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 |
|---|---|---|---|---|---|---|
For positive integers $n$ and $k$, let $\mho(n, k)$ be the number of distinct prime divisors of $n$ that are at least $k$. Find the closest integer to $$\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{\mho(n, k)}{3^{n+k-7}}$$ | 167 | A prime $p$ is counted in $\mho(n, k)$ if $p \mid n$ and $k \leq p$. Thus, for a given prime $p$, the total contribution from $p$ in the sum is $$3^{7} \sum_{m=1}^{\infty} \sum_{k=1}^{p} \frac{1}{3^{p m+k}}=3^{7} \sum_{i \geq p+1} \frac{1}{3^{i}}=\frac{3^{7-p}}{2}$$ Therefore, if we consider $p \in\{2,3,5,7, \ldots\}$ ... | 0 | 7,858.125 | -1 | 7,858.125 |
If $x = 151$ and $x^3y - 3x^2y + 3xy = 3423000$, what is the value of $y$? | \frac{3423000}{3375001} | 0 | 8,192 | -1 | 8,192 | |
Compute $\begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}^3.$ | \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}. | 0 | 4,647.1875 | -1 | 4,647.1875 | |
The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)
What is the sum of the $x$-coordinates of all points where $f(x) = 1.8$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsi... | 4.5 | 0 | 5,490.0625 | -1 | 5,490.0625 | |
Three workers can complete a certain task. The second and third worker together can complete it twice as fast as the first worker; the first and third worker together can complete it three times faster than the second worker. How many times faster can the first and second worker together complete the task compared to t... | 7/5 | 0.9375 | 4,560.25 | 4,318.133333 | 8,192 | |
A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \end{pmatrix}.\]A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ -9 \end{pmatrix} + u \begin{pmatrix} 4 \\ 2 \end{pmatrix}.\]Find the point ... | \begin{pmatrix} 7 \\ -8 \end{pmatrix} | 0.75 | 2,302.4375 | 2,269.166667 | 2,402.25 | |
Find the largest integer less than 80 that leaves a remainder of 3 when divided by 5. | 78 | 1 | 631.0625 | 631.0625 | -1 | |
Half the people in a room left. One third of those remaining started to dance. There were then $12$ people who were not dancing. The original number of people in the room was | 36 | 1. **Define the variable:**
Let $x$ be the original number of people in the room.
2. **Calculate the number of people left after half left:**
Since half the people left, the number of people remaining in the room is $\frac{x}{2}$.
3. **Calculate the number of people who started dancing:**
One third of the re... | 1 | 1,529.4375 | 1,529.4375 | -1 |
Yann writes down the first $n$ consecutive positive integers, $1,2,3,4, \ldots, n-1, n$. He removes four different integers $p, q, r, s$ from the list. At least three of $p, q, r, s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is 89.5625. What is the number of possible values of... | 22 | When Yann removes 4 of the $n$ integers from his list, there are $n-4$ integers left. Suppose that the sum of the $n-4$ integers left is $T$. The average of these $n-4$ integers is $89.5625=89.5+0.0625=89+\frac{1}{2}+\frac{1}{16}=89 \frac{9}{16}=\frac{1433}{16}$. Since the sum of the $n-4$ integers is $T$, then $\frac{... | 0 | 8,192 | -1 | 8,192 |
Let \( p(x) \) be a polynomial with integer coefficients such that \( p(m) - p(n) \) divides \( m^2 - n^2 \) for all integers \( m \) and \( n \). If \( p(0) = 1 \) and \( p(1) = 2 \), find the largest possible value of \( p(100) \). | 10001 | 0.125 | 8,124.3125 | 7,650.5 | 8,192 | |
A person rolled a fair six-sided die $100$ times and obtained a $6$ $19$ times. What is the approximate probability of rolling a $6$? | 0.19 | 0.875 | 2,769.8125 | 2,887.285714 | 1,947.5 | |
Some marbles in a bag are red and the rest are blue. If one red marble is removed, then one-seventh of the remaining marbles are red. If two blue marbles are removed instead of one red, then one-fifth of the remaining marbles are red. How many marbles were in the bag originally? | 22 | Let $r$ and $b$ be the number of red and blue marbles originally in the bag, respectively.
