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 |
|---|---|---|---|---|---|---|
Suppose $$h \cdot a \cdot r \cdot v \cdot a \cdot r \cdot d=m \cdot i \cdot t=h \cdot m \cdot m \cdot t=100$$ Find $(r \cdot a \cdot d) \cdot(t \cdot r \cdot i \cdot v \cdot i \cdot a)$. | 10000 | The answer is $$\frac{\text { harvard } \cdot \text { mit } \cdot \text { mit }}{h m m t}=100^{2}=10000$$ | 0.625 | 6,073.5 | 4,802.4 | 8,192 |
12 soccer teams participate in a round-robin tournament. Each pair of teams plays a match, with the winning team earning 2 points and the losing team earning 0 points. If the match results in a draw, each team earns 1 point. Three referees recorded the total points for all teams and obtained three different results: 3,... | 47 | 0.125 | 7,765.0625 | 6,337.5 | 7,969 | |
The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers? | 4 | Let the two nonzero real numbers be $x$ and $y$. According to the problem, the sum of these two numbers is 4 times their product. This can be expressed as:
\[ x + y = 4xy. \]
We are asked to find the sum of the reciprocals of $x$ and $y$. Let's denote the reciprocals by $a = \frac{1}{x}$ and $b = \frac{1}{y}$. The sum... | 1 | 2,105.6875 | 2,105.6875 | -1 |
What is the coefficient of $x^2$ when $-5x^3 - 5x^2 - 7x + 1$ is multiplied by $-x^2 - 6x + 1$ and the like terms are combined? | 36 | 0.9375 | 4,262 | 4,000 | 8,192 | |
Find all solutions $x$ of the inequality $$\frac{5}{24} + \left|x-\frac{11}{48}\right| < \frac{5}{16}.$$Express your answer in interval notation, simplifying all fractions in your answer. | \left(\frac{1}{8},\frac{1}{3}\right) | 0.875 | 3,197.9375 | 2,484.5 | 8,192 | |
At a school trip, there are 8 students and a teacher. They want to take pictures in groups where each group consists of either 4 or 5 students. How many different group combinations can they make? | 126 | 0.25 | 4,238.5625 | 4,090.25 | 4,288 | |
Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two parabolas with distinct directrices $\ell_1$ and $\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is known that $F_1F_2||\ell_1||\ell_2$ , $F_1$ lies on $\mathcal{P}_2$ , and $F_2$ lies on $\mathcal{P}_1$ . The two parabolas intersect at disti... | 1504 | 0.75 | 6,021.6875 | 5,562 | 7,400.75 | |
A school club buys 1200 candy bars at a price of four for $3 dollars, and sells all the candy bars at a price of three for $2 dollars, or five for $3 dollars if more than 50 are bought at once. Calculate their total profit in dollars. | -100 | 0.5 | 5,917.125 | 5,543.25 | 6,291 | |
Let \[p(x,y) =
\begin{cases} x + y &\quad \text{if } x \ge 0 \text{ and } y \ge 0, \\
x - 2y &\quad \text{if } x < 0 \text{ and } y < 0, \\
3x + y &\quad \text{otherwise}.
\end{cases}
\]What is $p(p(1,-1),p(-5,-2))$? | 5 | 0.875 | 2,127.4375 | 1,972.142857 | 3,214.5 | |
Given the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a>0, b>0$) with the right focus $F$ and left vertex $A$, where $|FA|=2+\sqrt{5}$, the distance from $F$ to the asymptote of $C$ is $1$. A line $l$ passing through point $B(4,0)$ intersects the right branch of the hyperbola $C$ at points $P$ and $Q$. The lin... | -\frac{1}{48} | 0.25 | 7,584.5 | 7,077 | 7,753.666667 | |
If $f(x) = -\dfrac{1}{x},$ what is $f(f(f(f(f(6)))))$? | -\dfrac{1}{6} | 1 | 2,270.25 | 2,270.25 | -1 | |
A large square has each side divided into four equal parts. A square is inscribed such that its vertices touch these division points, as illustrated below. Determine the ratio of the area of the inscribed square to the large square.
