}{3n}+\frac{1}{6n}$ | 0 | 6,268.8125 | -1 | 6,268.8125 |
Evaluate $|\omega^2+6\omega+58|$ if $\omega=9+2i$. | 195 | 1 | 2,404 | 2,404 | -1 | |
Simplify first, then evaluate: \\((x+2)^{2}-4x(x+1)\\), where \\(x= \sqrt {2}\\). | -2 | 1 | 1,696.0625 | 1,696.0625 | -1 | |
Let \(S\) be the set of all nonzero real numbers. Let \(f : S \to S\) be a function such that
\[f(x) + f(y) = cf(xyf(x + y))\]
for all \(x, y \in S\) such that \(x + y \neq 0\) and for some nonzero constant \(c\). Determine all possible functions \(f\) that satisfy this equation and calculate \(f(5)\). | \frac{1}{5} | 0.0625 | 8,158.875 | 7,662 | 8,192 | |
Let the function y=f(x) have the domain D. If for any x1, x2 ∈ D, when x1+x2=2a, it always holds that f(x1)+f(x2)=2b, then the point (a,b) is called the symmetry center of the graph of the function y=f(x). Investigate a symmetry center of the function f(x)=x+sinπx-3, and find the value of f(1/2016)+f(2/2016)+f(3/2016)+... | -8062 | 0.375 | 7,569.5 | 6,532 | 8,192 | |
Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $32d$ in base $n$ equals $263$, and the value of the numeral $324$ in base $n$ equals the value of the numeral $11d1$ in base six. What is $n + d$? | 11 | 1. **Convert $\underline{32d}$ in base $n$ to decimal:**
The numeral $\underline{32d}$ in base $n$ can be expressed in decimal as:
\[
3n^2 + 2n + d
\]
Given that this equals 263, we have the equation:
\[
3n^2 + 2n + d = 263
\]
2. **Convert $\underline{324}$ in base $n$ to decimal:**
The ... | 1 | 2,608.75 | 2,608.75 | -1 |
In $\triangle ABC$, if $A=120^{\circ}$, $AB=5$, $BC=7$, find the value of $\sin B$. | \frac{3 \sqrt{3}}{14} | 0 | 3,764.6875 | -1 | 3,764.6875 | |
The maximum value of the function $f(x) = 8\sin x - \tan x$, defined on $\left(0, \frac{\pi}{2}\right)$, is $\_\_\_\_\_\_\_\_\_\_\_\_$. | 3\sqrt{3} | 1 | 3,064.4375 | 3,064.4375 | -1 | |
Given the function $f(x)=\sin (\omega x+\varphi)$ with $\omega > 0$ and $|\varphi| < \frac {\pi}{2}$, the function has a minimum period of $4\pi$ and, after being shifted to the right by $\frac {2\pi}{3}$ units, becomes symmetric about the $y$-axis. Determine the value of $\varphi$. | -\frac{\pi}{6} | 0.9375 | 4,559.375 | 4,317.2 | 8,192 | |
What is the value of $a^3 + b^3$ given that $a+b=10$ and $ab=17$? | 490 | 1 | 2,709 | 2,709 | -1 | |
Given the line $l: x+2y+1=0$, and the set $A=\{n|n<6, n\in \mathbb{N}^*\}$, if we randomly select 3 different elements from set $A$ to be $a$, $b$, and $r$ in the circle equation $(x-a)^2+(y-b)^2=r^2$, then the probability that the line connecting the center $(a, b)$ of the circle to the origin is perpendicular to line... | \frac {1}{10} | 0.4375 | 7,304.8125 | 6,164.142857 | 8,192 | |
Four problems were attempted by 100 contestants in a Mathematics competition. The first problem was solved by 90 contestants, the second by 85 contestants, the third by 80 contestants, and the fourth by 75 contestants. What is the smallest possible number of contestants who solved all four problems? | 30 | 0.8125 | 5,752.5 | 5,189.538462 | 8,192 | |
A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse ha... | 67 | 0.75 | 4,981.8125 | 3,911.75 | 8,192 | |
When three positive integers are divided by $24$, the remainders are $10,$ $4,$ and $12,$ respectively.