1. **Removing one red marble:**
- After removing one red marble, the number of red marbles becomes $r-1$.
- The total number of marbles left is $r+b-1$.
- Given that one-seventh of the remaining marbles are red, we have... | 0.9375 | 2,213.4375 | 1,814.866667 | 8,192 |
We define a number as an ultimate mountain number if it is a 4-digit number and the third digit is larger than the second and fourth digit but not necessarily the first digit. For example, 3516 is an ultimate mountain number. How many 4-digit ultimate mountain numbers are there? | 204 | 0 | 6,906 | -1 | 6,906 | |
A right triangle $\triangle ABC$ has sides $AC=3$, $BC=4$, and $AB=5$. When this triangle is rotated around the right-angle side $BC$, the surface area of the resulting solid is ______. | 24\pi | 0.5625 | 6,254.75 | 4,748 | 8,192 | |
Express $0.\overline{1}+0.\overline{01}+0.\overline{0001}$ as a common fraction. | \frac{1213}{9999} | 0.75 | 6,147.375 | 5,522.25 | 8,022.75 | |
Suppose $x-3$ and $y+3$ are multiples of $7$.
What is the smallest positive integer, $n,$ for which $x^2+xy+y^2+n$ is a multiple of $7$? | 5 | 1 | 2,327.25 | 2,327.25 | -1 | |
When the base-10 integers 200 and 1200 are expressed in base 2, how many more digits does 1200 have than 200 (after being converted)? | 3 | 0.625 | 5,090.125 | 3,229 | 8,192 | |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{2}{b}.\] | 9 | 0.875 | 5,285.3125 | 4,870.071429 | 8,192 | |
**
How many non-similar regular 500-pointed stars are there?
** | 99 | 0.5 | 7,208.875 | 6,390 | 8,027.75 | |
Two circles of radius 2 are centered at $(2,0)$ and at $(0,2)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | 2\pi-4 | 0.8125 | 5,088.25 | 4,372 | 8,192 | |
Let $n = 2^4 \cdot 3^5 \cdot 4^6\cdot 6^7$. How many natural-number factors does $n$ have? | 312 | 1 | 1,872.5625 | 1,872.5625 | -1 | |
Find the largest integer value of $n$ such that $n^2-9n+18$ is negative. | 5 | 1 | 1,691.875 | 1,691.875 | -1 | |
A stone is dropped into a well and the report of the stone striking the bottom is heard $7.7$ seconds after it is dropped. Assume that the stone falls $16t^2$ feet in t seconds and that the velocity of sound is $1120$ feet per second. The depth of the well is: | 784 | 1. **Define Variables:**
Let $d$ be the depth of the well in feet, $t_1$ be the time in seconds it takes for the stone to fall to the bottom of the well, and $t_2$ be the time in seconds it takes for the sound to travel back up to the top of the well. We know that the total time for the stone to fall and the sound t... | 0.875 | 3,911.375 | 3,299.857143 | 8,192 |
In a right triangle \( A B C \) (with right angle at \( C \)), the medians \( A M \) and \( B N \) are drawn with lengths 19 and 22, respectively. Find the length of the hypotenuse of this triangle. | 29 | 0 | 3,157.875 | -1 | 3,157.875 | |
Given that $E, U, L, S, R, T$ represent the numbers $1, 2, 3, 4, 5, 6$ (with each letter representing a different number), and satisfying the following conditions:
(1) $E + U + L = 6$
(2) $S + R + U + T = 18$
(3) $U \times T = 15$
(4) $S \times L = 8$
What is the six-digit number $\overline{EULSRT}$? | 132465 | 0.9375 | 3,400.4375 | 3,434.533333 | 2,889 | |
Let \(ABCD\) be an isosceles trapezoid with \(AB=1, BC=DA=5, CD=7\). Let \(P\) be the intersection of diagonals \(AC\) and \(BD\), and let \(Q\) be the foot of the altitude from \(D\) to \(BC\). Let \(PQ\) intersect \(AB\) at \(R\). Compute \(\sin \angle RPD\). | \frac{4}{5} | Let \(M\) be the foot of the altitude from \(B\) to \(CD\). Then \(2CM+AB=CD \Longrightarrow CM=3\). Then \(DM=4\) and by the Pythagorean theorem, \(BM=4\). Thus \(BMD\) is a right isosceles triangle i.e. \(\angle BDM=\angle PDC=\frac{\pi}{4}\). Similarly, \(\angle PCD=\frac{\pi}{4}\). Thus \(\angle DPC=\frac{\pi}{2}\)... | 0.75 | 6,431.875 | 6,273.25 | 6,907.75 |
