problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
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Let $x_1,$ $x_2,$ $\dots,$ $x_{100}$ be real numbers such that $x_1 + x_2 + \dots + x_{100} = 1$ and
\[\frac{x_1}{1 - x_1} + \frac{x_2}{1 - x_2} + \dots + \frac{x_{100}}{1 - x_{100}} = 1.\]Find
\[\frac{x_1^2}{1 - x_1} + \frac{x_2^2}{1 - x_2} + \dots + \frac{x_{100}^2}{1 - x_{100}}.\] | 0 | 0.4375 | 7,030.1875 | 6,015.857143 | 7,819.111111 | |
Given an isosceles triangle DEF with DE = DF = 5√3, a circle with radius 6 is tangent to DE at E and to DF at F. If the altitude from D to EF intersects the circle at its center, find the area of the circle that passes through vertices D, E, and F. | 36\pi | 0 | 7,664.75 | -1 | 7,664.75 | |
Given the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$, given that $a^{2} + b^{2} - 3c^{2} = 0$, where $c$ is the semi-latus rectum, find the value of $\frac{a + c}{a - c}$. | 3 + 2\sqrt{2} | 0.4375 | 4,680.9375 | 3,411 | 5,668.666667 | |
Given a positive integer \( n \geqslant 2 \), positive real numbers \( a_1, a_2, \ldots, a_n \), and non-negative real numbers \( b_1, b_2, \ldots, b_n \), which satisfy the following conditions:
(a) \( a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n = n \);
(b) \( a_1 a_2 \cdots a_n + b_1 b_2 \cdots b_n = \frac{1... | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Find the distance between the vertices of the hyperbola $9x^2 + 54x - y^2 + 10y + 55 = 0.$ | \frac{2}{3} | 1 | 2,701.625 | 2,701.625 | -1 | |
Convert $12012_3$ to a base 10 integer. | 140 | 0.875 | 4,351.5625 | 3,802.928571 | 8,192 | |
Determine the number of integers $2 \leq n \leq 2016$ such that $n^{n}-1$ is divisible by $2,3,5,7$. | 9 | Only $n \equiv 1(\bmod 210)$ work. Proof: we require $\operatorname{gcd}(n, 210)=1$. Note that $\forall p \leq 7$ the order of $n$ $(\bmod p)$ divides $p-1$, hence is relatively prime to any $p \leq 7$. So $n^{n} \equiv 1(\bmod p) \Longleftrightarrow n \equiv 1(\bmod p)$ for each of these $p$. | 0.0625 | 8,084.4375 | 7,817 | 8,102.266667 |
How many odd positive $3$-digit integers are divisible by $3$ but do not contain the digit $3$? | 96 | 1. **Define the problem**: We need to find the count of odd positive $3$-digit integers divisible by $3$ but not containing the digit $3$.
2. **Set the constraints**:
- The number $\underline{ABC}$ must be divisible by $3$, which implies $A+B+C \equiv 0 \pmod{3}$.
- The digits $A$, $B$, and $C$ cannot be $3$.
