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 |
|---|---|---|---|---|---|---|
If $x = 2$ and $y = 1,$ what is the value of $2\times x - 3 \times y?$ | 1 | 1 | 445.25 | 445.25 | -1 | |
In a nursery group, there are two small Christmas trees and five children. The caregivers want to split the children into two dance circles around each tree, with at least one child in each circle. The caregivers distinguish between the children but not between the trees: two such groupings are considered identical if ... | 50 | 0 | 8,003.625 | -1 | 8,003.625 | |
Five friends all brought some cakes with them when they met. Each of them gave a cake to each of the others. They then ate all the cakes they had just been given. As a result, the total number of cakes they had between them decreased by half. How many cakes did the five friends have at the start? | 40 | 0.125 | 5,948.8125 | 6,198 | 5,913.214286 | |
Let $S$ be the set of all ordered triple of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n\ge 4$. Find the number of such sequences for which $a_n=0$ for some $n$.
| 494 | 0 | 8,092.625 | -1 | 8,092.625 | |
Olya, after covering one-fifth of the way from home to school, realized that she forgot her notebook. If she does not return for it, she will reach school 6 minutes before the bell rings, but if she returns, she will be 2 minutes late. How much time (in minutes) does the journey to school take? | 20 | 0 | 6,261 | -1 | 6,261 | |
There are $n$ students in the math club. When grouped in 4s, there is one incomplete group. When grouped in 3s, there are 3 more complete groups than with 4s, and one incomplete group. When grouped in 2s, there are 5 more complete groups than with 3s, and one incomplete group. What is the sum of the digits of $n^{2}-n$... | 12 | Suppose that, when the $n$ students are put in groups of 2, there are $g$ complete groups and 1 incomplete group. Since the students are being put in groups of 2, an incomplete group must have exactly 1 student in it. Therefore, $n=2g+1$. Since the number of complete groups of 2 is 5 more than the number of complete gr... | 0.25 | 6,405.75 | 3,829.25 | 7,264.583333 |
If $b>1$, $x>0$, and $(2x)^{\log_b 2}-(3x)^{\log_b 3}=0$, then $x$ is | \frac{1}{6} | Given the equation:
\[
(2x)^{\log_b 2} - (3x)^{\log_b 3} = 0
\]
We can rewrite the terms using the properties of exponents:
\[
(2x)^{\log_b 2} = 2^{\log_b 2} \cdot x^{\log_b 2} = b^{\log_b 2} \cdot x^{\log_b 2} = 2 \cdot x^{\log_b 2}
\]
\[
(3x)^{\log_b 3} = 3^{\log_b 3} \cdot x^{\log_b 3} = b^{\log_b 3} \cdot x^{\log_... | 1 | 3,204.6875 | 3,204.6875 | -1 |
Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms? | 15 | 1. **Understanding the problem**: Paula initially had enough paint for 30 rooms, but after losing three cans, she could only paint 25 rooms. We need to find out how many cans she used for these 25 rooms.
2. **Relating cans of paint to rooms**: The loss of three cans resulted in a decrease of 5 rooms that could be pain... | 1 | 1,902.5625 | 1,902.5625 | -1 |
Given that $\sin \alpha + \cos \alpha = -\frac{\sqrt{5}}{2}$ and $\frac{5\pi}{4} < \alpha < \frac{3\pi}{2}$, find the value of $\cos \alpha - \sin \alpha$. | \frac{\sqrt{3}}{2} | 0 | 6,181.8125 | -1 | 6,181.8125 | |
The diagram shows two unshaded circles which touch each other and also touch a larger circle. Chord \( PQ \) of the larger circle is a tangent to both unshaded circles. The length of \( PQ \) is 6 units. What is the area, in square units, of the shaded region? | \frac{9\pi}{2} | 0.0625 | 6,989.6875 | 5,271 | 7,104.266667 | |
A motorist left point A for point D, covering a distance of 100 km. The road from A to D passes through points B and C. At point B, the GPS indicated that 30 minutes of travel time remained, and the motorist immediately reduced speed by 10 km/h. At point C, the GPS indicated that 20 km of travel distance remained, and ... | 100 | 0.125 | 6,700.8125 | 5,276 | 6,904.357143 | |
Given the function $f(x)=\sin(2x+\frac{π}{6})+2\sin^2x$.