This was a highly dangerous car rally. It began with a small and very narrow bridge, where one out of five cars would fall into the water. Then followed a terrifying sharp turn, where three out of ten cars would go off the road. Next, there was a dark and winding tunnel where one out of ten cars would crash. The last p... | 69.76 | 0.25 | 1,811.9375 | 2,701.5 | 1,515.416667 | |
Given that the lengths of the three sides of $\triangle ABC$ form an arithmetic sequence with a common difference of 2, and the sine of the largest angle is $\frac{\sqrt{3}}{2}$, the sine of the smallest angle of this triangle is \_\_\_\_\_\_. | \frac{3\sqrt{3}}{14} | 0 | 4,861.375 | -1 | 4,861.375 | |
Find the units digit of the decimal expansion of $\left(15 + \sqrt{220}\right)^{19} + \left(15 + \sqrt{220}\right)^{82}$. | 9 | 1. **Define the expressions**:
Let $A = 15 + \sqrt{220}$ and $B = 15 - \sqrt{220}$.
2. **Use the properties of conjugates**:
Note that $A$ and $B$ are conjugates, and any even power of $\sqrt{220}$ in their binomial expansions will result in rational numbers, while odd powers will result in irrational numbers t... | 0.3125 | 5,499.625 | 6,394.8 | 5,092.727273 |
In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:
(a) Michael K had an average test score of $90$ , Michael M had an average test score of $91$ , and Michael R had an average test score of $92$ .
(b) M... | 413 | 0 | 8,192 | -1 | 8,192 | |
A store sells jellybeans at a fixed price per gram. The price for 250 g of jellybeans is $\$ 7.50$. What mass of jellybeans sells for $\$ 1.80$? | 60 \mathrm{~g} | The store sells 250 g of jellybeans for $\$ 7.50$, which is 750 cents. Therefore, 1 g of jellybeans costs $750 \div 250=3$ cents. This means that $\$ 1.80$, which is 180 cents, will buy $180 \div 3=60 \mathrm{~g}$ of jellybeans. | 0 | 623 | -1 | 623 |
How many distinct sequences of five letters can be made from the letters in "PROBLEMS" if each letter can be used only once and each sequence must begin with "S" and not end with "M"? | 720 | 0.6875 | 5,961.1875 | 4,947.181818 | 8,192 | |
Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and if the card is black, she simultaneously replaces the number on each vertex by the sum o... | 42648 | The order of the deck does not matter as black cards and white cards commute, therefore we can assume that the cards are alternating black and white, and only worry about the arrangement of the numbers. After each pair of black and white cards, each number is replaced by the sum of two times the edge neighbors and thre... | 0 | 8,192 | -1 | 8,192 |
In triangle $ABC,$ $D,$ $E,$ and $F$ are points on sides $\overline{BC},$ $\overline{AC},$ and $\overline{AB},$ respectively, so that $BD:DC = CE:EA = AF:FB = 1:2.$
[asy]
unitsize(0.8 cm);
pair A, B, C, D, E, F, P, Q, R;
A = (2,5);
B = (0,0);
C = (7,0);
D = interp(B,C,1/3);
E = interp(C,A,1/3);
F = interp(A,B,1/3);
... | \frac{1}{7} | 0.125 | 8,062.0625 | 7,152.5 | 8,192 | |
Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, respectively. Point $P$ is on the right branch of the hyperbola, and $|PF_{2}| = |F_{1}F_{2}|$. The distance from $F_{2}$ to the line $PF_{1}$ is equal to the length of the real axis of the hyperbola. Fi... | \frac{5}{3} | 0.125 | 8,192 | 8,192 | 8,192 | |
In a school event, there are 360 students participating. They need to be arranged in rows such that each row contains the same number of students. The arrangement must have at least 12 rows and at least 18 students in each row. Calculate the sum of all possible numbers of students per row. | 92 | 1 | 3,059.625 | 3,059.625 | -1 | |
A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle? | 2.37 | 1. **Identify the Geometry and Setup**: We are given a $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ with $AB=1$. The circle with center $O$ is tangent to the coordinate axes and the hypotenuse of triangle $ABC$. We need to find the radius $r$ of this circle.