... | 0.4375 | 7,058.9375 | 5,937.428571 | 7,931.222222 |
Given a set of data pairs (3,y_{1}), (5,y_{2}), (7,y_{3}), (12,y_{4}), (13,y_{5}) corresponding to variables x and y, the linear regression equation obtained is \hat{y} = \frac{1}{2}x + 20. Calculate the value of \sum\limits_{i=1}^{5}y_{i}. | 120 | 1 | 1,729.1875 | 1,729.1875 | -1 | |
From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______. | \frac{6}{35} | 0.5625 | 7,129.375 | 6,302.888889 | 8,192 | |
In 500 kg of ore, there is a certain amount of iron. After removing 200 kg of impurities, which contain on average 12.5% iron, the iron content in the remaining ore increased by 20%. What amount of iron remains in the ore? | 187.5 | 0 | 5,998.5625 | -1 | 5,998.5625 | |
Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\dots$ , $x_{2011}$ such that \[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\] | 16 | First of all, note that the nonnegative integer condition really does not matter, since even if we have a nonnegative power, there is always a power of $m$ we can multiply to get to non-negative powers. Now we see that our problem is just a matter of m-chopping blocks. What is meant by $m$-chopping is taking an existin... | 0 | 8,192 | -1 | 8,192 |
Suppose that \(a, b, c, d\) are real numbers satisfying \(a \geq b \geq c \geq d \geq 0\), \(a^2 + d^2 = 1\), \(b^2 + c^2 = 1\), and \(ac + bd = \frac{1}{3}\). Find the value of \(ab - cd\). | \frac{2\sqrt{2}}{3} | 0 | 6,117.3125 | -1 | 6,117.3125 | |
Calculate the product of $1101_2 \cdot 111_2$. Express your answer in base 2. | 10010111_2 | 0 | 7,948.1875 | -1 | 7,948.1875 | |
Let \mathbb{N} denote the natural numbers. Compute the number of functions $f: \mathbb{N} \rightarrow\{0,1, \ldots, 16\}$ such that $$f(x+17)=f(x) \quad \text { and } \quad f\left(x^{2}\right) \equiv f(x)^{2}+15 \quad(\bmod 17)$$ for all integers $x \geq 1$ | 12066 | By plugging in $x=0$, we get that $f(0)$ can be either $-1,2$. As $f(0)$ is unrelated to all other values, we need to remember to multiply our answer by 2 at the end. Similarly, $f(1)=-1$ or 2 . Consider the graph $x \rightarrow x^{2}$. It is a binary tree rooted at -1 , and there is an edge $-1 \rightarrow 1$, and a l... | 0 | 8,192 | -1 | 8,192 |
An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. Calculate the area of this triangle. | \frac{3}{2} | 0.875 | 4,932.875 | 4,740.071429 | 6,282.5 | |
Two cars, $A$ and $B$, depart from one city to another. In the first 5 minutes, they traveled the same distance. Then, due to an engine failure, $B$ had to reduce its speed to 2/5 of its original speed, and thus arrived at the destination 15 minutes after car $A$, which continued at a constant speed. If the failure had... | 18 | 0.0625 | 7,989.875 | 4,958 | 8,192 | |
Solve the inequality
\[|x - 1| + |x + 2| < 5.\] | (-3,2) | 1 | 2,947.625 | 2,947.625 | -1 | |
For how many integer values of $n$ between 1 and 500 inclusive does the decimal representation of $\frac{n}{2520}$ terminate? | 23 | 0 | 5,612.25 | -1 | 5,612.25 | |
The increasing sequence consists of all those positive integers which are either powers of 2, powers of 3, or sums of distinct powers of 2 and 3. Find the $50^{\rm th}$ term of this sequence. | 57 | 0 | 8,192 | -1 | 8,192 | |
Two circles $\Gamma_{1}$ and $\Gamma_{2}$ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\Gamma_{1}$. When it reaches $\Gamma_{1}$, it bounces off the circumference and heads back towards $\Gamma_{2}$. The particle continues bouncing off the two circles... | 403 | By symmetry, the particle must bounce off of $\Gamma_{2}$ at points that make angles of $60^{\circ}, 120^{\circ}, 180^{\circ}, 240^{\circ}$, and $300^{\circ}$ with the positive $x$-axis. Similarly, the particle must bounce off of $\Gamma_{1}$ at points that make angles of $30^{\circ}, 90^{\circ}, 150^{\circ}, 210^{\cir... | 0 | 8,192 | -1 | 8,192 |
In a $16 \times 16$ table of integers, each row and column contains at most 4 distinct integers. What is the maximum number of distinct integers that there can be in the whole table? | 49 | First, we show that 50 is too big. Assume for sake of contradiction that a labeling with at least 50 distinct integers exists. By the Pigeonhole Principle, there must be at least one row, say the first row, with at least 4 distinct integers in it; in this case, that is exactly 4 , since that is the maximum number of di... | 0 | 7,910.5625 | -1 | 7,910.5625 |
Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\] | 2, 3, 4, 6, 8, 12, 24 | To determine all positive integers \( n \geq 2 \) that satisfy the given condition, we need to analyze when \( a \equiv b \pmod{n} \) if and only if \( ab \equiv 1 \pmod{n} \) for all \( a \) and \( b \) that are relatively prime to \( n \).