[asy]
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((1,0)--(1,0.1)); draw((2,0)--(2,0.1... | \frac{5}{8} | 0 | 7,818.5625 | -1 | 7,818.5625 | |
Let \( A B C D \) be an isosceles trapezoid with \( [A B] \) as the larger base. It is given that the diagonals intersect at a point \( O \) such that \(\frac{O A}{O C}=2\). Given that the area of triangle \( B O C \) is 10, what is the area of the trapezoid \( A B C D \)? | 45 | 0.375 | 7,481.9375 | 6,298.5 | 8,192 | |
There are 99 positive integers, and their sum is 101101. The greatest possible value of the greatest common divisor of these 99 positive integers is: | 101 | 0.125 | 7,831.375 | 7,332 | 7,902.714286 | |
Given a segment \( AB \) of fixed length 3 with endpoints moving on the parabola \( y^2 = x \), find the shortest distance from the midpoint \( M \) of segment \( AB \) to the y-axis. | \frac{5}{4} | 0.5625 | 7,131 | 6,979.555556 | 7,325.714286 | |
Points \( M, N, P, Q \) are taken on the diagonals \( D_1A, A_1B, B_1C, C_1D \) of the faces of cube \( ABCD A_1B_1C_1D_1 \) respectively, such that:
\[ D_1M: D_1A = BA_1: BN = B_1P: B_1C = DQ: DC_1 = \mu, \]
and the lines \( MN \) and \( PQ \) are mutually perpendicular. Find \( \mu \). | \frac{1}{\sqrt{2}} | 0 | 7,416.3125 | -1 | 7,416.3125 | |
The average weight of 8 boys is 160 pounds, and the average weight of 6 girls is 130 pounds. Calculate the average weight of these 14 children. | 147 | 0 | 512.25 | -1 | 512.25 | |
Let the sample space be $\Omega =\{1,2,3,4,5,6,7,8\}$ with equally likely sample points, and events $A=\{1,2,3,4\}$, $B=\{1,2,3,5\}$, $C=\{1,m,n,8\}$, such that $p\left(ABC\right)=p\left(A\right)p\left(B\right)p\left(C\right)$, and satisfying that events $A$, $B$, and $C$ are not pairwise independent. Find $m+n$. | 13 | 0.375 | 6,627.8125 | 4,434.666667 | 7,943.7 | |
Solve the equation $\frac{2}{x}=\frac{1}{x+1}$. | -2 | 1 | 1,918.375 | 1,918.375 | -1 | |
A tetrahedron has a triangular base with sides all equal to 2, and each of its three lateral faces are squares. A smaller tetrahedron is placed within the larger one so that its base is parallel to the base of the larger tetrahedron and its vertices touch the midpoints of the lateral faces of the larger tetrahedron. Ca... | \frac{\sqrt{2}}{12} | 0 | 8,192 | -1 | 8,192 | |
A certain state issues license plates consisting of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates $\boxed{027592}$ and $\boxed{020592}$ cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can us... | \[ 10^5 \] | Consider license plates of $n$ digits, for some fixed $n$ , issued with the same criteria.
We first note that by the pigeonhole principle, we may have at most $10^{n-1}$ distinct plates. Indeed, if we have more, then there must be two plates which agree on the first $n-1$ digits; these plates thus differ only on one d... | 0 | 7,603.6875 | -1 | 7,603.6875 |
Let's consider two positive real numbers $a$ and $b$, where an operation $a \, \blacktriangle \, b$ is defined such that $(ab) \, \blacktriangle \, b = a(b \, \blacktriangle \, b)$ and $(a \, \blacktriangle \, 1) \, \blacktriangle \, a = a \, \blacktriangle \, 1$ for all $a,b>0$. Additionally, it is given that $1 \, \b... | 2070 | 0 | 8,187 | -1 | 8,187 | |
For the set $\{1,2,\cdots,n\}$ and each of its non-empty subsets, define a unique "alternating sum" as follows: Arrange the numbers in each subset in descending order, then start from the largest number and alternately subtract and add subsequent numbers to obtain the alternating sum (for example, the alternating sum o... | 1024 | 0 | 8,192 | -1 | 8,192 | |
There are $168$ primes below $1000$ . Then sum of all primes below $1000$ is, | 76127 | 0.5 | 6,594.1875 | 4,996.375 | 8,192 | |
Let $f(x) = 2x - 3$ and $g(x) = x + 1$. What is the value of $f(1 + g(2))$? | 5 | 1 | 2,468.625 | 2,468.625 | -1 | |
Find $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^{2}}\right)$. | \frac{1}{2} | $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^{2}}\right)=\prod_{n=2}^{\infty} \frac{n^{2}-1}{n^{2}}=\prod_{n=2}^{\infty} \frac{(n-1)(n+1)}{n \cdot n}=\frac{1 \cdot 3}{2 \cdot 2} \frac{2 \cdot 4}{3 \cdot 3} \frac{3 \cdot 5}{4 \cdot 4} \frac{4 \cdot 6}{5 \cdot 5} \frac{5 \cdot 7}{6 \cdot 6} \cdots=\frac{1 \cdot 2 \cdot 3 \cdo... | 0.8125 | 5,603.375 | 5,006 | 8,192 |
