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 |
|---|---|---|---|---|---|---|
There are exactly 120 ways to color five cells in a $5 \times 5$ grid such that exactly one cell in each row and each column is colored.
There are exactly 96 ways to color five cells in a $5 \times 5$ grid without the corner cell, such that exactly one cell in each row and each column is colored.
How many ways are th... | 78 | 0.375 | 6,421.4375 | 4,929.5 | 7,316.6 | |
A very large number $x$ is equal to $2^23^34^45^56^67^78^89^9$. What is the smallest positive integer that, when multiplied with $x$, produces a product that is a perfect square? | 105 | 0.9375 | 2,649.875 | 2,698.533333 | 1,920 | |
Two distinct primes, each greater than 20, are multiplied. What is the least possible product of these two primes? | 667 | 0.9375 | 2,221.6875 | 1,823.666667 | 8,192 | |
Let \( m \) be the product of all positive integer divisors of 360,000. Suppose the prime factors of \( m \) are \( p_{1}, p_{2}, \ldots, p_{k} \), for some positive integer \( k \), and \( m = p_{1}^{e_{1}} p_{2}^{e_{2}} \cdot \ldots \cdot p_{k}^{e_{k}} \), for some positive integers \( e_{1}, e_{2}, \ldots, e_{k} \).... | 630 | 0.875 | 3,691.5 | 3,048.571429 | 8,192 | |
Given that $f(x)= \frac {4}{4^{x}+2}$, $S\_n$ is the sum of the first $n$ terms of the sequence $\{a\_n\}$, and $\{a\_n\}$ satisfies $a\_1=0$, and when $n \geqslant 2$, $a\_n=f( \frac {1}{n})+f( \frac {2}{n})+f( \frac {3}{n})+…+f( \frac {n-1}{n})$, find the maximum value of $\frac {a_{n+1}}{2S\_n+a\_6}$. | \frac {2}{7} | 0.5 | 7,874.875 | 7,557.75 | 8,192 | |
The sequence starts with 800,000; each subsequent term is obtained by dividing the previous term by 3. What is the last integer in this sequence? | 800000 | 0.5625 | 5,627.4375 | 4,427.111111 | 7,170.714286 | |
In triangle $XYZ$, where $XY = 8$, $YZ = 12$, and $XZ = 14$. Points $D$ and $E$ are selected on $\overline{XY}$ and $\overline{XZ}$ respectively, such that $XD = 3$ and $XE = 9$. Calculate the area of triangle $XDE$. | \frac{405 \sqrt{17}}{112} | 0 | 7,660.25 | -1 | 7,660.25 | |
Two cards are placed in each of three different envelopes, and cards 1 and 2 are placed in the same envelope. Calculate the total number of different arrangements. | 18 | 0.1875 | 6,407.0625 | 5,058.333333 | 6,718.307692 | |
We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$ ? | 119 | 0.375 | 7,557.3125 | 6,499.5 | 8,192 | |
Given the function $f(x) = \sqrt{\log_{3}(4x-1)} + \sqrt{16-2^{x}}$, its domain is A.