### Step 1: Analyze the given condition
The problem requires:
- \( a \equiv ... | 0 | 8,162.125 | -1 | 8,162.125 |
Chester traveled from Hualien to Lukang in Changhua to participate in the Hua Luogeng Gold Cup Math Competition. Before leaving, his father checked the car’s odometer, which displayed a palindromic number of 69,696 kilometers (a palindromic number reads the same forward and backward). After driving for 5 hours, they ar... | 82.2 | 0.125 | 8,150.5 | 7,860 | 8,192 | |
Joanie takes a $\$6,\!000$ loan to pay for her car. The annual interest rate on the loan is $12\%$. She makes no payments for 4 years, but has to pay back all the money she owes at the end of 4 years. How much more money will she owe if the interest compounds quarterly than if the interest compounds annually? Expres... | \$187.12 | 0 | 7,835.875 | -1 | 7,835.875 | |
What is the least five-digit positive integer which is congruent to 7 (mod 21)? | 10,003 | 0 | 3,917.9375 | -1 | 3,917.9375 | |
If $N$ is represented as $11000_2$ in binary, what is the binary representation of the integer that comes immediately before $N$? | $10111_2$ | 0 | 810.0625 | -1 | 810.0625 | |
In a small reserve, a biologist counted a total of 300 heads comprising of two-legged birds, four-legged mammals, and six-legged insects. The total number of legs counted was 980. Calculate the number of two-legged birds. | 110 | 0.0625 | 7,266.5 | 8,192 | 7,204.8 | |
Given the inequality (e-a)e^x + x + b + 1 ≤ 0, where e is the natural constant, find the maximum value of $\frac{b+1}{a}$. | \frac{1}{e} | 0.5625 | 5,959.5 | 4,268.666667 | 8,133.428571 | |
Compute the square of 1085 without using a calculator. | 1177225 | 0.4375 | 896.625 | 1,364.857143 | 532.444444 | |
What is the least positive integer that is divisible by the first three prime numbers greater than 5? | 1001 | 1 | 1,384.0625 | 1,384.0625 | -1 | |
8. Andrey likes all numbers that are not divisible by 3, and Tanya likes all numbers that do not contain digits that are divisible by 3.
a) How many four-digit numbers are liked by both Andrey and Tanya?
b) Find the total sum of the digits of all such four-digit numbers. | 14580 | 0 | 7,591.625 | -1 | 7,591.625 | |
Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:
$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$,
$\mathcal{A} \cap \mathcal{B} = \emptyset$,
The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
The number ... | 772 | 0.125 | 7,381.625 | 5,359.5 | 7,670.5 | |
How can you measure 15 minutes using a 7-minute hourglass and an 11-minute hourglass? | 15 | 0.25 | 3,989.9375 | 1,680 | 4,759.916667 | |
What time is it 2017 minutes after 20:17? | 05:54 | 0.125 | 508.875 | 537 | 504.857143 | |
Given the following definition: We call a pair of rational numbers $a$ and $b$ that satisfy the equation $a-b=ab+1$ as "companion rational number pairs," denoted as $\left(a,b\right)$. For example, $3-\frac{1}{2}=3\times \frac{1}{2}+1$, $5-\frac{2}{3}=5\times \frac{2}{3}+1$, so the pairs $(3,\frac{1}{2})$ and $(5,\frac... | \frac{1}{2} | 0.9375 | 3,674.25 | 3,373.066667 | 8,192 | |
Given points $A(-2,-2)$, $B(-2,6)$, $C(4,-2)$, and point $P$ moving on the circle $x^{2}+y^{2}=4$, find the maximum value of $|PA|^{2}+|PB|^{2}+|PC|^{2}$. | 88 | 0.75 | 5,641 | 4,790.666667 | 8,192 | |
In the diagram, $COB$ is a sector of a circle with $\angle COB=45^\circ.$ $OZ$ is drawn perpendicular to $CB$ and intersects $CB$ at $W.$ What is the length of $WZ$? Assume the radius of the circle is 10 units. | 10 - 5 \sqrt{2+\sqrt{2}} | 0 | 8,169.9375 | -1 | 8,169.9375 | |
In the geometric sequence $\{a_n\}$, if $a_n > a_{n+1}$, and $a_7 \cdot a_{14} = 6, a_4 + a_{17} = 5$, calculate $\frac{a_5}{a_{18}}$. | \frac{3}{2} | 0.9375 | 4,657.6875 | 4,422.066667 | 8,192 | |
Given that the radius of a hemisphere is 2, calculate the maximum lateral area of the inscribed cylinder. | 4\pi | 0.875 | 4,086.5625 | 4,181.357143 | 3,423 | |
A function $g$ from the integers to the integers is defined as follows:
\[g(n) = \left\{
\begin{array}{cl}
n + 5 & \text{if $n$ is odd}, \\
n/2 & \text{if $n$ is even}.