If triangle $PQR$ has sides of length $PQ = 7,$ $PR = 6,$ and $QR = 8,$ then calculate
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | \frac{12}{7} | 0.625 | 5,018.75 | 4,626.4 | 5,672.666667 | |
In triangle \(A B C, A B=6, B C=7\) and \(C A=8\). Let \(D, E, F\) be the midpoints of sides \(B C\), \(A C, A B\), respectively. Also let \(O_{A}, O_{B}, O_{C}\) be the circumcenters of triangles \(A F D, B D E\), and \(C E F\), respectively. Find the area of triangle \(O_{A} O_{B} O_{C}\). | \frac{21 \sqrt{15}}{16} | Let \(A B=z, B C=x, C A=y\). Let \(X, Y, Z, O, N\) be the circumcenter of \(A E F, B F D, C D E, A B C, D E F\) respectively. Note that \(N\) is the nine-point center of \(A B C\), and \(X, Y, Z\) are the midpoints of \(O A, O B, O C\) respectively, and thus \(X Y Z\) is the image of homothety of \(A B C\) with center ... | 0 | 8,192 | -1 | 8,192 |
Kolya traveled on an electric scooter to a store in a neighboring village at a speed of 10 km/h. After covering exactly one-third of the total distance, he realized that if he continued at the same speed, he would arrive exactly at the store's closing time. He then doubled his speed. However, after traveling exactly tw... | 6.666666666666667 | 0 | 5,887.5625 | -1 | 5,887.5625 | |
Let
\[\mathbf{A} = \begin{pmatrix} 4 & 1 \\ -9 & -2 \end{pmatrix}.\]Compute $\mathbf{A}^{100}.$ | \begin{pmatrix} 301 & 100 \\ -900 & -299 \end{pmatrix} | 0.8125 | 5,665.6875 | 5,082.692308 | 8,192 | |
In a circle of radius $5$ units, $CD$ and $AB$ are perpendicular diameters. A chord $CH$ cutting $AB$ at $K$ is $8$ units long. The diameter $AB$ is divided into two segments whose dimensions are: | 2,8 | 1. **Identify the center and setup the problem**: Let $O$ be the center of the circle, and let $N$ be the intersection of chord $CH$ with diameter $AB$. Let $ON = a$ and $CN = x$. Since $CD$ and $AB$ are perpendicular diameters, $O$ is the midpoint of $AB$, and the radius of the circle is $5$ units.
2. **Use the Power... | 0 | 5,840.9375 | -1 | 5,840.9375 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. Given that $\cos B = \frac{1}{3}$, $ac = 6$, and $b = 3$.
$(1)$ Find the value of $\cos C$ for side $a$;
$(2)$ Find the value of $\cos (2C+\frac{\pi }{3})$. | \frac{17-56 \sqrt{6}}{162} | 0 | 7,117.6875 | -1 | 7,117.6875 | |
A person bequeathed an amount of money, slightly less than 1500 dollars, to be distributed as follows. His five children and the notary received amounts such that the square root of the eldest son's share, half of the second son's share, the third son's share minus 2 dollars, the fourth son's share plus 2 dollars, the ... | 1464 | 0.3125 | 6,857.75 | 6,085.4 | 7,208.818182 | |
Compute $\dbinom{50}{2}$. | 1225 | 1 | 1,545.5 | 1,545.5 | -1 | |
How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac14$? | 5 | 1 | 3,677.625 | 3,677.625 | -1 | |
Find the number of ordered quintuples $(a,b,c,d,e)$ of nonnegative real numbers such that:
\begin{align*}
a^2 + b^2 + c^2 + d^2 + e^2 &= 5, \\
(a + b + c + d + e)(a^3 + b^3 + c^3 + d^3 + e^3) &= 25.
\end{align*} | 31 | 0.0625 | 8,062.875 | 8,192 | 8,054.266667 | |
The union of sets \( A \) and \( B \), \( A \cup B = \{a_1, a_2, a_3\} \). When \( A \neq B \), pairs \((A, B)\) and \((B, A)\) are considered different. How many such pairs \((A, B)\) are there? | 27 | 0.625 | 5,361.5625 | 4,510 | 6,780.833333 | |
Given the function $f(x)=\sin \frac {x}{2}\cos \frac {x}{2}+\cos ^{2} \frac {x}{2}-1$.
$(1)$ Find the smallest positive period of the function $f(x)$ and the interval where it is monotonically decreasing;
$(2)$ Find the minimum value of the function $f(x)$ on the interval $\left[ \frac {\pi}{4}, \frac {3\pi}{2}\right]$... | - \frac { \sqrt {2}+1}{2} | 0 | 6,938.9375 | -1 | 6,938.9375 | |
In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?
[asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(... | 7 | To find the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry, we first need to understand the symmetry requirements. The figure must have both horizontal and vertical lines of symmetry.