2. **Label the Points of Tangency**: Draw radii $OE$ and ... | 0 | 7,851.1875 | -1 | 7,851.1875 |
The square was cut into 25 smaller squares, of which exactly one has a side length different from 1 (each of the others has a side length of 1).
Find the area of the original square. | 49 | 0.375 | 6,836.5 | 4,577.333333 | 8,192 | |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), the left and right foci coincide with the symmetric points about the two asymptotes, respectively. Then, the eccentricity of the hyperbola is __________. | \sqrt{2} | 0.0625 | 8,188.875 | 8,142 | 8,192 | |
A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points? | 1979 | 0.25 | 6,977.1875 | 4,326.25 | 7,860.833333 | |
A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: | 2(n-1) | 1. **Define the relationship**: Given that the speed $s_n$ for the $n^\text{th}$ mile varies inversely as the number of miles already traveled, we can express this relationship as:
\[ s_n = \frac{k}{d} \]
where $d$ is the number of miles already traveled and $k$ is a constant.
2. **Determine the constant $k$**: ... | 1 | 3,140.1875 | 3,140.1875 | -1 |
Let $A_{11}$ denote the answer to problem 11. Determine the smallest prime $p$ such that the arithmetic sequence $p, p+A_{11}, p+2 A_{11}, \ldots$ begins with the largest possible number of primes. | 7 | First, note that the maximal number of initial primes is bounded above by the smallest prime not dividing $A_{11}$, with equality possible only if $p$ is this prime. For, if $q$ is the smallest prime not dividing $A_{11}$, then the first $q$ terms of the arithmetic sequence determine a complete residue class modulo $q$... | 0.0625 | 8,130.1875 | 8,192 | 8,126.066667 |
If $8^x - 8^{x-1} = 60$, calculate the value of $(3x)^x$. | 58.9 | 0 | 8,192 | -1 | 8,192 | |
(1) If $\cos (\frac{\pi}{4}+x) = \frac{3}{5}$, and $\frac{17}{12}\pi < x < \frac{7}{4}\pi$, find the value of $\frac{\sin 2x + 2\sin^2 x}{1 - \tan x}$.
(2) Given the function $f(x) = 2\sqrt{3}\sin x\cos x + 2\cos^2 x - 1 (x \in \mathbb{R})$, if $f(x_0) = \frac{6}{5}$, and $x_0 \in [\frac{\pi}{4}, \frac{\pi}{2}]$, find... | \frac{3 - 4\sqrt{3}}{10} | 0 | 7,337.75 | -1 | 7,337.75 | |
Sara lists the whole numbers from 1 to 50. Lucas copies Sara's numbers, replacing each occurrence of the digit '3' with the digit '2'. Calculate the difference between Sara's sum and Lucas's sum. | 105 | 0.125 | 4,048.25 | 3,606 | 4,111.428571 | |
A certain integer has $4$ digits when written in base $8$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$? | 33 | 0.8125 | 4,491.375 | 3,810.461538 | 7,442 | |
Suppose that $A$, $B$, and $C$ are non-zero distinct digits less than $6$, and suppose we have ${AB_6}+{C_6}={C}0_6$ and ${AB_6}+{BA_6}={CC_6}$. Find the three-digit number ${ABC}$. (Interpret $AB_6$ as a base-6 number with digits $A$ and $B$, not as $A$ times $B$. The other expressions should be interpreted in this wa... | 415 | 0.9375 | 2,587.25 | 2,213.6 | 8,192 | |
Find the coefficient of $x^{90}$ in the expansion of
\[(x - 1)(x^2 - 2)(x^3 - 3) \dotsm (x^{12} - 12)(x^{13} - 13).\] | -1 | 0.0625 | 7,767.6875 | 6,988 | 7,819.666667 | |
An octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\triangle ABJ$ (where $J$ is the center of the octagon), shown in boldface in the diagram. By connecting every third vertex, we obtain a larger equilateral triangle $\triangle ADE$, which is also shown in boldface. Compute the ratio $[\... | \frac{1}{4} | 0 | 7,929.5625 | -1 | 7,929.5625 | |
An ordered pair $(n,p)$ is *juicy* if $n^{2} \equiv 1 \pmod{p^{2}}$ and $n \equiv -1 \pmod{p}$ for positive integer $n$ and odd prime $p$ . How many juicy pairs exist such that $n,p \leq 200$ ?