When the sum of the three integers is divided by $24$, what is the remainder? | 2 | 1 | 1,707.6875 | 1,707.6875 | -1 | |
In the Cartesian coordinate system $xOy$, a moving line $l$: $y=x+m$ intersects the parabola $C$: $x^2=2py$ ($p>0$) at points $A$ and $B$, and $\overrightarrow {OA}\cdot \overrightarrow {OB}=m^{2}-2m$.
1. Find the equation of the parabola $C$.
2. Let $P$ be the point where the line $y=x$ intersects $C$ (and $P$ is dif... | \frac{5}{2} | 0.875 | 5,331.125 | 4,922.428571 | 8,192 | |
Given the function $f(x) = e^{\sin x + \cos x} - \frac{1}{2}\sin 2x$ ($x \in \mathbb{R}$), find the difference between the maximum and minimum values of the function $f(x)$. | e^{\sqrt{2}} - e^{-\sqrt{2}} | 0 | 7,573.9375 | -1 | 7,573.9375 | |
For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,\ldots.$ For how many values of $k$ does $S_k$ contain the term $2005$? | 12 | Suppose that the $n$th term of the sequence $S_k$ is $2005$. Then $1+(n-1)k=2005$ so $k(n-1)=2004=2^2\cdot 3\cdot 167$. The ordered pairs $(k,n-1)$ of positive integers that satisfy the last equation are $(1,2004)$,$(2,1002)$, $(3,668)$, $(4,501)$, $(6,334)$, $(12,167)$, $(167,12)$,$(334,6)$, $(501,4)$, $(668,3)$, $(10... | 1 | 2,629.375 | 2,629.375 | -1 |
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family? | 6 | Let $m$ be the age of the mother.
Let $x$ be the number of children and $y$ be the average age of the children. The total age of the children is then $xy$.
We are given two key pieces of information:
1. The average age of the family (mother, father, and children) is $20$.
2. The average age of the mother and children... | 1 | 1,566 | 1,566 | -1 |
Given α ∈ (0,π), β ∈ (-π/2,π/2) satisfies sin(α + π/3) = 1/3, cos(β - π/6) = √6/6, determine sin(α + 2β). | \frac{2\sqrt{10}-2}{9} | 0 | 7,967.8125 | -1 | 7,967.8125 | |
The minimum value of $\frac{b^{2}+1}{\sqrt{3}a}$ is what occurs when the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is 2. | \frac {4 \sqrt {3}}{3} | 0 | 2,577.3125 | -1 | 2,577.3125 | |
Given that $\frac{{\cos 2\alpha}}{{\sin(\alpha+\frac{\pi}{4})}}=\frac{4}{7}$, find the value of $\sin 2\alpha$. | \frac{41}{49} | 0.5625 | 6,397.75 | 5,054.777778 | 8,124.428571 | |
Solve for $x$: $$2^x+6=3\cdot2^x-26$$ | 4 | 1 | 2,306.75 | 2,306.75 | -1 | |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed.