\end{array}
\right.\]
Suppose $m$ is odd and $g(g(g(m))) = 39.$ Find $m.$ | 63 | 0 | 5,020.125 | -1 | 5,020.125 | |
If parallelogram ABCD has area 48 square meters, and E and F are the midpoints of sides AB and CD respectively, and G and H are the midpoints of sides BC and DA respectively, calculate the area of the quadrilateral EFGH in square meters. | 24 | 0.375 | 7,712.0625 | 6,912.166667 | 8,192 | |
Determine the set of all real numbers $p$ for which the polynomial $Q(x)=x^{3}+p x^{2}-p x-1$ has three distinct real roots. | p>1 \text{ and } p<-3 | First, we note that $x^{3}+p x^{2}-p x-1=(x-1)(x^{2}+(p+1)x+1)$. Hence, $x^{2}+(p+1)x+1$ has two distinct roots. Consequently, the discriminant of this equation must be positive, so $(p+1)^{2}-4>0$, so either $p>1$ or $p<-3$. However, the problem specifies that the quadratic must have distinct roots (since the original... | 0 | 6,111.3125 | -1 | 6,111.3125 |
Given that \( x \) and \( y \) are positive numbers, determine the minimum value of \(\left(x+\frac{1}{y}\right)^{2}+\left(y+\frac{1}{2x}\right)^{2}\). | 3 + 2 \sqrt{2} | 0.375 | 7,091.3125 | 5,717 | 7,915.9 | |
Let $PQRST$ be a convex pentagon with $PQ \parallel RT, QR \parallel PS, QS \parallel PT, \angle PQR=100^\circ, PQ=4, QR=7,$ and $PT = 21.$ Given that the ratio between the area of triangle $PQR$ and the area of triangle $RST$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | 232 | 0 | 8,192 | -1 | 8,192 | |
A class has $50$ students. The math scores $\xi$ of an exam follow a normal distribution $N(100, 10^{2})$. Given that $P(90 \leqslant \xi \leqslant 100)=0.3$, estimate the number of students with scores of $110$ or higher. | 10 | 0.375 | 6,881.75 | 5,104.166667 | 7,948.3 | |
The writer Arthur has $n \ge1$ co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-author, each co-author writes on a note how many books they remember having written with Arthur. Inspecting the numbers on t... | $n\le6$ |
To solve the problem, we need to determine the values of \( n \) for which it is possible that each co-author accurately remembers the number of books written with Arthur, and these numbers correspond to the first \( n \) Fibonacci numbers.