1. **Identify the lines of symmetry**:
- The vertical line of s... | 0 | 8,174.1875 | -1 | 8,174.1875 |
Each side of the square grid is 15 toothpicks long. Calculate the total number of toothpicks used to construct the square grid. | 480 | 0.8125 | 4,864.4375 | 4,197.307692 | 7,755.333333 | |
Given the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ $(a > b > 0)$, where $M$ and $N$ are the left and right vertices of the ellipse, and $P$ is any point on the ellipse. The slopes of the lines $PM$ and $PN$ are ${k_{1}}$ and ${k_{2}}$ respectively, and ${k_{1}}{k_{2}} \neq 0$. If the minimum value of $|{k_... | \frac{\sqrt{3}}{2} | 0 | 5,195.3125 | -1 | 5,195.3125 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $\frac{b}{c} = \frac{2\sqrt{3}}{3}$ and $A + 3C = \pi$.
$(1)$ Find the value of $\cos C$;
$(2)$ Find the value of $\sin B$;
$(3)$ If $b = 3\sqrt{3}$, find the area of $\triangle ABC$. | \frac{9\sqrt{2}}{4} | 0 | 5,400.875 | -1 | 5,400.875 | |
Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \in \mathbb{Q}$,
\[ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). \]
Show that there exists an integer $c$ such that for any aquaesulian fun... | 1 |
Let \( \mathbb{Q} \) be the set of rational numbers. We have a function \( f: \mathbb{Q} \to \mathbb{Q} \) that satisfies the property such that for every \( x, y \in \mathbb{Q} \):
\[ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). \]
Our task is to show that there exists an integer \( c \) such th... | 0.125 | 7,787.0625 | 4,952.5 | 8,192 |
The Lions are competing against the Eagles in a seven-game championship series. The Lions have a probability of $\dfrac{2}{3}$ of winning a game whenever it rains and a probability of $\dfrac{1}{2}$ of winning when it does not rain. Assume it's forecasted to rain for the first three games and the remaining will have no... | 76\% | 0 | 7,283.5 | -1 | 7,283.5 | |
The polynomial $P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}$ has $34$ complex roots of the form $z_k = r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34,$ with $0 < a_1 \le a_2 \le a_3 \le \cdots \le a_{34} < 1$ and $r_k>0.$ Given that $a_1 + a_2 + a_3 + a_4 + a_5 = m/n,$ where $m$ and $n$ are relatively prime positi... | 482 | We see that the expression for the polynomial $P$ is very difficult to work with directly, but there is one obvious transformation to make: sum the geometric series:
\begin{align*} P(x) &= \left(\frac{x^{18} - 1}{x - 1}\right)^2 - x^{17} = \frac{x^{36} - 2x^{18} + 1}{x^2 - 2x + 1} - x^{17}\\ &= \frac{x^{36} - x^{19} - ... | 0.1875 | 7,861.3125 | 7,474.666667 | 7,950.538462 |
Given the line $y=kx+b$ is a tangent to the curve $f\left(x\right)=\ln x+2$ and also a tangent to the curve $g\left(x\right)=\ln \left(x+1\right)$, determine the value of $k-b$. | 1 + \ln 2 | 0.75 | 5,614.4375 | 4,755.25 | 8,192 | |
The Bulls are playing the Knicks in the NBA playoffs. To win this playoff series, a team must secure 4 victories before the other team. If the Knicks win each game with a probability of $\dfrac{3}{5}$ and there are no ties, what is the probability that the Bulls will win the playoff series and that the contest will req... | \frac{864}{15625} | 0 | 4,595.3125 | -1 | 4,595.3125 | |
Given the sequence ${a_n}$, $a_1=1$ and $a_n a_{n+1} + \sqrt{3}(a_n - a_{n+1}) + 1 = 0$. Determine the value of $a_{2016}$. | 2 - \sqrt{3} | 0.625 | 5,012.6875 | 4,417.6 | 6,004.5 | |
In triangle $ABC,$ angle bisectors $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ If $AB = 8,$ $AC = 6,$ and $BC = 4,$ find $\frac{BP}{PE}.$ | \frac{3}{2} | 0 | 5,005 | -1 | 5,005 | |
What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? | 63 | 0.0625 | 7,498.4375 | 8,192 | 7,452.2 | |
The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center? | 130 | 1. **Identify the distance between the third and tenth exits**:
The third exit is at milepost 40 and the tenth exit is at milepost 160. Therefore, the distance between these two exits is calculated as:
\[
160 - 40 = 120 \text{ miles}
\]
2. **Calculate the location of the service center**:
The service c... | 1 | 2,935.625 | 2,935.625 | -1 |
Cindy leaves school at the same time every day. If she cycles at \(20 \ \text{km/h}\), she arrives home at 4:30 in the afternoon. If she cycles at \(10 \ \text{km/h}\), she arrives home at 5:15 in the afternoon. At what speed, in \(\text{km/h}\), must she cycle to arrive home at 5:00 in the afternoon? | 12 | 0.1875 | 5,744.3125 | 3,787.666667 | 6,195.846154 | |
