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 |
|---|---|---|---|---|---|---|
Circles $\mathcal{C}_1, \mathcal{C}_2,$ and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ a... | 27 | Call the centers $O_1, O_2, O_3$, the points of tangency $r_1, r_2, s_1, s_2$ (with $r$ on $t_1$ and $s$ on $t_2$, and $s_2$ on $\mathcal{C}_2$), and the intersection of each common internal tangent to the X-axis $r, s$. $\triangle O_1r_1r \sim \triangle O_2r_2r$ since both triangles have a right angle and have vertica... | 0.0625 | 7,528.9375 | 3,802 | 7,777.4 |
Given $\triangle ABC$ and a point $P$ on one of its sides, call line $\ell$ the $\textit{splitting line}$ of $\triangle ABC$ through $P$ if $\ell$ passes through $P$ and divides $\triangle ABC$ into two polygons of equal perimeter. Let $\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive intege... | 459 | We wish to solve the Diophantine equation $a^2+ab+b^2=3^2 \cdot 73^2$. It can be shown that $3|a$ and $3|b$, so we make the substitution $a=3x$ and $b=3y$ to obtain $x^2+xy+y^2=73^2$ as our new equation to solve for.
Notice that $r^2+r+1=(r-\omega)(r-{\omega}^2)$, where $\omega=e^{i\frac{2\pi}{3}}$. Thus, \[x^2+xy+y^2... | 0 | 8,192 | -1 | 8,192 |
In how many ways can four people sit in a row of five chairs? | 120 | 1 | 2,330 | 2,330 | -1 | |
Find the integer $n$, $12 \le n \le 18$, such that \[n \equiv 9001 \pmod{7}.\] | 13 | 0.9375 | 3,818.875 | 3,527.333333 | 8,192 | |
Point \( O \) is located inside an isosceles right triangle \( ABC \). The distance from \( O \) to the vertex \( A \) of the right angle is 6, to the vertex \( B \) is 4, and to the vertex \( C \) is 8. Find the area of triangle \( ABC \). | 20 + 6\sqrt{7} | 0.5625 | 7,302 | 6,609.777778 | 8,192 | |
In $\triangle ABC$ with side lengths $AB = 13,$ $BC = 14,$ and $CA = 15,$ let $M$ be the midpoint of $\overline{BC}.$ Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}.$ There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ.$ Then $AQ$... | 247 | We use the law of Cosine and get \[AB^2 = AM^2 + BM^2 - 2 AM \cdot BM \cos \angle AMB,\] \[AC^2 = AM^2 + CM^2 + 2 AM \cdot CM \cos \angle AMB \implies\] \[AM^2 = \frac {AB^2 + AC^2}{2}- BM^2 = \sqrt{148} \approx 12.\] We use the power of point $M$ with respect circumcircle $\triangle ABC$ and get \[AM \cdot MP = BM \cd... | 0 | 8,192 | -1 | 8,192 |
There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.
All the students in this class sat at a round table.
- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers ... | 10 | 0 | 8,192 | -1 | 8,192 | |
The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams? | 88 | 1. **Calculate the number of games within the conference:**
- There are 8 teams in the conference.
- Each team plays every other team twice (once at home and once away).
- The number of ways to choose 2 teams from 8 is given by the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n$ is the tota... | 0.4375 | 4,792.4375 | 4,296.142857 | 5,178.444444 |
For every integer $n \ge 1$ , the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$ , $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$ . Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$ . Find the remainder when $\... | 26 | 0 | 8,161.625 | -1 | 8,161.625 | |
Lauren solved the equation $|x-5| = 2$. Meanwhile Jane solved an equation of the form $x^2+ bx + c = 0$ that had the same two solutions for $x$ as Lauren's equation. What is the ordered pair $(b, c)$? | (-10,21) | 1 | 1,293.8125 | 1,293.8125 | -1 | |
What is the largest $2$-digit prime factor of the integer $n = {300\choose 150}$? | 97 | 0.875 | 5,916 | 5,590.857143 | 8,192 | |
The product of three even consecutive positive integers is twenty times their sum. What is the sum of the three integers? | 24 | 1 | 2,529 | 2,529 | -1 | |
How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 38 | 0.0625 | 4,201.625 | 2,395 | 4,322.066667 | |
Mice built an underground house consisting of chambers and tunnels:
- Each tunnel leads from one chamber to another (i.e., none are dead ends).
- From each chamber, exactly three tunnels lead to three different chambers.
- From each chamber, it is possible to reach any other chamber through tunnels.