Proposed by Harry Chen (Extile) | 36 | 0.1875 | 7,915.625 | 6,718 | 8,192 | |
The number \( n \) has exactly six divisors (including 1 and itself). These divisors are arranged in ascending order. It turns out that the third divisor is seven times larger than the second one, and the fourth divisor is 10 larger than the third one. What is \( n \)? | 2891 | 0.9375 | 5,489.875 | 5,309.733333 | 8,192 | |
Given that a match between two people is played with a best-of-five-games format, where the winner is the first to win three games, and that the probability of person A winning a game is $\dfrac{2}{3}$, calculate the probability that person A wins with a score of $3:1$. | \dfrac{8}{27} | 0.3125 | 6,935.0625 | 6,230.4 | 7,255.363636 | |
Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\frac{\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region enclosed by the trajectory of the moving point $... | \frac{5\pi}{6} | 0.0625 | 8,106.6875 | 6,827 | 8,192 | |
Determine the value of $x$ that satisfies $\sqrt[4]{x\sqrt{x^4}}=4$. | 2^{8/3} | 0.0625 | 5,008.9375 | 3,945 | 5,079.866667 | |
From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected. | 14 | 0.125 | 5,845.1875 | 5,938.5 | 5,831.857143 | |
Simplify $$\sqrt{6+4\sqrt2}+\sqrt{6-4\sqrt2}.$$ | 4 | 1 | 2,199.5 | 2,199.5 | -1 | |
Given a box contains $5$ shiny pennies and $6$ dull pennies, determine the sum of the numerator and denominator of the probability that it will take exactly six draws to get the fourth shiny penny. | 236 | 0 | 5,808.3125 | -1 | 5,808.3125 | |
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$ , with $i = 1,2, ..., 2015$ . | 2016 | 0 | 8,192 | -1 | 8,192 | |
How many numbers are in the list $ 4, 6, 8, \ldots, 128, 130 ?$ | 64 | 1 | 2,150.5 | 2,150.5 | -1 | |
Lisa drew graphs of all functions of the form \( y = ax + b \), where \( a \) and \( b \) take all natural values from 1 to 100. How many of these graphs pass through the point \((3, 333)\)? | 33 | 0 | 2,846.5 | -1 | 2,846.5 | |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | 60 | 0.25 | 6,489.5625 | 2,585.25 | 7,791 | |
Gari is seated in a jeep, and at the moment, has one 10-peso coin, two 5-peso coins, and six 1-peso coins in his pocket. If he picks four coins at random from his pocket, what is the probability that these will be enough to pay for his jeepney fare of 8 pesos? | 37/42 | 0.3125 | 7,192.5 | 5,613.8 | 7,910.090909 | |
Suppose $f(x) = 6x - 9$ and $g(x) = \frac{x}{3} + 2$. Find $f(g(x)) - g(f(x))$. | 4 | 1 | 1,930.0625 | 1,930.0625 | -1 | |
Define a set \( \mathcal{T} \) of distinct positive integers such that for every integer \( y \) in \( \mathcal{T}, \) the geometric mean of the values obtained by omitting \( y \) from \( \mathcal{T} \) remains a positive integer. In addition, assume that 1 is a member of \( \mathcal{T} \) and the largest element is 2... | 15 | 0 | 7,898.875 | -1 | 7,898.875 | |
Let $w = \dfrac{\sqrt{3} + i}{2}$ and $z = \dfrac{-1 + i\sqrt{3}}{2},$ where $i = \sqrt{-1}.$ Find the number of ordered pairs $(r,s)$ of positive integers not exceeding $100$ that satisfy the equation $i \cdot w^r = z^s.$ | 834 | We rewrite $w$ and $z$ in polar form: \begin{align*} w &= e^{i\cdot\frac{\pi}{6}}, \\ z &= e^{i\cdot\frac{2\pi}{3}}. \end{align*} The equation $i \cdot w^r = z^s$ becomes \begin{align*} e^{i\cdot\frac{\pi}{2}} \cdot \left(e^{i\cdot\frac{\pi}{6}}\right)^r &= \left(e^{i\cdot\frac{2\pi}{3}}\right)^s \\ e^{i\left(\frac{\pi... | 0 | 8,021.9375 | -1 | 8,021.9375 |