Find the area of triangle $EFC$ given that $[EFC]=\left(\frac{5}{6}\right)[AEC]=\left(\frac{5}{6}\right)\left(\frac{4}{5}\right)[ADC]=\left(\frac{5}{6}\right)\left(\frac{4}{5}\right)\left(\frac{2}{3}\right)[ABC]$ and $[ABC]=20\sqrt{3}$. | \frac{80\sqrt{3}}{9} | By shared bases, we know that $$[EFC]=\left(\frac{5}{6}\right)[AEC]=\left(\frac{5}{6}\right)\left(\frac{4}{5}\right)[ADC]=\left(\frac{5}{6}\right)\left(\frac{4}{5}\right)\left(\frac{2}{3}\right)[ABC]$$ By Heron's formula, we find that $[ABC]=\sqrt{(15)(8)(2)(5)}=20\sqrt{3}$, so $[AEC]=\frac{80\sqrt{3}}{9}$ | 0 | 4,615.25 | -1 | 4,615.25 |
Let \( S = \{1, 2, \cdots, 2005\} \), and \( A \subseteq S \) with \( |A| = 31 \). Additionally, the sum of all elements in \( A \) is a multiple of 5. Determine the number of such subsets \( A \). | \frac{1}{5} \binom{2005}{31} | 0 | 8,178.125 | -1 | 8,178.125 | |
Let $D$ be the set of divisors of 100. Let $Z$ be the set of integers between 1 and 100, inclusive. Mark chooses an element $d$ of $D$ and an element $z$ of $Z$ uniformly at random. What is the probability that $d$ divides $z$? | \frac{217}{900} | As $100=2^{2} \cdot 5^{2}$, there are $3 \cdot 3=9$ divisors of 100, so there are 900 possible pairs of $d$ and $z$ that can be chosen. If $d$ is chosen, then there are $\frac{100}{d}$ possible values of $z$ such that $d$ divides $z$, so the total number of valid pairs of $d$ and $z$ is $\sum_{d \mid 100} \frac{100}{d}... | 0.8125 | 5,678.75 | 5,098.769231 | 8,192 |
In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.
| 306 | 0.0625 | 7,665.9375 | 4,210 | 7,896.333333 | |
Complete the table below, discover the patterns of square roots and cube roots, and apply the patterns to solve the problem.
| $x$ | $\ldots $ | $0.064$ | $0.64$ | $64$ | $6400$ | $64000$ | $\ldots $ |
|---------|-----------|---------|--------|-------|--------|---------|-----------|
| $\sqrt{x}$ | $\ldots $ | $0.... | 208.879 | 0 | 7,771.25 | -1 | 7,771.25 | |
Two positive integers \(m\) and \(n\) are chosen such that \(m\) is the smallest positive integer with only two positive divisors, and \(n\) is the largest integer less than 200 that has exactly four positive divisors. What is \(m+n\)? | 192 | 0 | 5,771.1875 | -1 | 5,771.1875 | |
Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. What is the probability that $abc + ab + a$ is divisible by $3$? | \frac{13}{27} | 1. **Understanding the Problem:**
We need to find the probability that the expression $abc + ab + a$ is divisible by $3$ when $a$, $b$, and $c$ are chosen randomly from the set $\{1, 2, 3, \dots, 2010\}$.
2. **Divisibility Analysis:**
We can simplify the expression modulo $3$. Notice that if $a \equiv 0 \pmod{3}... | 0.6875 | 6,617.3125 | 5,901.545455 | 8,192 |
Let $A$ and $B$ be the endpoints of a semicircular arc of radius $3$. The arc is divided into nine congruent arcs by eight equally spaced points $C_1$, $C_2$, $\dots$, $C_8$. All chords of the form $\overline {AC_i}$ or $\overline {BC_i}$ are drawn. Find the product of the lengths of these sixteen chords. | 387420489 | 0.125 | 7,796.125 | 5,363.5 | 8,143.642857 | |
In the trapezoid \(ABCD\), the lengths of the bases are \(AD = 24\) and \(BC = 8\), and the lengths of the diagonals are \(AC = 13\) and \(BD = 5\sqrt{17}\). Find the area of the trapezoid. | 80 | 1 | 2,788.25 | 2,788.25 | -1 | |
A package of milk with a volume of 1 liter cost 60 rubles. Recently, for the purpose of economy, the manufacturer reduced the package volume to 0.9 liters and increased its price to 81 rubles. By what percentage did the manufacturer's revenue increase? | 50 | 0 | 3,927.9375 | -1 | 3,927.9375 | |
Jacqueline has 40% less sugar than Liliane, and Bob has 30% less sugar than Liliane. Express the relationship between the amounts of sugar that Jacqueline and Bob have as a percentage. | 14.29\% | 0.0625 | 1,948.6875 | 599 | 2,038.666667 | |
Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$. | 19 | Note that the cube root is increasing for positive reals while its derivative is decreasing, so linear approximation gives \[\sqrt[3]{n^3+1} - n < \left.\frac{d\sqrt[3]{x}}{dx}\right|_{x=n^3} = \frac{1}{3n^2}\] and \[\sqrt[3]{n^3+1} - n > \left.\frac{d\sqrt[3]{x}}{dx}\right|_{x=n^3+1} = \frac{1}{3\sqrt[3]{(n^3+1)^2}}\]... | 0.3125 | 7,321.0625 | 5,852.8 | 7,988.454545 |