(1) Find the set A;
(2) If the function $g(x) = (\log_{2}x)^{2} - 2\log_{2}x - 1$, and $x \in A$, find the maximum and minimum values of the function $g(x)$ and the corresponding values of $x$. | -2 | 0.3125 | 4,040.25 | 3,660.4 | 4,212.909091 | |
There are 2019 numbers written on the board. One of them occurs more frequently than the others - 10 times. What is the minimum number of different numbers that could be written on the board? | 225 | 0.25 | 6,838.625 | 5,417.5 | 7,312.333333 | |
The first two terms of a given sequence are 1 and 1 respectively, and each successive term is the sum of the two preceding terms. What is the value of the first term which exceeds 1000? | 1597 | 1 | 2,665.5625 | 2,665.5625 | -1 | |
For his birthday, Piglet baked a big cake weighing 10 kg and invited 100 guests. Among them was Winnie-the-Pooh, who has a weakness for sweets. The birthday celebrant announced the cake-cutting rule: the first guest cuts themselves a piece of cake equal to \(1\%\) of the remaining cake, the second guest cuts themselves... | 10 | 0.0625 | 8,014.875 | 6,985 | 8,083.533333 | |
Given that the three interior angles $A$, $B$, $C$ of $\triangle ABC$ form an arithmetic sequence, and the side $b$ opposite to angle $B$ equals $\sqrt{3}$, and the function $f(x)=2 \sqrt{3}\sin ^{2}x+2\sin x\cos x- \sqrt{3}$ reaches its maximum value at $x=A$, then the area of $\triangle ABC$ is __________. | \frac{3+ \sqrt{3}}{4} | 0 | 6,851.5625 | -1 | 6,851.5625 | |
The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the $1$ at each end, row $4$ consists of only even numbers, as does row $2.$ How many of the first $20$ rows have this property? (Don't include row $0$ or row $1$). \begin{tabular}{ccccccccccc}
&&&&&1&&&&&\\
&&&&1&&1&&&&\\
&&&1... | 4 | 0.875 | 4,450.875 | 3,925.5 | 8,128.5 | |
A kitten bites off a quarter of a sausage from one end, then a puppy bites off a third of the remaining piece from the opposite end, then the kitten again bites off a quarter from its end, and the puppy bites off a third from its end, and so on. You need to tie a string around the sausage in advance to ensure that no o... | 1:1 | 0 | 7,968.9375 | -1 | 7,968.9375 | |
What is the greatest common divisor of $2^{1001}-1$ and $2^{1012}-1$? | 2047 | 0.9375 | 2,783.3125 | 2,422.733333 | 8,192 | |
Define the operation $\spadesuit$ as $a\,\spadesuit\,b = |a- b|$ . What is the value of $2\, \spadesuit\,(4\,\spadesuit\,7)$? | 1 | 1 | 1,731.25 | 1,731.25 | -1 | |
Shift the graph of the function $y = \sin\left(\frac{\pi}{3} - x\right)$ to obtain the graph of the function $y = \cos\left(x + \frac{2\pi}{3}\right)$. | \frac{\pi}{2} | 0.0625 | 7,732.75 | 6,927 | 7,786.466667 | |
The smallest possible value of $m$ for which Casper can buy exactly $10$ pieces of strawberry candy, $18$ pieces of lemon candy, and $20$ pieces of cherry candy, given that each piece of orange candy costs $15$ cents. | 12 | 0 | 7,300.125 | -1 | 7,300.125 | |
In square $EFGH$, $EF$ is 8 centimeters, and $N$ is the midpoint of $\overline{GH}$. Let $P$ be the intersection of $\overline{EC}$ and $\overline{FN}$, where $C$ is a point on segment $GH$ such that $GC = 6$ cm. What is the area ratio of triangle $EFP$ to triangle $EPG$? | \frac{2}{3} | 0 | 6,513.625 | -1 | 6,513.625 | |
Find the max. value of $ M$,such that for all $ a,b,c>0$:
$ a^{3}+b^{3}+c^{3}-3abc\geq M(|a-b|^{3}+|a-c|^{3}+|c-b|^{3})$ | \sqrt{9 + 6\sqrt{3}} |
To find the maximum value of \( M \) such that the inequality
\[
a^3 + b^3 + c^3 - 3abc \geq M(|a-b|^3 + |a-c|^3 + |c-b|^3)
\]
holds for all \( a, b, c > 0 \), we start by analyzing both sides of the inequality.
### Step 1: Understand the Expression on the Left
The left-hand side of the inequality is:
\[
a^3 + b^3... | 0 | 8,156 | -1 | 8,156 |
One day while Tony plays in the back yard of the Kubik's home, he wonders about the width of the back yard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally s... | 82 | 0.0625 | 6,889.6875 | 7,642 | 6,839.533333 | |
The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number $4$ brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of $55$ seats. What is the maximum number of contestants that can be seated in a single row with the ... | 30 |
To address the problem, we need to determine the maximum number of contestants that can be seated in a single row of 55 seats under the restriction that no two contestants are seated 4 seats apart.