$(1)$ Find the center of symmetry and the interval of monotonic decrease of the function $f(x)$;
$(2)$ If the graph of $f(x)$ is shifted to the right by $\frac{π}{12}$ units, resulting in the graph of the function $g(x)$, find the maximum and minimum values o... | -\frac{\sqrt{3}}{2}+1 | 0 | 6,722.875 | -1 | 6,722.875 | |
Let $\{a_n\}$ be an arithmetic sequence, and $S_n$ be the sum of its first $n$ terms, given that $S_5 < S_6$, $S_6=S_7 > S_8$, then the correct conclusion(s) is/are \_\_\_\_\_\_
$(1) d < 0$
$(2) a_7=0$
$(3) S_9 > S_5$
$(4) S_6$ and $S_7$ are both the maximum value of $S_n$. | (1)(2)(4) | 0 | 6,596 | -1 | 6,596 | |
A sphere is inscribed in a right cone with base radius $12$ cm and height $24$ cm, as shown. The radius of the sphere can be expressed as $a\sqrt{c} - a$ cm. What is the value of $a + c$? [asy]
import three; size(120); defaultpen(linewidth(1)); pen dashes = linetype("2 2") + linewidth(1);
currentprojection = orthograph... | 11 | 0.8125 | 4,684.6875 | 3,875.307692 | 8,192 | |
Let $ABCD$ be a square of side length $4$ . Points $E$ and $F$ are chosen on sides $BC$ and $DA$ , respectively, such that $EF = 5$ . Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$ .
*Proposed by Andrew Wu* | 16 | 0.5625 | 6,652.625 | 5,782 | 7,772 | |
Kevin Kangaroo begins his journey on a number line at 0 with a goal of reaching a rock located at 1. However, he hops only $\frac{1}{4}$ of the distance toward the rock with each leap. After each leap, he tires, reducing his subsequent hop to $\frac{1}{4}$ of the remaining distance to the rock. Calculate the total dist... | \frac{14197}{16384} | 0.6875 | 5,827.5 | 4,752.727273 | 8,192 | |
How many integers $n$ (with $1 \le n \le 2021$ ) have the property that $8n + 1$ is a perfect square? | 63 | 0.6875 | 6,315 | 5,461.818182 | 8,192 | |
What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$? | 10 | 1. **Rewrite the expression with negative exponents:**
Given the expression $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$, we first simplify the numerator. Recall that $a^{-1} = \frac{1}{a}$. Thus, the expression becomes:
\[
\frac{2\left(\frac{1}{a}\right) + \frac{\frac{1}{a}}{2}}{a}
\]
2. **Simplify the numerato... | 1 | 2,389.1875 | 2,389.1875 | -1 |
Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if all three triplets are in the starting lineup? | 165 | 1 | 2,581.625 | 2,581.625 | -1 | |
Rationalize the denominator of $\frac{\sqrt{8}+\sqrt{3}}{\sqrt{2}+\sqrt{3}}$. Express your answer in simplest form. | \sqrt{6}-1 | 0.9375 | 3,394.625 | 3,074.8 | 8,192 | |
Four years ago you invested some money at $10\%$ interest. You now have $\$439.23$ in the account. If the interest was compounded yearly, how much did you invest 4 years ago? | 300 | 1 | 2,080.1875 | 2,080.1875 | -1 | |
Mrs. Široká was expecting guests in the evening. First, she prepared 25 open-faced sandwiches. She then calculated that if each guest took two sandwiches, three of them would not have enough. She then thought that if she made 10 more sandwiches, each guest could take three, but four of them would not have enough. This ... | 11 | 0.0625 | 7,026.3125 | 8,192 | 6,948.6 | |
Given the function $f(x) = \sin x + \cos x$, where $x \in \mathbb{R}$,
- (I) Find the value of $f\left(\frac{\pi}{2}\right)$.
- (II) Determine the smallest positive period of the function $f(x)$.