Mathematical operation refers to the ability to solve mathematical problems based on clear operation objects and operation rules. Because of operations, the power of numbers is infinite; without operations, numbers are just symbols. Logarithmic operation and exponential operation are two important types of operations. ... | \frac{1}{2} | 0.3125 | 7,465.0625 | 5,865.8 | 8,192 | |
Simplify the expression $\frac{15b^4 - 45b^3}{75b^2}$ when $b=2$. | -\frac{2}{5} | 1 | 2,355.1875 | 2,355.1875 | -1 | |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).
Given points:
\( A_{1}(1, -1, 1) \)
\( A_{2}(-2, 0, 3) \)
\( A_{3}(2, 1, -1) \)
\( A_{4}(2, -2, -4) \) | \frac{33}{\sqrt{101}} | 0 | 6,707.125 | -1 | 6,707.125 | |
Given 6000 cards, each with a unique natural number from 1 to 6000 written on it. It is required to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done? | 179940 | 0.0625 | 7,863.1875 | 7,983 | 7,855.2 | |
A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $A, B$ and $C$, respectively, be $u, v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = w^2$? | $2 + \sqrt{2}$ | 1. **Position the Square and Define Coordinates:**
Place the square in the $xy$-plane such that $A$ is at the origin $(0,0)$, $B$ at $(1,0)$, $C$ at $(1,1)$, and $D$ at $(0,1)$.
2. **Translate the Given Condition:**
We are given that $u^2 + v^2 = w^2$, where $u = PA$, $v = PB$, and $w = PC$. Using the coordinate... | 0 | 5,808.25 | -1 | 5,808.25 |
Let $M = 123456789101112\dots5354$ be the number that results from writing the integers from $1$ to $54$ consecutively. What is the remainder when $M$ is divided by $55$? | 44 | 0 | 7,214.0625 | -1 | 7,214.0625 | |
Find all positive integers $n$ such that the inequality $$\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i$$ holds for any $n$ positive numbers $a_1, \dots, a_n$. | 3 |
To find all positive integers \( n \) such that the given inequality:
\[
\left( \sum_{i=1}^n a_i^2\right) \left(\sum_{i=1}^n a_i \right) -\sum_{i=1}^n a_i^3 \geq 6 \prod_{i=1}^n a_i
\]
holds for any \( n \) positive numbers \( a_1, \dots, a_n \), we proceed as follows:
1. **Case \( n = 1 \):**
- Substitute into ... | 0.1875 | 7,900.625 | 6,638 | 8,192 |
In an isosceles trapezoid, the longer base \(AB\) is 24 units, the shorter base \(CD\) is 12 units, and each of the non-parallel sides has a length of 13 units. What is the length of the diagonal \(AC\)? | \sqrt{457} | 0.8125 | 5,241.375 | 4,931.153846 | 6,585.666667 | |
Mike has two containers. One container is a rectangular prism with width 2 cm, length 4 cm, and height 10 cm. The other is a right cylinder with radius 1 cm and height 10 cm. Both containers sit on a flat surface. Water has been poured into the two containers so that the height of the water in both containers is the sa... | 7.2 | Suppose that the height of the water in each container is $h \mathrm{~cm}$. Since the first container is a rectangular prism with a base that is 2 cm by 4 cm, then the volume of the water that it contains, in $\mathrm{cm}^{3}$, is $2 \times 4 \times h=8h$. Since the second container is a right cylinder with a radius of... | 0 | 3,849.25 | -1 | 3,849.25 |
What is the result if you add the largest odd two-digit number to the smallest even three-digit number? | 199 | 0.9375 | 299.0625 | 297.933333 | 316 | |
The digits $0,1,2,3,4,5,6$ are randomly arranged in a sequence. What is the probability of obtaining a seven-digit number that is divisible by four? (The number cannot start with zero.) | 0.25 | 0 | 7,498.375 | -1 | 7,498.375 | |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $4\cos C \cdot \sin^2 \frac{C}{2} + \cos 2C = 0$.