- There is exactly... | 10 | 0 | 7,436.4375 | -1 | 7,436.4375 | |
Let $A=\{a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{10}, b_{10}\}$, and consider the 2-configuration $C$ consisting of \( \{a_{i}, b_{i}\} \) for all \( 1 \leq i \leq 10, \{a_{i}, a_{i+1}\} \) for all \( 1 \leq i \leq 9 \), and \( \{b_{i}, b_{i+1}\} \) for all \( 1 \leq i \leq 9 \). Find the number of subsets of $C$ that a... | 89 | Let \( A_{n}=\{a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{n}, b_{n}\} \) for \( n \geq 1 \), and consider the 2-configuration \( C_{n} \) consisting of \( \{a_{i}, b_{i}\} \) for all \( 1 \leq i \leq n, \{a_{i}, a_{i+1}\} \) for all \( 1 \leq i \leq n-1 \), and \( \{b_{i}, b_{i+1}\} \) for all \( 1 \leq i \leq n-1 \). Let ... | 0 | 7,650.4375 | -1 | 7,650.4375 |
Given a sequence $\{a_{n}\}$ that satisfies the equation: ${a_{n+1}}+{({-1})^n}{a_n}=3n-1$ ($n∈{N^*}$), calculate the sum of the first $60$ terms of the sequence $\{a_{n}\}$. | 2760 | 0.25 | 7,727.1875 | 6,628.5 | 8,093.416667 | |
In the diagram, if points $ A$ , $ B$ and $ C$ are points of tangency, then $ x$ equals:
[asy]unitsize(5cm);
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dotfactor=3;
pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);
pair O=(0,3/8);
draw((-2/3,9/16)--(2/3,9/16));
draw((-2/3,1/2)--(-sqrt(3)/6,1/2)-... | $\frac{1}{16}$ | 0 | 7,817.1875 | -1 | 7,817.1875 | |
A sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. When every second term is added up, starting with the first term and ending with the second last term, what is the sum? | 2151 | We label the terms $x_{1}, x_{2}, x_{3}, \ldots, x_{2009}, x_{2010}$. Suppose that $S$ is the sum of the odd-numbered terms in the sequence; that is, $S=x_{1}+x_{3}+x_{5}+\cdots+x_{2007}+x_{2009}$. We know that the sum of all of the terms is 5307; that is, $x_{1}+x_{2}+x_{3}+\cdots+x_{2009}+x_{2010}=5307$. Next, we pai... | 0.5 | 7,514.125 | 7,126.625 | 7,901.625 |
Yao Ming has a free throw shooting percentage of 90% during games. What is the probability that he misses one free throw out of three attempts? | 0.243 | 0.5 | 665.625 | 624.875 | 706.375 | |
The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to $3.15$. The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to $4.20$ at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the ... | $1.05 | 1. **Define Variables:**
Let $s$ be the cost of one sandwich, $c$ be the cost of one cup of coffee, and $p$ be the price of one piece of pie.
2. **Set Up Equations:**
From the problem, we have the following equations based on the given checks:
\[
3s + 7c + p = 3.15 \quad \text{(Equation 1)}
\]
\[
... | 0 | 4,142.5625 | -1 | 4,142.5625 |
How many integers are there from 1 to 16500 that
a) are not divisible by 5;
b) are not divisible by either 5 or 3;
c) are not divisible by 5, 3, or 11? | 8000 | 0.9375 | 3,976.875 | 3,918.533333 | 4,852 | |
Calculate the arc lengths of the curves given by equations in the rectangular coordinate system.
$$
y=1+\arcsin x-\sqrt{1-x^{2}}, 0 \leq x \leq \frac{3}{4}
$$ | \sqrt{2} | 0.875 | 5,060 | 4,612.571429 | 8,192 | |
Let $A = (8,0,0),$ $B = (0,-4,0),$ $C = (0,0,6),$ and $D = (0,0,0).$ Find the point $P$ such that
\[AP = BP = CP = DP.\] | (4,-2,3) | 1 | 3,067.0625 | 3,067.0625 | -1 | |
Given 95 numbers \( a_{1}, a_{2}, \cdots, a_{95} \) where each number can only be +1 or -1, find the minimum value of the sum of the products of each pair of these numbers, \( \sum_{1 \leq i<j \leq 95} a_{i} a_{j} \). | 13 | 0 | 5,776.9375 | -1 | 5,776.9375 | |
Given the function $f(x)=\ln x-xe^{x}+ax$ where $a\in \mathbb{R}$.
(Ⅰ) If the function $f(x)$ is monotonically decreasing on $\left[1,+\infty \right)$, find the range of real number $a$.
(Ⅱ) If $a=1$, find the maximum value of $f(x)$. | -1 | 0.375 | 6,186.9375 | 5,949.333333 | 6,329.5 | |
In $\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$? | 61 | 1. **Assign Variables:**
Let $x$ represent $CX$, and let $y$ represent $BX$. Since the circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$, we have $AB = AX = 86$.