- (III) Calculate the minimum value of the function $g(x) = f\left(x + \frac{\pi}{4}\right) + f\left(x + \frac{3\pi}{4}\righ... | -2 | 0.9375 | 4,765 | 4,717.2 | 5,482 | |
Find the $1314^{\text{th}}$ digit past the decimal point in the decimal expansion of $\dfrac{5}{14}$. | 2 | 0.625 | 3,622.3125 | 3,206.6 | 4,315.166667 | |
Let $r_{1}, \ldots, r_{n}$ be the distinct real zeroes of the equation $x^{8}-14 x^{4}-8 x^{3}-x^{2}+1=0$. Evaluate $r_{1}^{2}+\cdots+r_{n}^{2}$ | 8 | Observe that $x^{8}-14 x^{4}-8 x^{3}-x^{2}+1 =\left(x^{8}+2 x^{4}+1\right)-\left(16 x^{4}+8 x^{3}+x^{2}\right) =\left(x^{4}+4 x^{2}+x+1\right)\left(x^{4}-4 x^{2}-x+1\right)$. The polynomial $x^{4}+4 x^{2}+x+1=x^{4}+\frac{15}{4} x^{2}+\left(\frac{x}{2}+1\right)^{2}$ has no real roots. On the other hand, let $P(x)=x^{4}-... | 0 | 8,192 | -1 | 8,192 |
Given the function \( y = \frac{1}{2}\left(x^{2}-100x+196+\left|x^{2}-100x+196\right|\right) \), calculate the sum of the function values when the variable \( x \) takes on the 100 natural numbers \( 1, 2, 3, \ldots, 100 \). | 390 | 0.75 | 5,392.0625 | 4,903.166667 | 6,858.75 | |
A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $R$ is the ratio of the lesser part to the greater part, then the value of
\[R^{\left(R^{(R^2+R^{-1})}+R^{-1}\right)}+R^{-1}\]is | 2 | 1. **Identify the Relationship**: Given that a line segment is divided into two parts, $w$ (lesser part) and $l$ (greater part), such that the ratio of the lesser part to the greater part is the same as the ratio of the greater part to the whole segment. This can be expressed as:
\[
\frac{w}{l} = \frac{l}{w+l}
... | 0.625 | 5,429.875 | 3,772.6 | 8,192 |
The circumference of the axial cross-section of a cylinder is $90 \text{ cm}$. What is the maximum possible volume of the cylinder? | 3375\pi | 0.625 | 5,757.875 | 4,525.1 | 7,812.5 | |
A machine that records the number of visitors to a museum shows 1,879,564. Note that this number has all distinct digits. What is the minimum number of additional visitors needed for the machine to register another number that also has all distinct digits?
(a) 35
(b) 36
(c) 38
(d) 47
(e) 52 | 38 | 0 | 7,292.1875 | -1 | 7,292.1875 | |
In a meeting room, the first row has a total of 8 seats. Now 3 people are seated, and the requirement is that there should be empty seats to the left and right of each person. Calculate the number of different seating arrangements. | 24 | 0.0625 | 6,961.5625 | 6,205 | 7,012 | |
Find a four-digit number that is a perfect square, knowing that the first two digits, as well as the last two digits, are each equal to each other. | 7744 | 0.875 | 4,846.375 | 4,368.428571 | 8,192 | |
Given that F<sub>1</sub>(-c, 0) and F<sub>2</sub>(c, 0) are the left and right foci of the ellipse G: $$\frac{x^2}{a^2}+ \frac{y^2}{4}=1 \quad (a>0),$$ point M is a point on the ellipse, and MF<sub>2</sub> is perpendicular to F<sub>1</sub>F<sub>2</sub>, with |MF<sub>1</sub>|-|MF<sub>2</sub>|= $$\frac{4}{3}a.$$
(1) Fi... | \frac{9}{2} | 0.5625 | 7,038.9375 | 6,142.111111 | 8,192 | |
Let $a_0=-2,b_0=1$, and for $n\geq 0$, let
\begin{align*}a_{n+1}&=a_n+b_n+\sqrt{a_n^2+b_n^2},\\b_{n+1}&=a_n+b_n-\sqrt{a_n^2+b_n^2}.\end{align*}Find $\frac{1}{a_{2012}} + \frac{1}{b_{2012}}.$ | \frac{1}{2} | 0.625 | 7,009.6875 | 6,300.3 | 8,192 | |
Alice has six magical pies in her pocket - two that make you grow and the rest make you shrink. When Alice met Mary Ann, she blindly took three pies out of her pocket and gave them to Mary. Find the probability that one of the girls has no growth pies. | 0.4 | 0 | 4,353.8125 | -1 | 4,353.8125 | |
In a certain class, there are 28 boys and 22 girls. If 5 students are to be elected to different class committee positions, and it's desired that both boys and girls are represented among the 5 students, how many different election outcomes are possible? | 239297520 | 0.1875 | 5,898.1875 | 5,210 | 6,057 | |
The side of a square has the length $(x-2)$, while a rectangle has a length of $(x-3)$ and a width of $(x+4)$. If the area of the rectangle is twice the area of the square, what is the sum of the possible values of $x$? | 9 | 1 | 1,783.125 | 1,783.125 | -1 | |