Given that the probability of player A winning a single game is $\frac{2}{3}$, calculate the probability that A wins the match with a score of 3:1 in a best of five games format. | \frac{8}{27} | 0.3125 | 6,741.125 | 5,925.8 | 7,111.727273 | |
In triangle $ABC$, the angle bisectors are $AD$, $BE$, and $CF$, which intersect at the incenter $I$. If $\angle ACB = 38^\circ$, then find the measure of $\angle AIE$, in degrees. | 71^\circ | 0.0625 | 7,668.9375 | 6,187 | 7,767.733333 | |
Let \( M \) and \( m \) be the maximum and minimum elements, respectively, of the set \( \left\{\left.\frac{3}{a}+b \right\rvert\, 1 \leq a \leq b \leq 2\right\} \). Find the value of \( M - m \). | 5 - 2\sqrt{3} | 1 | 4,536.1875 | 4,536.1875 | -1 | |
Six cards numbered $1$ through $6$ are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order. | 52 | Similar to above, a $1-1$ correspondence between ascending and descending is established by subtracting each number from $7$.
We note that the given condition is equivalent to "cycling" $123456$ for a contiguous subset of it. For example,
$12(345)6 \rightarrow 125346, 124536$
It's not hard to see that no overcount is... | 0 | 8,192 | -1 | 8,192 |
Given \\(a > 0\\), the function \\(f(x)= \frac {1}{3}x^{3}+ \frac {1-a}{2}x^{2}-ax-a\\).
\\((1)\\) Discuss the monotonicity of \\(f(x)\\);
\\((2)\\) When \\(a=1\\), let the function \\(g(t)\\) represent the difference between the maximum and minimum values of \\(f(x)\\) on the interval \\([t,t+3]\\). Find the minimum v... | \frac {4}{3} | 0 | 8,112.5 | -1 | 8,112.5 | |
Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 5$ and $xy + xz + yz = 8.$ Find the largest possible value of $x.$ | \frac{7}{3} | 1 | 3,493.75 | 3,493.75 | -1 | |
Let $g_{1}(x)=\frac{1}{3}\left(1+x+x^{2}+\cdots\right)$ for all values of $x$ for which the right hand side converges. Let $g_{n}(x)=g_{1}\left(g_{n-1}(x)\right)$ for all integers $n \geq 2$. What is the largest integer $r$ such that $g_{r}(x)$ is defined for some real number $x$ ? | 5 | Notice that the series is geometric with ratio $x$, so it converges if $-1<x<1$. Also notice that where $g_{1}(x)$ is defined, it is equal to $\frac{1}{3(1-x)}$. The image of $g_{1}(x)$ is then the interval $\left(\frac{1}{6}, \infty\right)$. The image of $g_{2}(x)$ is simply the values of $g_{1}(x)$ for $x$ in $\left(... | 0 | 8,192 | -1 | 8,192 |
Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school? | 9 |
Let's denote the distance to school as $d$ miles and the usual speed during rush hour as $v$ miles per hour.
1. **Convert time to hours**:
- The time taken in rush hour traffic is 20 minutes, which is $\frac{20}{60} = \frac{1}{3}$ hours.
- The time taken without traffic is 12 minutes, which is $\frac{12}{60} ... | 1 | 1,570.6875 | 1,570.6875 | -1 |
A bug moves in the coordinate plane, starting at $(0,0)$. On the first turn, the bug moves one unit up, down, left, or right, each with equal probability. On subsequent turns the bug moves one unit up, down, left, or right, choosing with equal probability among the three directions other than that of its previous move.... | 1/54 | 0 | 8,192 | -1 | 8,192 | |
In a modified version of a walking game, I play by different rules. On the first move, I stand still, but for each subsequent move $n$ where $2 \le n \le 25$, I take two steps forward if $n$ is prime and three steps backward if $n$ is composite. After completing all 25 moves, I must return to my original starting point... | 27 | 0.9375 | 2,663.6875 | 2,734 | 1,609 | |
Given that the general term of the sequence $\{a_{n}\}$ is ${a}_{n}=97-3n(n∈{N}^{*})$, find the value of $n$ for which the sum of the first $n$ terms of the sequence $\{{a}_{n}{a}_{n+1}{a}_{n+2}\}(n∈{N}^{*})$ reaches its maximum value. | 32 | 0.25 | 7,726.5625 | 6,330.25 | 8,192 | |
Find the number of permutations \(a_1, a_2, \ldots, a_{10}\) of the numbers \(1, 2, \ldots, 10\) such that \(a_{i+1}\) is not less than \(a_i - 1\) for \(i = 1, 2, \ldots, 9\). | 512 | 0 | 8,192 | -1 | 8,192 | |
Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$. What is the ratio of the area of $\triangle DFE$ to the area of quadrilateral $ABEF$? | 1/5 | 1. **Assumption of Coordinates**: Assume $ABCD$ is a unit square for simplicity, with coordinates $A(0, 0)$, $B(0, 1)$, $C(1, 1)$, and $D(1, 0)$.