(1) If $\tan A = 2\tan B$, find the value of $\sin(A-B)$;
(2) If $3ab = 25 - c^2$, find the maximum area of $\triangle ABC$. | \frac{25\sqrt{3}}{16} | 0 | 7,559.375 | -1 | 7,559.375 | |
The function \( f(n) \) is defined on the set of natural numbers \( N \) as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{ if } n \geqslant 1000, \\
f[f(n + 7)], & \text{ if } n < 1000.
\end{cases} \]
What is the value of \( f(90) \)? | 999 | 0.0625 | 8,116.9375 | 8,192 | 8,111.933333 | |
Find the product of all possible real values for $k$ such that the system of equations $$ x^2+y^2= 80 $$ $$ x^2+y^2= k+2x-8y $$ has exactly one real solution $(x,y)$ .
*Proposed by Nathan Xiong* | 960 | 0.9375 | 4,953.6875 | 4,737.8 | 8,192 | |
For positive integers $n$, the number of pairs of different adjacent digits in the binary (base two) representation of $n$ can be denoted as $D(n)$. Determine the number of positive integers less than or equal to 97 for which $D(n) = 2$. | 26 | 0 | 8,192 | -1 | 8,192 | |
Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$
| 742 | 0.3125 | 6,212.9375 | 5,401.8 | 6,581.636364 | |
In the diagram, \( AB \parallel EF \parallel DC \). Given that \( AC + BD = 250 \), \( BC = 100 \), and \( EC + ED = 150 \), find \( CF \). | 60 | 0.0625 | 7,874.5625 | 3,113 | 8,192 | |
Given 1985 sets, each consisting of 45 elements, and the union of any two sets contains exactly 89 elements. How many elements are in the union of all these 1985 sets? | 87341 | 0 | 8,192 | -1 | 8,192 | |
Compute
\[e^{2 \pi i/13} + e^{4 \pi i/13} + e^{6 \pi i/13} + \dots + e^{24 \pi i/13}.\] | -1 | 0.4375 | 7,275.6875 | 6,097.571429 | 8,192 | |
$(1)$ Calculate: $\sqrt{12}-(-\frac{1}{2})^{-1}-|\sqrt{3}+3|+(2023-\pi)^0$<br/>$(2)$ Simplify the algebraic expression $\frac{3x-8}{x-1}-\frac{x+1}{x}÷\frac{{x}^{2}-1}{{x}^{2}-3x}$, and select appropriate integers from $0 \lt x\leqslant 3$ to substitute and find the value. | -1 | 0.5 | 3,592.5625 | 3,266.25 | 3,918.875 | |
Define an ordered quadruple of integers $(a, b, c, d)$ as captivating if $1 \le a < b < c < d \le 15$, and $a+d > 2(b+c)$. How many captivating ordered quadruples are there? | 200 | 0 | 8,149.5625 | -1 | 8,149.5625 | |
A quadrilateral has vertices at $(0,1)$, $(3,4)$, $(4,3)$ and $(3,0)$. Its perimeter can be expressed in the form $a\sqrt2+b\sqrt{10}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$? | 6 | 1 | 2,742.875 | 2,742.875 | -1 | |
Find the integer \( m \) so that
\[ 10^{m} < \frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times \ldots \times \frac{99}{100} < 10^{m+1} \] | -2 | 0.0625 | 8,123.875 | 7,102 | 8,192 | |
A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction. [asy]
size(7cm);
defaultpen(linewidth... | \dfrac{1001}{2003} | 0.4375 | 6,512.5 | 5,250.571429 | 7,494 | |
Calculate the integrals:
1) \(\int_{0}^{\frac{\pi}{2}} \sin ^{3} x \, dx\);
2) \(\int_{0}^{\ln 2} \sqrt{e^{x}-1} \, dx\);
3) \(\int_{-a}^{a} x^{2} \sqrt{a^{2}-x^{2}} \, dx\);
4) \(\int_{1}^{2} \frac{\sqrt{x^{2}-1}}{x} \, dx\). | \sqrt{3} - \frac{\pi}{3} | 0.75 | 6,099.875 | 5,653.916667 | 7,437.75 | |