Let's denote the seats in the row as positions \(1, 2, 3, \ldots, 55\). The condition that no two contestants are 4 seat... | 0 | 7,958.8125 | -1 | 7,958.8125 |
10 times 0.1 equals to ____, 10 times 0.01 equals to ____, 10 times 0.001 equals to ____. | 0.01 | 0.5 | 382.25 | 385.625 | 378.875 | |
There are two circles: one centered at point \(A\) with a radius of 5, and another centered at point \(B\) with a radius of 15. Their common internal tangent touches the circles at points \(C\) and \(D\) respectively. The lines \(AB\) and \(CD\) intersect at point \(E\). Find \(CD\) if \(BE = 39\). | 48 | 0.5 | 6,493.4375 | 4,794.875 | 8,192 | |
Given \( |z|=2 \) and \( u=\left|z^{2}-z+1\right| \), find the minimum value of \( u \) where \( z \in \mathbf{C} \). | \frac{3}{2} \sqrt{3} | 0 | 7,690.9375 | -1 | 7,690.9375 | |
Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? Express your answer as a common fraction. | \dfrac{1}{8} | 0 | 4,432.0625 | -1 | 4,432.0625 | |
Given that D is a point on the hypotenuse BC of right triangle ABC, and $AC= \sqrt {3}DC$, $BD=2DC$. If $AD=2 \sqrt {3}$, then $DC=\_\_\_\_\_\_$. | \sqrt {6} | 0 | 4,158.0625 | -1 | 4,158.0625 | |
Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median? | 18 | 1 | 2,934.8125 | 2,934.8125 | -1 | |
Can you use the four basic arithmetic operations (addition, subtraction, multiplication, division) and parentheses to write the number 2016 using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 in sequence? | 2016 | 0 | 8,038.8125 | -1 | 8,038.8125 | |
Given two lines $l_{1}: ax-y+a=0$ and $l_{2}: (2a-3)x+ay-a=0$ are parallel, determine the value of $a$. | -3 | 0.4375 | 5,513 | 4,577.285714 | 6,240.777778 | |
Express as a fraction in lowest terms: $0.\overline{1} + 0.\overline{01}$ | \frac{4}{33} | 1 | 2,208.5 | 2,208.5 | -1 | |
Find all positive real numbers $x$ that satisfy
\[x \sqrt{12 - x} + \sqrt{12x - x^3} \ge 12.\]Enter all solutions, separated by commas. | 3 | 0.625 | 6,646.0625 | 5,718.5 | 8,192 | |
Given 5 balls with 2 identical black balls and one each of red, white, and blue, calculate the number of different arrangements of 4 balls in a row. | 60 | 0.25 | 7,156.4375 | 5,165.75 | 7,820 | |
Let $P(x)=x^{4}+2 x^{3}-13 x^{2}-14 x+24$ be a polynomial with roots $r_{1}, r_{2}, r_{3}, r_{4}$. Let $Q$ be the quartic polynomial with roots $r_{1}^{2}, r_{2}^{2}, r_{3}^{2}, r_{4}^{2}$, such that the coefficient of the $x^{4}$ term of $Q$ is 1. Simplify the quotient $Q\left(x^{2}\right) / P(x)$, leaving your answer... | $x^{4}-2 x^{3}-13 x^{2}+14 x+24$ | We note that we must have $$Q(x)=\left(x-r_{1}^{2}\right)\left(x-r_{2}^{2}\right)\left(x-r_{3}^{2}\right)\left(x-r_{4}^{2}\right) \Rightarrow Q\left(x^{2}\right)=\left(x^{2}-r_{1}^{2}\right)\left(x^{2}-r_{2}^{2}\right)\left(x^{2}-r_{3}^{2}\right)\left(x^{2}-r_{4}^{2}\right)$$. Since $P(x)=\left(x-r_{1}\right)\left(x-r_... | 0 | 7,456.5625 | -1 | 7,456.5625 |
How many integers are common solutions to these three inequalities? \[
\begin{array}{cccc}
(1) & -3y & \geq & y+7 \\
(2) & -2y & \leq & 12 \\
(3) & -4y & \geq & 2y+17
\end{array}
\] | 4 | 0.9375 | 3,545.25 | 3,235.466667 | 8,192 | |
In base 10, compute the result of \( (456_{10} + 123_{10}) - 579_{10} \). Express your answer in base 10. | 0_{10} | 0 | 3,389.875 | -1 | 3,389.875 | |
An equilateral triangle ABC has a side length of 4. A right isosceles triangle DBE, where $DB=EB=1$ and angle $D\hat{B}E = 90^\circ$, is cut from triangle ABC. Calculate the perimeter of the remaining quadrilateral. | 10 + \sqrt{2} | 0 | 7,599.6875 | -1 | 7,599.6875 | |
There is infinite sequence of composite numbers $a_1,a_2,...,$ where $a_{n+1}=a_n-p_n+\frac{a_n}{p_n}$ ; $p_n$ is smallest prime divisor of $a_n$ . It is known, that $37|a_n$ for every $n$ .