The Fibonacci sequence is defined by:
\[
F_1 = 1, \quad F_2 = 1, \quad F_{k+... | 0 | 7,440.1875 | -1 | 7,440.1875 |
Two positive real numbers have geometric mean $\sqrt{3}$ and harmonic mean $\frac{3}{2}.$ Enter the two numbers, separated by commas. | 1,3 | 0.1875 | 1,996.4375 | 1,671.666667 | 2,071.384615 | |
What is $\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?$ | \frac{7}{12} |
#### Step 1: Evaluate each sum in the fractions
First, we calculate the sums in the numerators and denominators:
- Numerator of the first fraction: $2 + 4 + 6 = 12$
- Denominator of the first fraction: $1 + 3 + 5 = 9$
- Numerator of the second fraction: $1 + 3 + 5 = 9$
- Denominator of the second fraction: $2 + 4 + 6 ... | 1 | 1,722.375 | 1,722.375 | -1 |
Given points $A(-1, 2, 0)$, $B(5, 2, -1)$, $C(2, -1, 4)$, and $D(-2, 2, -1)$ in space, find:
a) the distance from vertex $D$ of tetrahedron $ABCD$ to the intersection point of the medians of the base $ABC$;
b) the equation of the plane $ABC$;
c) the height of the tetrahedron from vertex $D$;
d) the angle between li... | \frac{3}{5} | 0.8125 | 6,779.3125 | 6,492 | 8,024.333333 | |
A furniture store received a batch of office chairs that were identical except for their colors: 15 chairs were black and 18 were brown. The chairs were in demand and were being bought in a random order. At some point, a customer on the store's website discovered that only two chairs were left for sale. What is the pro... | 0.489 | 0 | 5,934.3125 | -1 | 5,934.3125 | |
A permutation of a finite set is a one-to-one function from the set to itself; for instance, one permutation of $\{1,2,3,4\}$ is the function $\pi$ defined such that $\pi(1)=1, \pi(2)=3$, $\pi(3)=4$, and $\pi(4)=2$. How many permutations $\pi$ of the set $\{1,2, \ldots, 10\}$ have the property that $\pi(i) \neq i$ for ... | 945 | For each such $\pi$, the elements of $\{1,2, \ldots, 10\}$ can be arranged into pairs $\{i, j\}$ such that $\pi(i)=j ; \pi(j)=i$. Choosing a permutation $\pi$ is thus tantamount to choosing a partition of $\{1,2, \ldots, 10\}$ into five disjoint pairs. There are 9 ways to pair off the number 1, then 7 ways to pair off ... | 0.9375 | 3,598.3125 | 3,292.066667 | 8,192 |
A $7 \times 7$ board is either empty or contains an invisible $2 \times 2$ ship placed "on the cells." Detectors can be placed on certain cells of the board and then activated simultaneously. An activated detector signals if its cell is occupied by the ship. What is the minimum number of detectors needed to guarantee d... | 16 | 0 | 7,738.8125 | -1 | 7,738.8125 | |
Compute $\arccos(\cos 9).$ All functions are in radians. | 9 - 2\pi | 0.3125 | 6,501.9375 | 3,826 | 7,718.272727 | |
Let the focus of the parabola $y^{2}=8x$ be $F$, and its directrix be $l$. Let $P$ be a point on the parabola, and $PA\perpendicular l$ with $A$ being the foot of the perpendicular. If the angle of inclination of the line $PF$ is $120^{\circ}$, then $|PF|=$ ______. | \dfrac{8}{3} | 0.125 | 7,004.0625 | 5,583 | 7,207.071429 | |
In parallelogram \(ABCD\), the angle at vertex \(A\) is \(60^{\circ}\), \(AB = 73\) and \(BC = 88\). The angle bisector of \(\angle ABC\) intersects segment \(AD\) at point \(E\) and ray \(CD\) at point \(F\). Find the length of segment \(EF\).
1. 9
2. 13
3. 12
4. 15 | 15 | 0 | 6,786.3125 | -1 | 6,786.3125 | |
A line is drawn through the left focus $F_1$ of a hyperbola at an angle of $30^{\circ}$, intersecting the right branch of the hyperbola at point P. If a circle with diameter PF_1 passes through the right focus of the hyperbola, calculate the eccentricity of the hyperbola. | \sqrt{3} | 0.5625 | 6,592 | 5,347.555556 | 8,192 | |
If real numbers \(a\), \(b\), and \(c\) satisfy \(a^{2} + b^{2} + c^{2} = 9\), then what is the maximum value of the algebraic expression \((a - b)^{2} + (b - c)^{2} + (c - a)^{2}\)? | 27 | 0.75 | 6,959.25 | 6,548.333333 | 8,192 | |
Given $f(x)=\cos x+\cos (x+ \frac {π}{2}).$
(1) Find $f( \frac {π}{12})$;
(2) Suppose $α$ and $β∈(- \frac {π}{2},0)$, $f(α+ \frac {3π}{4})=- \frac {3 \sqrt {2}}{5}$, $f( \frac {π}{4}-β)=- \frac {5 \sqrt {2}}{13}$, find $\cos (α+β)$. | \frac {16}{65} | 0.75 | 5,328.4375 | 4,525.333333 | 7,737.75 | |
Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords. | 127 | 0.0625 | 7,383.8125 | 6,927 | 7,414.266667 | |
The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? | 0 | 1 | 1,419.9375 | 1,419.9375 | -1 | |
Find the area of a triangle with side lengths 14, 48, and 50. | 336 | Note that this is a multiple of the 7-24-25 right triangle. The area is therefore $$\frac{14(48)}{2}=336$$. | 1 | 2,459.3125 | 2,459.3125 | -1 |
A picture $3$ feet across is hung in the center of a wall that is $19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture? | 8 | 1. **Identify the total width and picture width**: The wall is $19$ feet wide and the picture is $3$ feet wide.