| 71 | 0 | 8,192 | -1 | 8,192 | |
In the sequence $\{a_{n}\}$, where $a_{1}=1$, $a_{n} \gt 0$, and the sum of the first $n$ terms is $S_{n}$. If ${a_n}=\sqrt{{S_n}}+\sqrt{{S_{n-1}}}$ for $n \in \mathbb{N}^*$ and $n \geqslant 2$, then the sum of the first $15$ terms of the sequence $\{\frac{1}{{{a_n}{a_{n+1}}}}\}$ is ____. | \frac{15}{31} | 1 | 3,546.25 | 3,546.25 | -1 | |
Find the minimum value of
\[2x^2 + 2xy + y^2 - 2x + 2y + 4\]over all real numbers $x$ and $y.$ | -1 | 0.75 | 5,261.125 | 4,284.166667 | 8,192 | |
What is the smallest \( n > 1 \) for which the average of the first \( n \) (non-zero) squares is a square? | 337 | 0.25 | 8,024.625 | 7,522.5 | 8,192 | |
A trapezoid is divided into seven strips of equal width. What fraction of the trapezoid's area is shaded? Explain why your answer is correct. | 4/7 | 0 | 8,192 | -1 | 8,192 | |
Barney Schwinn noted that his bike's odometer showed a reading of $2332$, a palindrome. After riding for $5$ hours one day and $4$ hours the next day, he observed that the odometer displayed another palindrome, $2552$. Calculate Barney's average riding speed during this period. | \frac{220}{9} | 0.875 | 3,351.8125 | 3,350.142857 | 3,363.5 | |
Find $\left|\left(1+i\right)^6\right|$. | 8 | 1 | 2,080.4375 | 2,080.4375 | -1 | |
Chloe is baking muffins for a school event. If she divides the muffins equally among 8 of her friends, she'll have 3 muffins left over. If she divides the muffins equally among 5 of her friends, she'll have 2 muffins left over. Assuming Chloe made fewer than 60 muffins, what is the sum of the possible numbers of muffin... | 118 | 0 | 5,132.9375 | -1 | 5,132.9375 | |
In a flood control emergency, a large oil tank drifting downstream from upstream needs to be exploded by shooting. It is known that there are only $5$ bullets. The first hit can only cause the oil to flow out, and the second hit can cause an explosion. Each shot is independent, and the probability of hitting each time ... | \frac{7}{27} | 0 | 8,046.5625 | -1 | 8,046.5625 | |
For a finite sequence $B=(b_1,b_2,\dots,b_{50})$ of numbers, the Cesaro sum of $B$ is defined to be
\[\frac{T_1 + \cdots + T_{50}}{50},\] where $T_k = b_1 + \cdots + b_k$ and $1 \leq k \leq 50$.
If the Cesaro sum of the 50-term sequence $(b_1,\dots,b_{50})$ is 500, what is the Cesaro sum of the 51-term sequence $(2, b... | 492 | 0 | 7,924.375 | -1 | 7,924.375 | |
Let triangle $ABC$ with incenter $I$ satisfy $AB = 10$ , $BC = 21$ , and $CA = 17$ . Points $D$ and E lie on side $BC$ such that $BD = 4$ , $DE = 6$ , and $EC = 11$ . The circumcircles of triangles $BIE$ and $CID$ meet again at point $P$ , and line $IP$ meets the altitude from $A$ to $BC$ at ... | 85 | 0.25 | 8,021.5625 | 7,510.25 | 8,192 | |
Nyusha has 2022 coins, and Barash has 2023. Nyusha and Barash toss all their coins simultaneously and count how many heads each gets. The one who gets more heads wins, and in case of a tie, Nyusha wins. What is the probability that Nyusha wins? | 0.5 | 0.125 | 7,530.0625 | 6,108.5 | 7,733.142857 | |
Calculate:<br/>$(1)-3+\left(-9\right)+10-\left(-18\right)$;<br/>$(2)12÷2×(-\frac{1}{2})-75÷(-5)$;<br/>$(3)-4^3-2×(-5)^2+6÷(-\frac{1}{3})^2$;<br/>$(4)(-1\frac{1}{4}-1\frac{5}{6}+2\frac{8}{9})÷(-\frac{1}{6})^2$. | -7 | 1 | 2,796.25 | 2,796.25 | -1 | |
The roots of the equation $2x^2-mx+n=0$ sum to 6 and multiply to 10. What is the value of $m+n$? | 32 | 1 | 1,400.875 | 1,400.875 | -1 | |