Find possible values of $a_1$ | 37^2 | 0 | 8,181.6875 | -1 | 8,181.6875 | |
What is the value of the sum $\frac{2}{3}+\frac{2^2}{3^2}+\frac{2^3}{3^3}+ \ldots +\frac{2^{10}}{3^{10}}$? Express your answer as a common fraction. | \frac{116050}{59049} | 0.75 | 5,633.5 | 4,780.666667 | 8,192 | |
The arithmetic mean of several consecutive natural numbers is 5 times greater than the smallest of them. How many times is the arithmetic mean smaller than the largest of these numbers? | 1.8 | 0 | 4,736.0625 | -1 | 4,736.0625 | |
A line passing through the focus of the parabola $y^2=4x$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $|AB|=12$, then $x_1+x_2=$ ___. | 10 | 0.625 | 6,183.3125 | 5,170.2 | 7,871.833333 | |
Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron each of whose edges measures 2 meters. A bug, starting from vertex $A$, follows the rule that at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex... | \frac{20}{81} | 0.25 | 7,383.8125 | 4,959.25 | 8,192 | |
What is the coefficient of $x^5$ in the expansion of $(1 + x + x^2)^9$ ? | 882 | 0.625 | 6,356.9375 | 5,255.9 | 8,192 | |
Expand the product $$(x^2-2x+2)(x^2+2x+2).$$ | x^4+4 | 1 | 2,759.1875 | 2,759.1875 | -1 | |
Compute the sum of all positive integers $a \leq 26$ for which there exist integers $b$ and $c$ such that $a+23 b+15 c-2$ and $2 a+5 b+14 c-8$ are both multiples of 26. | 31 | Assume $b$ and $c$ exist. Considering the two values modulo 13, we find $$\begin{cases}a+10 b+2 c \equiv 2 & (\bmod 13) \\ 2 a+5 b+c \equiv 8 & (\bmod 13)\end{cases}$$ Subtracting twice the second equation from the first, we get $-3 a \equiv-14(\bmod 13)$. So, we have $a \equiv 9$ $(\bmod 13)$. Therefore we must either... | 0.3125 | 7,546.6875 | 6,127 | 8,192 |
Let $S=\left\{p_{1} p_{2} \cdots p_{n} \mid p_{1}, p_{2}, \ldots, p_{n}\right.$ are distinct primes and $\left.p_{1}, \ldots, p_{n}<30\right\}$. Assume 1 is in $S$. Let $a_{1}$ be an element of $S$. We define, for all positive integers $n$ : $$ \begin{gathered} a_{n+1}=a_{n} /(n+1) \quad \text { if } a_{n} \text { is d... | 512 | If $a_{1}$ is odd, then we can see by induction that $a_{j}=(j+1) a_{1}$ when $j$ is even and $a_{j}=a_{1}$ when $j$ is odd (using the fact that no even $j$ can divide $a_{1}$ ). So we have infinitely many $j$ 's for which $a_{j}=a_{1}$. If $a_{1}>2$ is even, then $a_{2}$ is odd, since $a_{2}=a_{1} / 2$, and $a_{1}$ ma... | 0 | 8,192 | -1 | 8,192 |
In a right triangle $ABC$ with legs $AB = 3$ and $BC = 4$, a circle is drawn through the midpoints of sides $AB$ and $AC$, touching the leg $BC$. Find the length of the segment of the hypotenuse $AC$ that lies inside this circle. | 11/10 | 0.5625 | 6,152 | 5,611.555556 | 6,846.857143 | |
Given that a multiple-choice math question has incorrect options that cannot be adjacent, calculate the total number of arrangements of $4$ different options that meet the requirements. | 36 | 0 | 6,282.375 | -1 | 6,282.375 | |
Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length? | 1333 | 0 | 8,192 | -1 | 8,192 | |
Around the outside of a $4$ by $4$ square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles. The area of... | 64 | 1. **Identify the radius of the semicircles**: The original square has a side length of $4$. Each semicircle is constructed with the side of the square as its diameter. Therefore, the radius of each semicircle is half the side length of the square, which is $\frac{4}{2} = 2$.