2. **Set up the equation**: The picture is centered, so the distances from each end of the wall to the nearest edge of the picture are equal. Let $x$ be the distance from the end of the wall to the nearest e... | 1 | 1,957.0625 | 1,957.0625 | -1 |
Find the smallest natural number with 6 as the last digit, such that if the final 6 is moved to the front of the number it is multiplied by 4. | 153846 | 0.8125 | 4,416.875 | 3,545.692308 | 8,192 | |
Given that bag A contains 3 white balls and 5 black balls, and bag B contains 4 white balls and 6 black balls, calculate the probability that the number of white balls in bag A does not decrease after a ball is randomly taken from bag A and put into bag B, and a ball is then randomly taken from bag B and put back into ... | \frac{35}{44} | 0 | 7,908.625 | -1 | 7,908.625 | |
A regular $n$-gon is inscribed in a circle with radius $R$, and its area is equal to $3 R^{2}$. Find $n$. | 12 | 1 | 3,287.4375 | 3,287.4375 | -1 | |
Given a real number \(a\), and for any \(k \in [-1, 1]\), when \(x \in (0, 6]\), the inequality \(6 \ln x + x^2 - 8x + a \leq kx\) always holds. Determine the maximum value of \(a\). | 6 - 6 \ln 6 | 0.1875 | 6,831.6875 | 5,958 | 7,033.307692 | |
Andrey drove a car to the airport in a neighboring city. After one hour of driving at a speed of 60 km/h, he realized that if he did not change his speed, he would be 20 minutes late. He then sharply increased his speed, and as a result, covered the remaining distance at an average speed of 90 km/h and arrived at the a... | 180 | 0.625 | 4,449.625 | 3,217.4 | 6,503.333333 | |
Let point $P$ be a moving point on the ellipse $x^{2}+4y^{2}=36$, and let $F$ be the left focus of the ellipse. The maximum value of $|PF|$ is _________. | 6 + 3\sqrt{3} | 1 | 5,987.25 | 5,987.25 | -1 | |
What is the positive difference of the solutions of $\dfrac{s^2 - 4s - 22}{s + 3} = 3s + 8$? | \frac{27}{2} | 0 | 3,448.4375 | -1 | 3,448.4375 | |
How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$? | 6 | 1. **Use the Pythagorean Theorem**: Given a right triangle with legs $a$ and $b$ and hypotenuse $b+1$, we apply the Pythagorean theorem:
\[
a^2 + b^2 = (b+1)^2
\]
Expanding the square on the right side, we get:
\[
a^2 + b^2 = b^2 + 2b + 1
\]
Simplifying this, we find:
\[
a^2 = 2b + 1
\]... | 0.25 | 7,099.6875 | 3,822.75 | 8,192 |
Points \( A \), \( B \), \( C \), \( D \), and \( E \) are located in 3-dimensional space with \( AB = BC = CD = DE = 3 \) and \( \angle ABC = \angle CDE = 60^\circ \). Additionally, the plane of triangle \( ABC \) is perpendicular to line \( \overline{DE} \). Determine the area of triangle \( BDE \). | 4.5 | 0 | 7,915.4375 | -1 | 7,915.4375 | |
Given that $m$ is an integer and $0 < 3m < 27$, what is the sum of all possible integer values of $m$? | 36 | 0.875 | 1,285.1875 | 1,417 | 362.5 | |
What is the greatest possible value of $x+y$ such that $x^{2} + y^{2} =90$ and $xy=27$? | 12 | 0.9375 | 3,595.4375 | 3,289 | 8,192 | |
The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is | \frac{1}{3} | 1. **Identify the common difference**: Given an arithmetic sequence, the common difference $d$ can be calculated using any two consecutive terms. From the terms $x$ and $2x$, we have:
\[
d = 2x - x = x
\]
However, since $2x$ is the fourth term and $x$ is the second term, the difference $x$ accounts for two ... | 1 | 2,195.9375 | 2,195.9375 | -1 |