2. **Locating Point $E$**: Since $E$ is the midpoint of diagonal $BD$, we find $E$ by averaging the coordinates of $B$ and $D$. Thus, $E = \left(\frac{0+1}{2}, \frac{1+0}{2... | 0.75 | 6,957.6875 | 6,546.25 | 8,192 |
Find all integers satisfying the equation $ 2^x\cdot(4\minus{}x)\equal{}2x\plus{}4$. | 0, 1, 2 |
To solve the equation \(2^x \cdot (4 - x) = 2x + 4\) for integer values of \(x\), we will analyze the equation step-by-step.
### Step 1: Simplification and Possible Inspection
First, it's often useful to inspect possible simple integer solutions that might satisfy the given equation, especially small integers. We st... | 0 | 8,192 | -1 | 8,192 |
Consider a rectangular array of single digits \(d_{i,j}\) with 10 rows and 7 columns, such that \(d_{i+1, j} - d_{i, j}\) is always 1 or -9 for all \(1 \leq i \leq 9\) and all \(1 \leq j \leq 7\). For \(1 \leq i \leq 10\), let \(m_{i}\) be the median of \(d_{i,1}, \ldots, d_{i, 7}\). Determine the least and greatest po... | 4.5 | 0 | 8,192 | -1 | 8,192 | |
Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying $f(1)=1$ and for $n \in \mathbb{N}, f(2 n)=2 f(n)$ and $f(2 n+1)=2 f(n)-1$. Determine the sum of all positive integer solutions to $f(x)=19$ that do not exceed 2019. | 1889 | For $n=2^{a_{0}}+2^{a_{1}}+\cdots+2^{a_{k}}$ where $a_{0}>a_{1}>\cdots>a_{k}$, we can show that $f(n)=2^{a_{0}}-2^{a_{1}}-\cdots-2^{a_{k}}=2^{a_{0}+1}-n$ by induction: the base case $f(1)=1$ clearly holds; for the inductive step, when $n$ is even we note that $f(n)=2 f\left(\frac{n}{2}\right)=2\left(2^{a_{0}}-\frac{n}{... | 0.125 | 8,094.8125 | 7,414.5 | 8,192 |
Below is pictured a regular seven-pointed star. Find the measure of angle \(a\) in radians. | \frac{5\pi}{7} | 0 | 6,405.3125 | -1 | 6,405.3125 | |
Given that line $l\_1$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l\_2$ passes through points $C(1,m)$ and $D(-1,m+1)$, find the values of the real number $m$ when $l\_1$ is parallel to $l\_2$ or $l\_1$ is perpendicular to $l\_2$. | -\frac{9}{2} | 0.9375 | 2,773.5 | 2,412.266667 | 8,192 | |
In the diagram, pentagon \( PQRST \) has \( PQ = 13 \), \( QR = 18 \), \( ST = 30 \), and a perimeter of 82. Also, \( \angle QRS = \angle RST = \angle STP = 90^\circ \). The area of the pentagon \( PQRST \) is: | 270 | 0.0625 | 8,046 | 5,856 | 8,192 | |
For a sample of size \( n = 41 \), a biased estimate \( D_{\text{в}} = 3 \) of the population variance is found. Find the unbiased estimate of the population variance. | 3.075 | 0.8125 | 2,803.1875 | 2,312.923077 | 4,927.666667 | |
Calculate the probability that in a family where there is already one child who is a boy, the next child will also be a boy. | 1/3 | 0.125 | 3,336.0625 | 498 | 3,741.5 | |
Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x)=x^{3}+a x^{2}+b x+c\) has three real roots \(x_{1}, x_{2}, x_{3}\), satisfying:
(1) \(x_{2}-x_{1}=\lambda\);
(2) \(x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)\).
Find the maximum value of \(\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}\). | \frac{3\sqrt{3}}{2} | 0 | 8,192 | -1 | 8,192 | |
If $y+4 = (x-2)^2$ and $x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$? | 15 |
We are given the equations:
1. \( y + 4 = (x-2)^2 \)
2. \( x + 4 = (y-2)^2 \)
We are also given that \( x \neq y \) and need to find \( x^2 + y^2 \).