2. **Determine the position of square $ABC... | 0.5 | 7,445.8125 | 6,699.625 | 8,192 |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 | 0 | 7,990 | -1 | 7,990 | |
Find an integer $n$, where $100 \leq n \leq 1997$, such that
\[ \frac{2^n+2}{n} \]
is also an integer. | 946 | To find an integer \( n \) such that \( 100 \leq n \leq 1997 \) and
\[
\frac{2^n + 2}{n}
\]
is an integer, we need to ensure that \( n \mid (2^n + 2) \). This means that the expression can be rewritten using divisibility:
\[
2^n + 2 \equiv 0 \pmod{n}.
\]
This simplifies to:
\[
2^n \equiv -2 \equiv n-2 \pmod{n}.
\]
... | 0 | 8,192 | -1 | 8,192 |
Let $OX, OY$ and $OZ$ be three rays in the space, and $G$ a point "[i]between these rays[/i]" (i. e. in the interior of the part of the space bordered by the angles $Y OZ, ZOX$ and $XOY$). Consider a plane passing through $G$ and meeting the rays $OX, OY$ and $OZ$ in the points $A, B, C$, respectively. There are infini... |
To solve for the plane that minimizes the volume of the tetrahedron \( OABC \), where the plane meets the rays \( OX, OY, \) and \( OZ \) at points \( A, B, \) and \( C \) respectively, we need to strategically place these intersection points. To achieve the minimum volume for the tetrahedron \( OABC \), we should mak... | 0 | 7,534.375 | -1 | 7,534.375 | |
$(1)$ Find the value of $x$: $4\left(x+1\right)^{2}=49$;<br/>$(2)$ Calculate: $\sqrt{9}-{({-1})^{2018}}-\sqrt[3]{{27}}+|{2-\sqrt{5}}|$. | \sqrt{5} - 3 | 0.75 | 1,455.75 | 1,682.083333 | 776.75 | |
Given an arithmetic sequence $\{a\_n\}$, the sum of its first $n$ terms, $S\_n$, satisfies $S\_3=0$ and $S\_5=-5$. The sum of the first 2016 terms of the sequence $\{ \frac{1}{a_{2n-1}a_{2n+1}} \}$ is $\_\_\_\_\_\_\_\_.$ | -\frac{2016}{4031} | 0.1875 | 7,992.3125 | 7,735 | 8,051.692308 | |
Given that $F\_1$ and $F\_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the hyperbola such that $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = 0$ and $|\overrightarrow{PF\_1}| \cdot |\overrightarrow{PF\_2}| = 2ac (c$ is the semi-focal distance$)$... | \frac{\sqrt{5} + 1}{2} | 0 | 7,115.625 | -1 | 7,115.625 | |
For any positive integer $n$, we define the integer $P(n)$ by :
$P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$.
Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$. | 510510 |
To find the greatest common divisor (GCD) of the integers \( P(1), P(2), P(3), \ldots, P(2016) \), where \( P(n) = n(n+1)(2n+1)(3n+1)\cdots(16n+1) \), we will first consider each part of the product and determine if there is a consistent factor across all \( P(n) \).
### Step 1: Analyze the Form of \( P(n) \)
The ex... | 0.0625 | 8,170.5625 | 7,849 | 8,192 |
Evaluate $\left\lceil\sqrt{3}\,\right\rceil+\left\lceil\sqrt{33}\,\right\rceil+\left\lceil\sqrt{333}\,\right\rceil$. | 27 | 0.9375 | 3,645.1875 | 3,342.066667 | 8,192 | |
The radius of the Earth measures approximately $6378 \mathrm{~km}$ at the Equator. Suppose that a wire is adjusted exactly over the Equator.