Now a ball is launched from a vertex of an equilateral triangle with side length 5. It strikes the opposite side after traveling a distance of $\sqrt{19}$. How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.) | 7 | The key idea is that, instead of reflecting the line $AY$ off of $BC$, we will reflect $ABC$ about $BC$ and extend $AY$ beyond $\triangle ABC$. We keep doing this until the extension of $AY$ hits a vertex of one of our reflected triangles. This is illustrated in the diagram below: We can calculate that the line $AY$ ha... | 0 | 8,137.4375 | -1 | 8,137.4375 |
If $f(x) = 3x+2$ and $g(x) = (x-1)^2$, what is $f(g(-2))$? | 29 | 1 | 1,579.8125 | 1,579.8125 | -1 | |
In quadrilateral \(ABCD\), we have \(AB=5\), \(BC=6\), \(CD=5\), \(DA=4\), and \(\angle ABC=90^\circ\). Let \(AC\) and \(BD\) meet at \(E\). Compute \(\frac{BE}{ED}\). | \sqrt{3} | 0.0625 | 8,159.6875 | 7,675 | 8,192 | |
Triangle $A B C$ satisfies $\angle B>\angle C$. Let $M$ be the midpoint of $B C$, and let the perpendicular bisector of $B C$ meet the circumcircle of $\triangle A B C$ at a point $D$ such that points $A, D, C$, and $B$ appear on the circle in that order. Given that $\angle A D M=68^{\circ}$ and $\angle D A C=64^{\circ... | 86^{\circ} | Extend $D M$ to hit the circumcircle at $E$. Then, note that since $A D E B$ is a cyclic quadrilateral, $\angle A B E=180^{\circ}-\angle A D E=180^{\circ}-\angle A D M=180^{\circ}-68^{\circ}=112^{\circ}$. We also have that $\angle M E C=\angle D E C=\angle D A C=64^{\circ}$. But now, since $M$ is the midpoint of $B C$ ... | 0 | 8,046.4375 | -1 | 8,046.4375 |
Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence of real numbers satisfying
\[a_n = a_{n - 1} a_{n + 1}\]for all $n \ge 2.$ If $a_1 = 1 + \sqrt{7}$ and $a_{1776} = 13 + \sqrt{7},$ then determine $a_{2009}.$ | -1 + 2 \sqrt{7} | 0.75 | 5,165.6875 | 4,989.75 | 5,693.5 | |
Observe the following set of equations:
\\(S_{1}=1\\),
\\(S_{2}=2+3=5\\),
\\(S_{3}=4+5+6=15\\),
\\(S_{4}=7+8+9+10=34\\),
\\(S_{5}=11+12+13+14+15=65\\),
\\(\ldots\\)
Based on the equations above, guess that \\(S_{2n-1}=(2n-1)(an^{2}+bn+c)\\), then \\(a\cdot b\cdot c=\\) \_\_\_\_\_\_. | -4 | 0.625 | 5,899.9375 | 4,882.4 | 7,595.833333 | |
A student's score on a 150-point test is directly proportional to the hours she studies. If she scores 90 points after studying for 2 hours, what would her score be if she studied for 5 hours? | 225 | 1 | 431.8125 | 431.8125 | -1 | |
There are three pastures full of grass. The first pasture is 33 acres and can feed 22 cows for 27 days. The second pasture is 28 acres and can feed 17 cows for 42 days. How many cows can the third pasture, which is 10 acres, feed for 3 days (assuming the grass grows at a uniform rate and each acre produces the same amo... | 20 | 0.125 | 7,408.9375 | 5,873 | 7,628.357143 | |
A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 3 meters from its base. When Leah walks 2.5 meters from the base of the pole toward the point where the cable is attached to the ground, her head just touches the cable. Leah is 1.5 meters tall. How many meter... | 9 | 0.875 | 4,629.6875 | 4,120.785714 | 8,192 | |
Given that $\frac{a}{36-a}+\frac{b}{48-b}+\frac{c}{72-c}=9$, evaluate $\frac{4}{36-a}+\frac{6}{48-b}+\frac{9}{72-c}$. | \frac{13}{3} | 0 | 7,882.25 | -1 | 7,882.25 | |
Determine the number of digits in the value of $2^{12} \times 5^8 $. | 10 | 0.9375 | 2,557.4375 | 2,699.533333 | 426 | |
Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2016} \right| = \frac{1}{2016}.\] | \frac{1007}{504} | 0.9375 | 6,454.5 | 6,338.666667 | 8,192 | |
Given the convex pentagon $ABCDE$, where each pair of neighboring vertices must have different colors and vertices at the ends of each diagonal must not share the same color, determine the number of possible colorings using 5 available colors. | 240 | 0 | 7,477.8125 | -1 | 7,477.8125 | |