#### Step 1: Manipulate the equations
First, we rewrite the equations:
- From equation 1: \( y = (x-2)^2 - 4 \)
- From equation 2: \( x = (y-2)^2 - 4 \)
#### Step 2: ... | 1 | 4,772.875 | 4,772.875 | -1 |
Raashan, Sylvia, and Ted play the following game. Each starts with $1. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1 to that player. What is the probability that after the bell has ... | \frac{1}{4} | 1. **Initial Setup**: Each player starts with $1. The possible states of money distribution are $(1-1-1)$ and $(2-1-0)$, as $(3-0-0)$ is not possible due to the rules of the game.
2. **State $(1-1-1)$ Analysis**:
- Each player has two choices of whom to give their dollar, leading to $2^3 = 8$ possible outcomes.
... | 0 | 8,192 | -1 | 8,192 |
How many even divisors does $9!$ have? | 140 | 1 | 3,426.625 | 3,426.625 | -1 | |
Evaluate: $81^2 - (x+9)^2$ where $x=45$. | 3645 | 1 | 2,326.75 | 2,326.75 | -1 | |
A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of... | 394 | Let $P_k$ be the probability of emptying the bag when it has $k$ pairs in it. Let's consider the possible draws for the first three cards:
Case 1. We draw a pair on the first two cards. The second card is the same as the first with probability $\frac {1}{2k - 1}$, then we have $k - 1$ pairs left. So this contributes pr... | 0 | 8,192 | -1 | 8,192 |
Given the relationship $P={P}_{0}{e}^{-kt}$ between the concentration of toxic and harmful substances $P$ (unit: $mg/L$) in the exhaust gas and time $t$ (unit: $h$) during the filtration process, determine the percentage of the original toxic and harmful substances that will remain after $5$ hours, given that $20\%$ of... | 57\% | 0.125 | 6,722.25 | 5,784.5 | 6,856.214286 | |
Let $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If no one of these numbers is more than twice any other, then find the minimum value of the product $xyz.$ | \frac{1}{32} | 0.0625 | 8,114.6875 | 6,955 | 8,192 | |
Olga purchases a rectangular mirror (the shaded region) that fits exactly inside a frame. The outer perimeter of the frame measures 60 cm by 80 cm. The width of each side of the frame is 10 cm. What is the area of the mirror?
[asy]
unitsize(0.15inch);
defaultpen(black);
draw(((0,0)--(8,0)--(8,6)--(0,6)--cycle));
draw(... | 2400 \mbox{ cm}^2 | 0 | 1,684.5 | -1 | 1,684.5 | |
Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an observer at $A$, the angle determined by ... | 96 | Let $O$ be the intersection of $BC$ and $AD$. By the Angle Bisector Theorem, $\frac {5}{BO}$ = $\frac {13}{CO}$, so $BO$ = $5x$ and $CO$ = $13x$, and $BO$ + $OC$ = $BC$ = $12$, so $x$ = $\frac {2}{3}$, and $OC$ = $\frac {26}{3}$. Let $P$ be the foot of the altitude from $D$ to $OC$. It can be seen that triangle $DOP$ i... | 0.0625 | 7,515.25 | 6,357 | 7,592.466667 |
In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. The number of cows was: | 7 | 1. **Define Variables:**
Let $x$ be the number of cows and $y$ be the number of chickens.
2. **Set Up Equations:**
- Each cow has 4 legs and each chicken has 2 legs.
- The total number of legs is given by $4x + 2y$.
- The total number of heads (since each animal has one head) is $x + y$.
3. **Translate th... | 0.75 | 3,242.6875 | 1,638.416667 | 8,055.5 |
When simplified, $\log_{16}{32} \cdot \log_{16}{\frac{1}{2}}$ becomes:
**A)** $-\frac{1}{4}$
**B)** $-\frac{5}{16}$
**C)** $\frac{5}{16}$
**D)** $-\frac{1}{16}$
**E)** $0$ | -\frac{5}{16} | 0 | 2,482.0625 | -1 | 2,482.0625 | |
For each positive integer $m$ and $n$ define function $f(m, n)$ by $f(1, 1) = 1$ , $f(m+ 1, n) = f(m, n) +m$ and $f(m, n + 1) = f(m, n) - n$ . Find the sum of all the values of $p$ such that $f(p, q) = 2004$ for some $q$ . | 3007 | 0.3125 | 7,743.1875 | 6,755.8 | 8,192 | |
In the diagram \(PQRS\) is a rhombus. Point \(T\) is the midpoint of \(PS\) and point \(W\) is the midpoint of \(SR\). What is the ratio of the unshaded area to the shaded area? | 1:1 | 0.0625 | 8,035 | 8,192 | 8,024.533333 | |
Each interior angle of a regular polygon measures $140^\circ$. How many sides does the polygon have? | 9 | 1 | 1,306.3125 | 1,306.3125 | -1 |
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