Next, suppose that the length of the wire is increased by $1 \mathrm{~m}$, so that the wire and the Equator are concentric circles around the Earth. Can a standing man, an ant, o... | 0.159 | 0 | 8,166.25 | -1 | 8,166.25 | |
In an isosceles trapezoid \(ABCD\), the larger base \(AD = 12\) and \(AB = 6\). Find the distance from point \(O\), the intersection of the diagonals, to point \(K\), the intersection of the extensions of the lateral sides, given that the extensions of the lateral sides intersect at a right angle. | \frac{12(3 - \sqrt{2})}{7} | 0 | 6,237.125 | -1 | 6,237.125 | |
A cuckoo clock chimes "cuckoo" on the hour, with the number of "cuckoo" calls equal to the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand with his finger until he moved the clock forward by 7 hours. How... | 43 | 0.0625 | 5,911.3125 | 4,222 | 6,023.933333 | |
What is the base five sum of the numbers $212_{5}$ and $12_{5}$? | 224_5 | 0.9375 | 2,735.5 | 2,371.733333 | 8,192 | |
The sum \( b_{6} + b_{7} + \ldots + b_{2018} \) of the terms of the geometric progression \( \left\{b_{n}\right\} \) with \( b_{n}>0 \) is equal to 6. The sum of the same terms taken with alternating signs \( b_{6} - b_{7} + b_{8} - \ldots - b_{2017} + b_{2018} \) is equal to 3. Find the sum of the squares of these ter... | 18 | 0 | 8,166.3125 | -1 | 8,166.3125 | |
Compute
\[\frac{5}{3^2 \cdot 7^2} + \frac{9}{7^2 \cdot 11^2} + \frac{13}{11^2 \cdot 15^2} + \dotsb.\] | \frac{1}{72} | 0.6875 | 5,615.875 | 4,444.909091 | 8,192 | |
Given that the sequence $\{a_n\}$ forms a geometric sequence, and $a_n > 0$.
(1) If $a_2 - a_1 = 8$, $a_3 = m$. ① When $m = 48$, find the general formula for the sequence $\{a_n\}$. ② If the sequence $\{a_n\}$ is unique, find the value of $m$.
(2) If $a_{2k} + a_{2k-1} + \ldots + a_{k+1} - (a_k + a_{k-1} + \ldots + a... | 32 | 0.375 | 7,389.375 | 6,455.333333 | 7,949.8 | |
Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? | 18 | Notice that in each $2 \times 2$ square contained in the grid, we can form 4 L-trominoes. By the pigeonhole principle, some color appears twice among the four squares, and there are two trominoes which contain both. Therefore each $2 \times 2$ square contains at most 2 L-trominoes with distinct colors. Equality is achi... | 0 | 8,079.8125 | -1 | 8,079.8125 |
In an organization with 200 employees, those over the age of 50 account for 20%, those aged 40-50 make up 30%, and those under 40 account for 50%. If 40 employees are to be sampled, and the systematic sampling method is used—where all employees are randomly numbered 1-200 and evenly divided into 40 groups (numbers 1-5,... | 20 | 0.5 | 5,708.875 | 3,724.25 | 7,693.5 | |
After the year 2002, which is a palindrome, identify the next year where the sum of the product of its digits is greater than 15. Find the sum of the product of the digits of that year. | 16 | 0.375 | 6,924.25 | 6,285.166667 | 7,307.7 | |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $A=\frac{\pi}{4}$ and $b=\frac{\sqrt{2}}{2}a$.