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a \rfloor, \lfloor 2a \rfloor) = 0$ for all real numbers $a$. (Note: $\lfloor \nu \rfloor$ is the greatest integer less than or equal to $\nu$.) | (y-2x)(y-2x-1) | Take $P(x,y) = (y-2x)(y-2x-1)$. To see that this works, first note that if $m = \lfloor a \rfloor$, then $2m$ is an integer less than or equal to $2a$, so $2m \leq \lfloor 2a \rfloor$. On the other hand, $m+1$ is an integer strictly greater than $a$, so $2m+2$ is an integer strictly greater than $2a$, so $\lfloor 2a \r... | 0.875 | 5,116.5 | 4,877.785714 | 6,787.5 |
Let \( x \in \left(0, \frac{\pi}{2}\right) \). Find the minimum value of the function \( y = \frac{1}{\sin^2 x} + \frac{12\sqrt{3}}{\cos x} \). | 28 | 0.4375 | 7,370.625 | 6,314.571429 | 8,192 | |
The diagonals of rectangle $PQRS$ intersect at point $X$. If $PS = 10$ and $RS=24$, then what is $\cos \angle PXS$? | \frac{119}{169} | 0.9375 | 3,577.875 | 3,270.266667 | 8,192 | |
Given that the line $y=kx+b$ is tangent to the graph of the function $f\left(x\right)=\frac{1}{2}x^{2}+\ln x$, calculate the minimum value of $k-b$. | \frac{7}{2} | 1 | 3,923.8125 | 3,923.8125 | -1 | |
In the sequence $\{a_n\}$, $a_1=1$, $a_2=2$, and $a_{n+2} - a_n = 1 + (-1)^n$ ($n \in \mathbb{N}^*$), then the sum $a_1 + a_2 + \ldots + a_{51} =$ ? | 676 | 0.4375 | 7,635 | 6,918.857143 | 8,192 | |
For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$ | 51 | Since the coefficients of the polynomial are real, it follows that the non-real roots must come in complex conjugate pairs. Let the first two roots be $m,n$. Since $m+n$ is not real, $m,n$ are not conjugates, so the other pair of roots must be the conjugates of $m,n$. Let $m'$ be the conjugate of $m$, and $n'$ be the c... | 0.5 | 6,988.1875 | 5,784.375 | 8,192 |
14 students attend the IMO training camp. Every student has at least $k$ favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least $k$ , such that this is always possible if: $a)$ The students can be arranged in a circle such tha... | k = 2 | 0.0625 | 8,058.8125 | 6,061 | 8,192 | |
A triangle in a Cartesian coordinate plane has vertices (5, -2), (10, 5) and (5, 5). How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth. | 17.5 | 0.9375 | 3,269.6875 | 2,941.533333 | 8,192 | |
Given the function $f(x)=ax^{2}-2x+1$.
$(1)$ When $a\neq 0$, discuss the monotonicity of the function $f(x)$;
$(2)$ If $\frac {1}{3}\leqslant a\leqslant 1$, and the maximum value of $f(x)$ on $[1,3]$ is $M(a)$, the minimum value is $N(a)$, let $g(a)=M(a)-N(a)$, find the expression of $g(a)$;
$(3)$ Under the condition o... | \frac {1}{2} | 0.375 | 6,696.8125 | 5,032.5 | 7,695.4 | |
What's the coefficient of the $m^4n^4$ term in the expansion of $(m+n)^8$? | 70 | 1 | 2,003.3125 | 2,003.3125 | -1 | |
Given \(\sin \alpha + \sin (\alpha + \beta) + \cos (\alpha + \beta) = \sqrt{3}\), where \(\beta \in \left[\frac{\pi}{4}, \pi\right]\), find the value of \(\beta\). | \frac{\pi}{4} | 0.5625 | 7,563.8125 | 7,075.222222 | 8,192 | |
Let \(ABC\) be a triangle with \(AB=7\), \(BC=9\), and \(CA=4\). Let \(D\) be the point such that \(AB \parallel CD\) and \(CA \parallel BD\). Let \(R\) be a point within triangle \(BCD\). Lines \(\ell\) and \(m\) going through \(R\) are parallel to \(CA\) and \(AB\) respectively. Line \(\ell\) meets \(AB\) and \(BC\) ... | 180 | 0 | 8,192 | -1 | 8,192 | |
Let $p,$ $q,$ $r,$ $s$ be real numbers such that
\[\frac{(p - q)(r - s)}{(q - r)(s - p)} = \frac{3}{4}.\]Find the sum of all possible values of
\[\frac{(p - r)(q - s)}{(p - q)(r - s)}.\] | -1 | 0 | 7,556.4375 | -1 | 7,556.4375 | |
The limiting sum of the infinite series, $\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots$ whose $n$th term is $\frac{n}{10^n}$ is: | \frac{10}{81} | To find the sum of the series $\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots$, where the $n$th term is $\frac{n}{10^n}$, we can use a technique that involves rewriting the series in a more manageable form.