There exist unique nonnegative integers $A, B$ between 0 and 9, inclusive, such that $(1001 \cdot A+110 \cdot B)^{2}=57,108,249$. Find $10 \cdot A+B$. | 75 | We only need to bound for $A B 00$; in other words, $A B^{2} \leq 5710$ but $(A B+1)^{2} \geq 5710$. A quick check gives $A B=75$. (Lots of ways to get this...) | 1 | 3,806.9375 | 3,806.9375 | -1 |
A line with a slope of $-3$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(10,0)$ and the $y$-axis at $D$. The lines intersect at $E(5,5)$. What is the area of the shaded quadrilateral $OBEC$? | 25 | 0 | 4,893.375 | -1 | 4,893.375 | |
Given the random variable $\eta\sim B(n,p)$, and $E(2\eta)=8$, $D(4\eta)=32$, find the respective values of $n$ and $p$. | 0.5 | 1 | 1,659.875 | 1,659.875 | -1 | |
In a hypothetical scenario, a small country is planning an international event in 2023. Let \( A \), \( B \), and \( C \) be distinct positive integers such that their product \( A \cdot B \cdot C = 2023 \). Determine the largest possible value of the sum \( A + B + C \). | 297 | 0.75 | 4,980.75 | 4,273.5 | 7,102.5 | |
Find the smallest positive integer $N$ with the following property: of the three numbers $N$, $N+1$, and $N+2$, one of them is divisible by $2^2$, one of them is divisible by $3^2$, one is divisible by $5^2$, and one is divisible by $7^2$. | 98 | 0 | 8,192 | -1 | 8,192 | |
The number $101$ is the smallest three-digit palindromic prime. What is the second-smallest one? | 131 | 0.875 | 5,357.6875 | 4,952.785714 | 8,192 | |
The number halfway between $\dfrac{1}{8}$ and $\dfrac{1}{3}$ is
A) $\dfrac{11}{48}$
B) $\dfrac{11}{24}$
C) $\dfrac{5}{24}$
D) $\dfrac{1}{4}$
E) $\dfrac{1}{5}$ | \dfrac{11}{48} | 0 | 2,238.3125 | -1 | 2,238.3125 | |
Given that 7,999,999,999 has at most two prime factors, find its largest prime factor. | 4,002,001 | 7,999,999,999=8 \cdot 10^{9}-1=2000^{3}-1=(2000-1)\left(2000^{2}+2000+1\right)$, so \left(2000^{2}+2000+1\right)=4,002,001$ is its largest prime factor. | 0 | 8,061.0625 | -1 | 8,061.0625 |
In the diagram, $\triangle ABC$ is right-angled at $A,$ with $AB=45$ and $AC=108.$ The point $D$ is on $BC$ so that $AD$ is perpendicular to $BC.$ Determine the length of $AD$ and the ratio of the areas of triangles $ABD$ and $ADC$. | 5:12 | 0.0625 | 5,454.875 | 6,309 | 5,397.933333 | |
Compute $\dbinom{8}{0}$. | 1 | 1 | 1,439.375 | 1,439.375 | -1 | |
A hyperbola is centered at the origin and opens either horizontally or vertically. It passes through the points $(-3, 4),$ $(-2, 0),$ and $(t, 2).$ Find $t^2.$ | \frac{21}{4} | 1 | 2,351.0625 | 2,351.0625 | -1 | |
Points A, B, C, and D lie along a line, in that order. If $AB:AC=1:5$, and $BC:CD=2:1$, what is the ratio $AB:CD$? | 1:2 | Suppose that $AB=x$ for some $x>0$. Since $AB:AC=1:5$, then $AC=5x$. This means that $BC=AC-AB=5x-x=4x$. Since $BC:CD=2:1$ and $BC=4x$, then $CD=2x$. Therefore, $AB:CD=x:2x=1:2$. | 1 | 2,827.6875 | 2,827.6875 | -1 |
How many integers between $100$ and $150$ have three different digits in increasing order? One such integer is $129$. | 18 | 0.75 | 5,869 | 5,338.916667 | 7,459.25 | |
On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process conti... | 60 | 1. **Identify the pattern and constraints**: The students are arranged in different configurations each day, with a unique number of students per row for 12 consecutive days. On the 13th day, no new configuration is possible. This implies that the total number of students must have exactly 12 divisors (one for each day... | 0.4375 | 6,455.0625 | 4,221.857143 | 8,192 |
Let $ABCDEF$ be a regular hexagon. Let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively. The segments $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ bound a smaller regular hexagon. Let the ratio of ... | 11 | Let $BC=2$ (without loss of generality).