(Ⅰ) Find the magnitude of $B$;
(Ⅱ) If $a=\sqrt{2}$, find the area of $\Delta ABC$. | \frac{\sqrt{3}+1}{4} | 0 | 5,521 | -1 | 5,521 | |
Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with the length of the minor axis being $2$ and the eccentricity being $\frac{\sqrt{2}}{2}$, the line $l: y = kx + m$ intersects the ellipse $C$ at points $A$ and $B$, and the perpendicular bisector of segment $AB$ passes through the point $(0, -\... | \frac{\sqrt{2}}{2} | 0 | 8,088.0625 | -1 | 8,088.0625 | |
Three six-sided dice (each numbered 1 through 6) are tossed. What is the probability that the sum of the numbers shown on the top faces is even? Express your answer as a common fraction. | \frac{1}{2} | 0.875 | 4,292.3125 | 3,735.214286 | 8,192 | |
Abigail, Beatrice, and Carson combine their eggs to sell them at the market. If Abigail has 37 eggs, Beatrice has 49 eggs, and Carson has 14 eggs, and if eggs can only be sold in cartons of 12, how many eggs will be left over if all cartons are sold? | 4 | 1 | 1,051.5625 | 1,051.5625 | -1 | |
There are 15 cards, each with 3 different Chinese characters. No two cards have the exact same set of Chinese characters, and in any set of 6 cards, there are always at least 2 cards that share a common Chinese character. What is the maximum number of different Chinese characters that can be on these 15 cards? | 35 | 0 | 7,892.1875 | -1 | 7,892.1875 | |
Simplify $\sqrt{245}$. | 7\sqrt{5} | 1 | 1,311.5 | 1,311.5 | -1 | |
Given that $α$ is an angle in the second quadrant, let point $P(x, \sqrt {5})$ be a point on the terminal side of $α$, and $\cos α= \frac { \sqrt {2}}{4}x$. Find the value of $4\cos (α+ \frac {π}{2})-3\tan α$. | \sqrt {15}- \sqrt {10} | 0 | 3,951.125 | -1 | 3,951.125 | |
Find all integers \( n \) for which \( n^2 + 20n + 11 \) is a perfect square. | 35 | 0.25 | 4,468.625 | 4,259.75 | 4,538.25 | |
When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes: | -1 | 1. **Rewrite the fraction inside the logarithm**: We start by recognizing that $\frac{1}{8}$ can be expressed as $8^{-1}$. Therefore, the expression $\log{\frac{1}{8}}$ can be rewritten using the property of logarithms that $\log{a^{-b}} = -b \log{a}$:
\[
\log{\frac{1}{8}} = \log{8^{-1}} = -\log{8}
\]
2. **Si... | 1 | 2,124.9375 | 2,124.9375 | -1 |
In $\triangle ABC, AB = 8, BC = 7, CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$. The length of $PC$ is
[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, P=(1.5,5), B=(8,0), C=P+2.5*dir(P--B); draw(A--P--C--A--B--C); label("A"... | 9 | 0 | 5,259 | -1 | 5,259 | |
Let $\mathbf{a}$ and $\mathbf{b}$ be unit vectors such that $\mathbf{a} + 2 \mathbf{b}$ and $5 \mathbf{a} - 4 \mathbf{b}$ are orthogonal. Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.
Note: A unit vector is a vector of magnitude 1. | 60^\circ | 1 | 1,769.375 | 1,769.375 | -1 | |
My school's Chess Club has 24 members. It needs to select 3 officers: president, secretary, and treasurer. Each person can hold at most one office. Two of the members, Alice and Bob, will only serve together as officers. In how many ways can the club choose its officers? | 9372 | 0.0625 | 7,072.875 | 5,305 | 7,190.733333 | |
Express .$\overline{28}$ as a common fraction. | \frac{28}{99} | 1 | 1,349.5 | 1,349.5 | -1 | |
Let $f(x)=(x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left |\sum_{k=1}^n\log_{10}f(k)\right|=1.\] | 21 | Note that $\cos(\pi x)$ is $-1$ when $x$ is odd and $1$ when $x$ is even. Also note that $x^2+3x+2=(x+1)(x+2)$ for all $x$. Therefore \[\log_{10}f(x)=\log_{10}(x+1)+\log_{10}(x+2)\ \ \ \text{if }x \text{ is even}\] \[\log_{10}f(x)=-\log_{10}(x+1)-\log_{10}(x+2)\ \ \ \text{if }x \text{ is odd}\] Because of this, $\sum_{... | 0.125 | 7,685.25 | 6,520.5 | 7,851.642857 |
Given \( a_{0}=1, a_{1}=2 \), and \( n(n+1) a_{n+1}=n(n-1) a_{n}-(n-2) a_{n-1} \) for \( n=1, 2, 3, \ldots \), find \( \frac{a_{0}}{a_{1}}+\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{50}}{a_{51}} \). | 51 | 0 | 7,679.875 | -1 | 7,679.875 | |
Suppose that $x$ is an integer that satisfies the following congruences:
\[
4 + x \equiv 3^2 \pmod{2^3}, \\
6 + x \equiv 2^3 \pmod{3^3}, \\
8 + x \equiv 7^2 \pmod{5^3}.