1. **Rewrite the series**: Notice that each term in the series can be expressed as a sum of several term... | 1 | 2,584.625 | 2,584.625 | -1 |
Grandma's garden has three types of apples: Antonovka, Grushovka, and White Naliv. If the amount of Antonovka apples were tripled, the total number of apples would increase by 70%. If the amount of Grushovka apples were tripled, the total number of apples would increase by 50%. By what percentage would the total number... | 80 | 0.9375 | 4,497.5 | 4,251.2 | 8,192 | |
Given the function $f(x) = \cos x \cdot \sin\left(\frac{\pi}{6} - x\right)$,
(1) Find the interval where $f(x)$ is monotonically decreasing;
(2) In $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. If $f(C) = -\frac{1}{4}$, $a=2$, and the area of $\triangle ABC$ i... | 2\sqrt{3} | 0.5 | 7,240.9375 | 6,430.75 | 8,051.125 | |
An 8 by 8 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 5 black squares, can be drawn on the checkerboard?
[asy]
draw((0,0)--(8,0)--(8,8)--(0,8)--cycle);
draw((1,8)--(1,0));
draw((7,8)-... | 73 | 0.0625 | 7,631.875 | 7,534 | 7,638.4 | |
Two cones share a common base and the vertices of both cones and the circumference of the base are all on the same sphere. If the area of the base of the cone is $\frac{3}{16}$ of the area of the sphere, calculate the ratio of the volumes of the two cones. | 1:3 | 0 | 8,192 | -1 | 8,192 | |
The sequence $\{a\_n\}$ is a geometric sequence with the first term $a\_1=4$, and $S\_3$, $S\_2$, $S\_4$ form an arithmetic sequence.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) If $b\_n=\log \_{2}|a\_n|$, let $T\_n$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{b\_n b\_{n+1}}\}$. ... | \frac{1}{16} | 0.5 | 6,692 | 5,192 | 8,192 | |
Ilya Muromets encounters the three-headed Dragon, Gorynych. Each minute, Ilya chops off one head of the dragon. Let $x$ be the dragon's resilience ($x > 0$). The probability $p_{s}$ that $s$ new heads will grow in place of a chopped-off one ($s=0,1,2$) is given by $\frac{x^{s}}{1+x+x^{2}}$. During the first 10 minutes ... | \frac{1 + \sqrt{97}}{8} | 0 | 5,189.25 | -1 | 5,189.25 | |
A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16.$ The set of all possible values of $r$ is an interval $[a,b).$ What is $a+b?$ | \sqrt{15}+8 | To solve this problem, we first need to understand the properties of the ellipse and the circle described in the problem.
1. **Identify the ellipse properties:**
The given ellipse equation is \(x^2 + 16y^2 = 16\). We can rewrite this equation in standard form:
\[
\frac{x^2}{16} + \frac{y^2}{1} = 1
\]
Fr... | 0 | 8,192 | -1 | 8,192 |
In $\Delta ABC$, $AC = BC$, $m\angle DCB = 40^{\circ}$, and $CD \parallel AB$. What is the number of degrees in $m\angle ECD$?
[asy] pair A,B,C,D,E; B = dir(-40); A = dir(-140); D = (.5,0); E = .4 * dir(40);
draw(C--B--A--E,EndArrow); draw(C--D,EndArrow);
label("$A$",A,W); label("$C$",C,NW);label("$B$",B,E);label("$D$... | 40 | 0.5625 | 7,216.875 | 6,523.222222 | 8,108.714286 | |
If $x$, $y$, and $z$ are positive with $xy=20\sqrt[3]{2}$, $xz = 35\sqrt[3]{2}$, and $yz=14\sqrt[3]{2}$, then what is $xyz$? | 140 | 0.875 | 4,037.3125 | 3,443.785714 | 8,192 | |
Let $T$ be a right triangle with sides having lengths $3$ , $4$ , and $5$ . A point $P$ is called *awesome* if P is the center of a parallelogram whose vertices all lie on the boundary of $T$ . What is the area of the set of awesome points? | 1.5 | 0 | 8,192 | -1 | 8,192 | |
Given the ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $(a>b>0)$, the “companion point” of a point M$(x_0, y_0)$ on the ellipse C is defined as $$N\left(\frac{x_0}{a}, \frac{y_0}{b}\right)$$.