Two skiers started from the same point one after another with an interval of 9 minutes. The second skier caught up with the first one 9 km from the starting point. After reaching the “27 km” mark, the second skier turned back and met the first skier at a distance of 2 km from the turning point. Find the speed of the se... | 15 | 0.0625 | 7,911 | 5,708 | 8,057.866667 | |
Acute triangle $A B C$ has circumcenter $O$. The bisector of $\angle A B C$ and the altitude from $C$ to side $A B$ intersect at $X$. Suppose that there is a circle passing through $B, O, X$, and $C$. If $\angle B A C=n^{\circ}$, where $n$ is a positive integer, compute the largest possible value of $n$. | 67 | We have $\angle X B C=B / 2$ and $\angle X C B=90^{\circ}-B$. Thus, $\angle B X C=90^{\circ}+B / 2$. We have $\angle B O C=2 A$, so $$90^{\circ}+B / 2=2 A$$ This gives $B=4 A-180^{\circ}$, which gives $C=360^{\circ}-5 A$. In order for $0^{\circ}<B<90^{\circ}$, we need $45^{\circ}<A<67.5^{\circ}$. In order for $0^{\circ... | 0 | 8,192 | -1 | 8,192 |
How many positive integers \( n \) satisfy \[ (n + 9)(n - 4)(n - 13) < 0 \]? | 11 | 0 | 3,019.625 | -1 | 3,019.625 | |
Given $a \in \mathbb{R}$, the function $f(x) = ax^3 - 3x^2$, and $x = 2$ is an extreme point of the function $y = f(x)$.
1. Find the value of $a$.
2. Find the extreme values of the function $f(x)$ in the interval $[-1, 5]$. | 50 | 0.5625 | 3,417.6875 | 3,910.444444 | 2,784.142857 | |
Let the ellipse \\(C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, (a > b > 0)\\) have an eccentricity of \\(\dfrac{2\sqrt{2}}{3}\\), and it is inscribed in the circle \\(x^2 + y^2 = 9\\).
\\((1)\\) Find the equation of ellipse \\(C\\).
\\((2)\\) A line \\(l\\) (not perpendicular to the x-axis) passing through point \\(Q... | -\dfrac{9}{4} | 0.375 | 7,810.8125 | 7,175.5 | 8,192 | |
A pair of natural numbers is called "good" if one of the numbers is divisible by the other. The numbers from 1 to 30 are divided into 15 pairs. What is the maximum number of good pairs that could be formed? | 13 | 0 | 8,192 | -1 | 8,192 | |
Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$ . Let $A_0, A_1, A_2,\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that
\[\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+... | 2\pi | 0 | 8,192 | -1 | 8,192 | |
$ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$ , with $AB=6$ , $BC=7$ , $CD=8$ . Find $AD$ . | \sqrt{51} | 0.125 | 7,756.5625 | 5,190 | 8,123.214286 | |
Tom, Dick, and Harry each flip a fair coin repeatedly until they get their first tail. Calculate the probability that all three flip their coins an even number of times and they all get their first tail on the same flip. | \frac{1}{63} | 0.3125 | 7,288.9375 | 7,011.2 | 7,415.181818 | |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ $\mathbf{d}$ be four distinct unit vectors in space such that
\[\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{c} =\mathbf{b} \cdot \mathbf{d} = \mathbf{c} \cdot \mathbf{d} = -\frac{1}{11}.\]Find $\mathbf{a} \cdot \mathbf{d}.$ | -\frac{53}{55} | 0 | 8,076.8125 | -1 | 8,076.8125 |
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