\]
What is the remainder when $x$ is divided by $30$? | 17 | 0 | 6,224.4375 | -1 | 6,224.4375 | |
Find the value of \[\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).\] | \frac{3}{2} | 0.875 | 5,000.375 | 4,544.428571 | 8,192 | |
Given that there are 5 people standing in a row, calculate the number of ways for person A and person B to stand such that there is exactly one person between them. | 36 | 0.375 | 7,152.9375 | 6,359 | 7,629.3 | |
Let $a$, $b$, and $c$ be the roots of the equation $x^3 - 2x - 5 = 0$. Find $\frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2}$. | 10 | 0.6875 | 5,226.125 | 3,878 | 8,192 | |
Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, and $g(x) \neq 0$, $f''(x)g(x) < f(x)g''(x)$, $f(x)=a^{x}g(x)$, $\frac{f(1)}{g(1)}+ \frac{f(-1)}{g(-1)}= \frac{5}{2}$, determine the probability that the sum of the first $k$ terms of the sequence $\left\{ \frac{f(n)}{g(n)}\right\} (n=1,2,…,10)$ is gre... | \frac{3}{5} | 0.0625 | 7,211.125 | 7,397 | 7,198.733333 | |
Is
\[f(x) = \log (x + \sqrt{1 + x^2})\]an even function, odd function, or neither?
Enter "odd", "even", or "neither". | \text{odd} | 0.5625 | 2,999.125 | 2,755.888889 | 3,311.857143 | |
Given two circles $C\_1$: $x^{2}+y^{2}=1$ and $C\_2$: $(x-2)^{2}+(y-4)^{2}=1$, a moving point $P(a,b)$ passes through and forms tangent lines $PM$ and $PN$ to circles $C\_1$ and $C\_2$ respectively with $M$ and $N$ being the points of tangency. If $PM=PN$, find the minimum value of $\sqrt{a^{2}+b^{2}}+\sqrt{(a-5)^{2}+(... | \sqrt{34} | 0.625 | 7,097.375 | 6,440.6 | 8,192 | |
Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits. | 397 | 0.3125 | 7,870.6875 | 7,163.8 | 8,192 | |
Given \( f(x)=\frac{2x+3}{x-1} \), the graph of the function \( y=g(x) \) is symmetric with the graph of the function \( y=f^{-1}(x+1) \) with respect to the line \( y=x \). Find \( g(3) \). | \frac{7}{2} | 0.625 | 6,443.25 | 5,843.4 | 7,443 | |
In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE=6$, and $AF=2$. Find the area of $ABCD$.
[asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
real length = 2 * (6*sqrt(3) - 2), wid... | 108\sqrt{3}-36 | 0 | 7,981.8125 | -1 | 7,981.8125 | |
Calculate: $(10 \times 19 \times 20 \times 53 \times 100 + 601) \div 13 = \ ?$ | 1549277 | 0.125 | 577.375 | 483 | 590.857143 | |
Quadrilateral $ABCD$ has right angles at $A$ and $C$, with diagonal $AC = 5$. If $AB = BC$ and sides $AD$ and $DC$ are of distinct integer lengths, what is the area of quadrilateral $ABCD$? Express your answer in simplest radical form. | 12.25 | 0 | 8,146.9375 | -1 | 8,146.9375 | |
Given that the probability of bus No. 3 arriving at the bus stop within 5 minutes is 0.20 and the probability of bus No. 6 arriving within 5 minutes is 0.60, calculate the probability that the passenger can catch the bus he needs within 5 minutes. | 0.80 | 0 | 2,567.5625 | -1 | 2,567.5625 | |
Let \( S_1, S_2, \ldots, S_{10} \) be the first ten terms of an arithmetic progression (A.P.) consisting of positive integers. If \( S_1 + S_2 + \ldots + S_{10} = 55 \) and \( \left(S_{10} - S_{8}\right) + \left(S_{9} - S_{7}\right) + \ldots + \left(S_{3} - S_{1}\right) = d \), find \( d \). | 16 | 0.5 | 6,023.8125 | 4,585 | 7,462.625 | |
What is the smallest whole number $b$ such that 101 can be expressed in base $b$ using only two digits? | 10 | 0 | 5,440.375 | -1 | 5,440.375 |
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