(1) Find the equation of the trajectory of the “companion point” N of point M on the ellipse C;
(2) If the “companio... | \sqrt{3} | 0.3125 | 7,939.25 | 7,383.2 | 8,192 | |
Given the island of Zenith has 32500 acres of usable land, each individual requires 2 acres for sustainable living, and the current population of 500 people increases by a factor of 4 every 30 years, calculate the number of years from 2022 when the population reaches its maximum capacity. | 90 | 0 | 7,627.8125 | -1 | 7,627.8125 | |
Read the text below and answer the questions. Everyone knows that $\sqrt{2}$ is an irrational number, and irrational numbers are infinite non-repeating decimals. Therefore, we cannot write out all the decimal parts of $\sqrt{2}$, but since $1 \lt \sqrt{2} \lt 2$, the integer part of $\sqrt{2}$ is $1$. Subtracting the i... | \sqrt{3} - 14 | 0.9375 | 2,469.5625 | 2,358.933333 | 4,129 | |
A test has ten questions. Points are awarded as follows:
- Each correct answer is worth 3 points.
- Each unanswered question is worth 1 point.
- Each incorrect answer is worth 0 points.
A total score that is not possible is: | 29 | 0.1875 | 8,077 | 7,578.666667 | 8,192 | |
Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,100\pi).$ | 100 | 0.25 | 7,954.875 | 7,243.5 | 8,192 | |
Let $f(x) = 4\cos(wx+\frac{\pi}{6})\sin(wx) - \cos(2wx) + 1$, where $0 < w < 2$.
1. If $x = \frac{\pi}{4}$ is a symmetry axis of the function $f(x)$, find the period $T$ of the function.
2. If the function $f(x)$ is increasing on the interval $[-\frac{\pi}{6}, \frac{\pi}{3}]$, find the maximum value of $w$. | \frac{3}{4} | 0.5 | 7,038.625 | 5,935.75 | 8,141.5 | |
Point \( M \) lies on the side of a regular hexagon with side length 12. Find the sum of the distances from point \( M \) to the lines containing the remaining sides of the hexagon. | 36\sqrt{3} | 0.4375 | 7,360.75 | 6,292 | 8,192 | |
In the geometric sequence ${a_n}$ where $q=2$, if the sum of the series $a_2 + a_5 + \dots + a_{98} = 22$, calculate the sum of the first 99 terms of the sequence $S_{99}$. | 77 | 0.625 | 6,103.9375 | 5,246.1 | 7,533.666667 | |
Given that a high school senior year has 12 classes, with exactly 8 classes to be proctored by their own homeroom teachers, find the number of different proctoring arrangements for the math exam. | 4455 | 0.0625 | 5,615.1875 | 5,422 | 5,628.066667 | |
A school plans to purchase two brands of soccer balls, brand A and brand B. It is known that the unit price of brand A soccer balls is $30 less than the unit price of brand B soccer balls. The quantity of brand A soccer balls that can be purchased with $1000 is the same as the quantity of brand B soccer balls that can ... | 60 | 1 | 2,572.4375 | 2,572.4375 | -1 | |
When the vectors $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 3 \end{pmatrix}$ are both projected onto the same vector $\mathbf{v},$ the result is $\mathbf{p}$ in both cases. Find $\mathbf{p}.$ | \begin{pmatrix} 26/29 \\ 65/29 \end{pmatrix} | 0 | 4,521.6875 | -1 | 4,521.6875 | |
In an old estate, the house is surrounded by tall trees arranged in a circle, including spruces, pines, and birches. There are 96 trees in total. These trees have a peculiar property: for any coniferous tree, among the two trees that are two trees away from it, one is coniferous and the other is deciduous; also, among ... | 32 | 0.0625 | 8,192 | 8,192 | 8,192 | |
To survive the coming Cambridge winter, Chim Tu doesn't wear one T-shirt, but instead wears up to FOUR T-shirts, all in different colors. An outfit consists of three or more T-shirts, put on one on top of the other in some order, such that two outfits are distinct if the sets of T-shirts used are different or the sets ... | 144 | We note that there are 4 choices for Chim Tu's innermost T-shirt, 3 choices for the next, and 2 choices for the next. At this point, he has exactly 1 T-shirt left, and 2 choices: either he puts that one on as well or he discards it. Thus, he has a total of $4 \times 3 \times 2 \times 2=48$ outfits, and can survive for ... | 0.3125 | 5,913.125 | 4,638.8 | 6,492.363636 |