Note that $\angle BMH$ is the vertical angle to an angle of regular hexagon, and so has degree $120^\circ$.
Because $\triangle ABH$ and $\triangle BCI$ are rotational images of one another, we get that $\angle{MBH}=\angle{HAB}$ and hence $\triangle ABH \sim \triangle BMH \sim ... | 0.25 | 8,144.25 | 8,001 | 8,192 |
Adam and Sarah start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Sarah travels west at 5 mph. After how many hours are they 85 miles apart? | 7.6 | 0 | 3,741.1875 | -1 | 3,741.1875 | |
Among the five-digit numbers formed using the digits 0, 1, 2, 3, 4, how many have the first and last digits the same, and the three middle digits all different? | 240 | 0 | 6,040.3125 | -1 | 6,040.3125 | |
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$ , for integers $i,j$ with $0\leq i,j\leq n$ , such that:
1. for all $0\leq i,j\leq n$ , the set $S_{i,j}$ has $i+j$ elements; and
2. $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\l... | \[
(2n)! \cdot 2^{n^2}
\] | Note that there are $(2n)!$ ways to choose $S_{1, 0}, S_{2, 0}... S_{n, 0}, S_{n, 1}, S_{n, 2}... S{n, n}$ , because there are $2n$ ways to choose which number $S_{1, 0}$ is, $2n-1$ ways to choose which number to append to make $S_{2, 0}$ , $2n-2$ ways to choose which number to append to make $S_{3, 0}$ ... After that,... | 0 | 8,192 | -1 | 8,192 |
Compute \(104 \times 96\). | 9984 | 0.9375 | 944.625 | 986.066667 | 323 | |
The set $\{[x]+[2x]+[3x] \mid x \in \mathbf{R}\} \bigcap \{1, 2, \cdots, 100\}$ contains how many elements, where $[x]$ represents the greatest integer less than or equal to $x$. | 67 | 0.1875 | 7,736.6875 | 6,131.333333 | 8,107.153846 | |
Compute \[\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\] | 373 | In both the numerator and the denominator, each factor is of the form $N^4+324=N^4+18^2$ for some positive integer $N.$
We factor $N^4+18^2$ by completing the square, then applying the difference of squares: \begin{align*} N^4+18^2&=\left(N^4+36N^2+18^2\right)-36N^2 \\ &=\left(N^2+18\right)^2-(6N)^2 \\ &=\left(N^2-6N+... | 0.1875 | 7,250.9375 | 4,307.333333 | 7,930.230769 |
A regular 15-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$? | 39 | 1. **Determine the Lines of Symmetry, $L$:**
A regular polygon with an odd number of sides, such as a regular 15-gon, has lines of symmetry that pass through each vertex and bisect the opposite side. This is because each line of symmetry aligns a vertex with the midpoint of the side directly opposite, maintaining th... | 0.9375 | 1,514.4375 | 1,069.266667 | 8,192 |
Let $a$ and $b$ be positive integers such that $(2a+b)(2b+a)=4752$ . Find the value of $ab$ .
*Proposed by James Lin* | 520 | 0.4375 | 7,063.375 | 5,612.285714 | 8,192 | |
How many positive integer multiples of \(3003\) can be expressed in the form \(10^j - 10^i\), where \(i\) and \(j\) are integers and \(0 \leq i < j \leq 50\)? | 192 | 0.6875 | 6,824.375 | 6,340.090909 | 7,889.8 | |
Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$. | 8.625 | 0 | 4,758.625 | -1 | 4,758.625 | |
The function $y=\frac{x^3+8x^2+21x+18}{x+2}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$? | 14 | 0.9375 | 2,515.75 | 2,137.333333 | 8,192 |
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