Tom's graduating class has 288 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 10 rows and at least 15 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$? | 58 | 0.9375 | 3,234.5 | 3,112.266667 | 5,068 | |
Tam created the mosaic shown using a regular hexagon, squares, and equilateral triangles. If the side length of the hexagon is \( 20 \text{ cm} \), what is the outside perimeter of the mosaic? | 240 | 0.125 | 6,916.3125 | 5,671 | 7,094.214286 | |
In a circle, an inscribed hexagon has three consecutive sides each of length 3, and the other three sides each of length 5. A chord of the circle splits the hexagon into two quadrilaterals: one quadrilateral has three sides each of length 3, and the other quadrilateral has three sides each of length 5. If the length of... | 409 | 0.3125 | 7,374.5 | 5,964.4 | 8,015.454545 | |
If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$ , compute the value of the expression
\[
\left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
\left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
\left(
\frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac... | -8 | 0.3125 | 7,261.25 | 5,854.2 | 7,900.818182 | |
Express the number $15.7$ billion in scientific notation. | 1.57\times 10^{9} | 0.0625 | 409 | 489 | 403.666667 | |
A triangle with side lengths in the ratio 2:3:4 is inscribed in a circle of radius 4. What is the area of the triangle? | 12 | 0 | 7,111.625 | -1 | 7,111.625 | |
Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals | \sqrt{4q+1} | 1. **Identify the roots of the quadratic equation**: Given the quadratic equation $x^2 + px + q = 0$, the roots can be found using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, $a = 1$, $b = p$, and $c = q$. Thus, the roots are:
\[
x = \frac{-p \pm \sqrt{p^2 - 4q}}{2}
\]... | 0.625 | 7,421.3125 | 6,969 | 8,175.166667 |
The average (mean) of a list of 10 numbers is 17. When one number is removed from the list, the new average is 16. What number was removed? | 26 | When 10 numbers have an average of 17, their sum is $10 \times 17=170$. When 9 numbers have an average of 16, their sum is $9 \times 16=144$. Therefore, the number that was removed was $170-144=26$. | 1 | 1,431.0625 | 1,431.0625 | -1 |
Rationalize the denominator of $\frac{5}{4\sqrt{7} + 3\sqrt{13}}$ and express your answer in the form $\frac{A\sqrt{B} + C\sqrt{D}}{E}$ where $B < D$, the fraction is in lowest terms, and all radicals are in simplest radical form. What is $A+B+C+D+E$? | 20 | 0.875 | 4,666 | 4,522.928571 | 5,667.5 | |
Yvan and Zoé play the following game. Let \( n \in \mathbb{N} \). The integers from 1 to \( n \) are written on \( n \) cards arranged in order. Yvan removes one card. Then, Zoé removes 2 consecutive cards. Next, Yvan removes 3 consecutive cards. Finally, Zoé removes 4 consecutive cards.
What is the smallest value of ... | 14 | 0 | 8,080.6875 | -1 | 8,080.6875 | |
Regular decagon $P_1 P_2 \dotsb P_{10}$ is drawn in the coordinate plane with $P_1$ at $(1,0)$ and $P_6$ at $(3,0).$ If $P_n$ is the point $(x_n,y_n),$ compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{10} + y_{10} i).\] | 1023 | 0.625 | 6,446.375 | 5,737 | 7,628.666667 | |
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squ... | \sqrt{2}+\frac{1}{2} | 1. **Understanding the Problem Setup**: We have three 1-inch squares aligned horizontally. The middle square is rotated by 45 degrees and then lowered until it touches the other two squares. We need to find the vertical distance from point $B$ (the top vertex of the rotated square) to the original horizontal line.