How many distinct sequences of five letters can be made from the letters in COMPUTER if each letter can be used only once, each sequence must begin with M, end with R, and the third letter must be a vowel (A, E, I, O, U)? | 36 | 0 | 5,698 | -1 | 5,698 | |
Let point $P$ be a point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$. Let $F_1$ and $F_2$ respectively be the left and right foci of the ellipse, and let $I$ be the incenter of $\triangle PF_1F_2$. If $S_{\triangle IF_1P} + S_{\triangle IPF_2} = 2S_{\triangle IF_1F_2}$, determine the eccentricity... | \frac{1}{2} | 0.25 | 7,702.75 | 6,235 | 8,192 | |
Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is along side $EG$ of the triangle, and side $AB$ is now one-third the length of side $AD$. The altitude from $F$ to side $EG$ is 12 inches, and side $EG$ is 15 inches. Determine the area of rectangle $ABCD$. | \frac{10800}{289} | 0.3125 | 7,224.5625 | 5,551 | 7,985.272727 | |
What is the value of \( \frac{2018-18+20}{2} \)? | 1010 | Evaluating, \( \frac{2018-18+20}{2} = \frac{2000+20}{2} = \frac{2020}{2} = 1010 \). | 0.25 | 3,361.125 | 2,942 | 3,500.833333 |
Let \(C\) be the circle with the equation \(x^2 - 4y - 18 = -y^2 + 6x + 26\). Find the center \((a, b)\) and radius \(r\) of the circle, and compute \(a + b + r\). | 5 + \sqrt{57} | 1 | 3,902.625 | 3,902.625 | -1 | |
Determine the constant term in the expansion of $(x-2+ \frac {1}{x})^{4}$. | 70 | 0.5625 | 6,622.1875 | 5,401.222222 | 8,192 | |
Given that $\overset{⇀}{m}=(2,1)$, $\overset{⇀}{n}=(\sin θ,\cos θ)$, where $θ∈(0, \dfrac{π}{2})$ is the inclination angle of line $l$ passing through point $A(1,4)$, if $\overset{⇀}{m}· \overset{⇀}{n}$ is at its maximum when line $l$ is tangent to the circle $(x+1)^{2}+(y-2)^{2}={r}^{2}(r > 0)$, then $r=$ . | \dfrac{2 \sqrt{5}}{5} | 0 | 5,435.3125 | -1 | 5,435.3125 | |
A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to regulations, the stove should not be located at a distance closer than 50 cm from the main gas pipeline. | 1/16 | 0.375 | 7,943 | 7,728.5 | 8,071.7 | |
The number of circular pipes with an inside diameter of $1$ inch which will carry the same amount of water as a pipe with an inside diameter of $6$ inches is: | 36 | 1. **Assumption of Equal Height**: We assume that all pipes have the same height, which allows us to focus on comparing their cross-sectional areas to determine their water carrying capacities.
2. **Cross-Sectional Area of the Larger Pipe**:
- The diameter of the larger pipe is $6$ inches, so its radius is half of ... | 1 | 1,594.875 | 1,594.875 | -1 |
Of the numbers $\frac{7}{10}, \frac{4}{5}$ and $\frac{3}{4}$, which number is the arithmetic mean of the other two? | \frac34 | 0.9375 | 3,013 | 2,912.733333 | 4,517 | |
In the grid made up of $1 \times 1$ squares, four digits of 2015 are written in the shaded areas. The edges are either horizontal or vertical line segments, line segments connecting the midpoints of adjacent sides of $1 \times 1$ squares, or the diagonals of $1 \times 1$ squares. What is the area of the shaded portion ... | $47 \frac{1}{2}$ | 0 | 4,052.3125 | -1 | 4,052.3125 | |
Yvon has 4 different notebooks and 5 different pens. Determine the number of different possible combinations of notebooks and pens he could bring. | 20 | 0.4375 | 998.875 | 704.714286 | 1,227.666667 | |
Given that construction teams A and B each have a certain number of workers. If team A transfers 90 workers to team B, the total number of workers in team B will be twice that of team A. If team B transfers a certain number of workers to team A, then the total number of workers in team A will be six times that of team ... | 153 | 0.9375 | 4,785.625 | 4,558.533333 | 8,192 | |
Let $\pi$ be a permutation of $\{1,2, \ldots, 2015\}$. With proof, determine the maximum possible number of ordered pairs $(i, j) \in\{1,2, \ldots, 2015\}^{2}$ with $i<j$ such that $\pi(i) \cdot \pi(j)>i \cdot j$. | \[
\binom{2014}{2}
\] | Let $n=2015$. The only information we will need about $n$ is that $n>5 \sqrt[4]{4}$. For the construction, take $\pi$ to be the $n$-cycle defined by $\pi(k)= \begin{cases}k+1 & \text { if } 1 \leq k \leq n-1 \\ 1 & \text { if } k=n\end{cases}$. Then $\pi(i)>i$ for $1 \leq i \leq n-1$. So $\pi(i) \pi(j)>i j$ for at leas... | 0 | 8,192 | -1 | 8,192 |
In $\triangle ABC$, $\angle A= \frac {\pi}{3}$, $BC=3$, $AB= \sqrt {6}$, find $\angle C=$ \_\_\_\_\_\_ and $AC=$ \_\_\_\_\_\_. | \frac{\sqrt{6} + 3\sqrt{2}}{2} | 0 | 4,825.9375 | -1 | 4,825.9375 | |
For a positive integer $n$ , let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$ . For example, $f(96)=1+2+3+6+9=21$ . Find the largest positive integer $n$ such that for all positive integers $k$ , there is some positive integer $a$ such that $f^k... | 15 | 0 | 8,041.25 | -1 | 8,041.25 | |
There are several soldiers forming a rectangular formation with exactly eight columns. If adding 120 people or removing 120 people from the formation can both form a square formation, how many soldiers are there in the original rectangular formation? | 136 | 0.0625 | 8,188.4375 | 8,135 | 8,192 | |
Given the common difference of the arithmetic sequence $\{a_n\}$ is $d$, and the sum of the first $n$ terms is $S_n$, with $a_1=d=1$, find the minimum value of $\frac{S_n+8}{a_n}$. | \frac{9}{2} | 1 | 3,798.3125 | 3,798.3125 | -1 | |
Let $k$ be a real number such that $k > 1$ and
\[\sum_{n=1}^{\infty} \frac{5n-1}{k^n} = \frac{13}{4}.\]Find $k.$ | 3 | 1 | 3,128.4375 | 3,128.4375 | -1 | |
Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$ . Find $w_1+w_2$ . | 27 | 0.125 | 7,965.5 | 6,380 | 8,192 | |
In parallelogram $ABCD$, $\overline{DE}$ is the altitude to the base $\overline{AB}$ and $\overline{DF}$ is the altitude to the base $\overline{BC}$. [Note: Both pictures represent the same parallelogram.] If $DC=12$, $EB=4$, and $DE=6$, then $DF=$ | 6.4 | To solve for $DF$, we first need to find the length of $AB$ using the information given about $EB$ and the properties of a parallelogram.
1. **Identify the length of $AB$**:
Since $ABCD$ is a parallelogram, opposite sides are equal, so $AB = DC = 12$.
2. **Determine the full length of $AE$**:
Since $EB = 4$ and... | 0 | 4,076.0625 | -1 | 4,076.0625 |
How many prime numbers are between 30 and 40? | 2 | 1 | 1,782.1875 | 1,782.1875 | -1 | |
How many positive odd integers greater than 1 and less than 200 are square-free? | 81 | 0 | 7,808.9375 | -1 | 7,808.9375 | |
If $x$ is $20 \%$ of $y$ and $x$ is $50 \%$ of $z$, then what percentage is $z$ of $y$? | 40 \% | Since $x$ is $20 \%$ of $y$, then $x=\frac{20}{100} y=\frac{1}{5} y$. Since $x$ is $50 \%$ of $z$, then $x=\frac{1}{2} z$. Therefore, $\frac{1}{5} y=\frac{1}{2} z$ which gives $\frac{2}{5} y=z$. Thus, $z=\frac{40}{100} y$ and so $z$ is $40 \%$ of $y$. | 1 | 1,762.1875 | 1,762.1875 | -1 |
What is the greatest common divisor of $2^{2024}-1$ and $2^{2015}-1$? | 511 | 0 | 2,586.125 | -1 | 2,586.125 | |
Let \( m \) and \( n \) be positive integers such that \( m > n \). If the last three digits of \( 2012^m \) and \( 2012^n \) are identical, find the smallest possible value of \( m+n \). | 104 | 0.1875 | 7,868.75 | 6,468 | 8,192 | |
The side \( AB \) of a regular hexagon \( ABCDEF \) is equal to \( \sqrt{3} \) and serves as a chord of a certain circle, while the other sides of the hexagon lie outside this circle. The length of the tangent \( CM \), drawn to the same circle from vertex \( C \), is 3. Find the diameter of the circle. | 2\sqrt{3} | 0.3125 | 7,524.875 | 6,057.2 | 8,192 |
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