2. ... | 0 | 8,046.4375 | -1 | 8,046.4375 |
Alli rolls a fair $8$-sided die twice. What is the probability of rolling numbers that differ by $3$ in her first two rolls? Express your answer as a common fraction. | \frac{1}{8} | 0 | 5,920.3125 | -1 | 5,920.3125 | |
Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 units in length. These lines became the wall... | 21 | 0 | 8,012.1875 | -1 | 8,012.1875 | |
On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3$. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this b... | \frac{35}{72} | 1. **Identify the regions and their point values**: The dartboard is divided into six regions by three radii and two concentric circles. The inner circle is divided into three regions each with a radius of $3$, and the outer ring (between the inner circle and the outer circle with radius $6$) is also divided into three... | 0.375 | 7,048.1875 | 5,717.166667 | 7,846.8 |
Let $[ABCD]$ be a convex quadrilateral with area $2014$ , and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$ . Let $R$ be the intersection of $[AC]$ and $[PQ]$ . Determine $\frac{\overline{RC}}{\overline{RA}}$ . | 2013 | 0.5625 | 5,881.6875 | 4,931.666667 | 7,103.142857 | |
Let tetrahedron $ABCD$ have $AD=BC=30$, $AC=BD=40$, and $AB=CD=50$. For any point $X$ in space, suppose $g(X)=AX+BX+CX+DX$. Determine the least possible value of $g(X)$, expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Report the sum $p+q$. | 101 | 0.125 | 7,919.125 | 6,862.5 | 8,070.071429 | |
A repunit is a positive integer, all of whose digits are 1s. Let $a_{1}<a_{2}<a_{3}<\ldots$ be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute $a_{111}$. | 1223456 | Let $\left\{r_{n}\right\}_{n \geq 0}$ be the repunits (so $r_{0}=1, r_{1}=11$, and so on). We see that for any $n$, there is $$r_{n-1}+r_{n-2}+\cdots+r_{0}<\frac{r_{n}}{10}+\frac{r_{n}}{100}+\cdots<\frac{r_{n}}{9}<r_{n}$$ so $r_{n}$ is only needed when all possible combinations of the first $n$ repunits are exhausted (... | 0 | 8,192 | -1 | 8,192 |
Given the function $f(x) = \ln x + \ln (ax+1) - \frac {3a}{2}x + 1$ ($a \in \mathbb{R}$).
(1) Discuss the intervals of monotonicity for the function $f(x)$.
(2) When $a = \frac {2}{3}$, if the inequality $xe^{x-\frac {1}{2}} + m \geqslant f(x)$ holds true for all $x$, find the minimum value of $m$, where $e$ is the b... | \ln \frac {2}{3} | 0 | 8,192 | -1 | 8,192 | |
Yan is at a point between his house and a park. He has two options to reach the park: He can either walk directly to the park or he can walk back to his house and use his scooter to reach the park. He scoots 10 times faster than he walks, and both routes take the same amount of time. Determine the ratio of Yan's distan... | \frac{9}{11} | 0.375 | 4,026.375 | 2,003.333333 | 5,240.2 | |
Let $a$ and $b$ be angles such that $\sin (a + b) = \frac{3}{4}$ and $\sin (a - b) = \frac{1}{2}.$ Find $\frac{\tan a}{\tan b}.$ | 5 | 0.875 | 4,213.375 | 3,645 | 8,192 | |
Given a list of the first 12 positive integers such that for each $2\le i\le 12$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list, calculate the number of such lists. | 2048 | 0.25 | 8,096.75 | 7,811 | 8,192 | |
Given the numbers \(-2, -1, 0, 1, 2\), arrange them in some order. Compute the difference between the largest and smallest possible values that can be obtained using the iterative average procedure. | 2.125 | 0 | 8,097.625 | -1 | 8,097.625 | |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 3.$ Find the minimum value of
\[\frac{4}{x} + \frac{9}{y} + \frac{16}{z}.\] | 27 | 0.8125 | 5,764.5 | 5,204.307692 | 8,192 | |
Let's call a natural number "remarkable" if all of its digits are different, it does not start with the digit 2, and by removing some of its digits, the number 2018 can be obtained. How many different seven-digit "remarkable" numbers exist? | 1800 | 0.0625 | 7,900.5 | 5,063 | 8,089.666667 |
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