2. **Apply Stewart's Theorem:**
Stewart's Theorem states that for a point $X$ on side $BC$ of $\triangle ABC$, th... | 0.625 | 4,795.8125 | 3,707.5 | 6,609.666667 |
If $a$ and $b$ are additive inverses, $c$ and $d$ are multiplicative inverses, and the absolute value of $m$ is 1, find $(a+b)cd-2009m=$ \_\_\_\_\_\_. | 2009 | 0.3125 | 7,922.625 | 8,192 | 7,800.181818 | |
If $a$, $b$, and $c$ are positive integers satisfying $ab+c = bc+a = ac+b = 41$, what is the value of $a+b+c$? | 42 | 1 | 3,935.75 | 3,935.75 | -1 | |
For any number $x$, we are told that $x\&=7-x$ and $\&x = x -7$. What is the value of $\&(12\&)$? | -12 | 0.8125 | 4,089.9375 | 3,574.230769 | 6,324.666667 | |
Let points $A = (0,0)$, $B = (2,3)$, $C = (5,4)$, and $D = (6,0)$. Quadrilateral $ABCD$ is divided into two equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$, where these fractions are in lowest terms. Determine $p + q + r + s$.
A) ... | 58 | 0 | 8,192 | -1 | 8,192 | |
Let $\omega$ be a circle with radius $1$ . Equilateral triangle $\vartriangle ABC$ is tangent to $\omega$ at the midpoint of side $BC$ and $\omega$ lies outside $\vartriangle ABC$ . If line $AB$ is tangent to $\omega$ , compute the side length of $\vartriangle ABC$ . | \frac{2 \sqrt{3}}{3} | 0 | 5,706.0625 | -1 | 5,706.0625 | |
How many degrees are in the sum of the measures of the six numbered angles pictured? [asy]
draw((3,8)--(10,4)--(1,0)--cycle,linewidth(1));
draw((7,8)--(9,0)--(0,4)--cycle,linewidth(1));
label("1",(3,8),SSE);
label("2",(7,8),SSW);
label("3",(10,4),2W);
label("4",(9,0),NW+NNW);
label("5",(1,0),NE+NNE);
label("6",(0,4),2E... | 360^\circ | 0.9375 | 3,208.4375 | 2,876.2 | 8,192 | |
For how many positive integers $a$ does the polynomial $x^{2}-a x+a$ have an integer root? | 1 | Let $r, s$ be the roots of $x^{2}-a x+a=0$. By Vieta's, we have $r+s=r s=a$. Note that if one root is an integer, then both roots must be integers, as they sum to an integer $a$. Then, $$r s-(r+s)+1=1 \Longrightarrow(r-1)(s-1)=1$$ Because we require $r, s$ to be both integers, we have $r-1=s-1= \pm 1$, which yields $r=... | 0.9375 | 6,271.5625 | 6,143.533333 | 8,192 |
Line $a$ is parallel to line $y=2x+4$ and passes through the point $(2,5)$. What is the y-intercept of line $a$? | 1 | 1 | 1,004.25 | 1,004.25 | -1 | |
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$ . Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by $\triangle '$ . Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from t... | 45 | 0.5 | 6,339.0625 | 4,789.75 | 7,888.375 | |
Let the function $f\left( x \right)=\sin \left( wx-\frac{\pi }{6} \right)+\sin \left( wx-\frac{\pi }{2} \right)$, where $0 < w < 3$, and it is known that $f\left( \frac{\pi }{6} \right)=0$,
$(I)$ Find $w$
$(II)$ Stretch the x-coordinates of the points on the graph of $y=f\left( x \right)$ by a factor of $2$ (the y-co... | -\frac{3}{2} | 0.6875 | 5,854.4375 | 5,185.090909 | 7,327 | |
In the Cartesian coordinate system, the center of circle $C$ is at $(2,0)$, and its radius is $\sqrt{2}$. Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. The parametric equation of line $l$ is:
$$
\begin{cases}
x=-t \\
y=1+t
\end{cases} \quad (t \tex... | 3\sqrt{2} | 0.875 | 5,805.75 | 5,464.857143 | 8,192 | |
Pablo is solving a quadratic equation and notices that the ink has smeared over the coefficient of $x$, making it unreadable. He recalls that the equation has two distinct negative, integer solutions. He needs to calculate the sum of all possible coefficients that were under the smeared ink.
Given the equation form: ... | -60 | 0.1875 | 2,697.25 | 2,002 | 2,857.692308 | |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 0.375 | 6,832.8125 | 5,751.5 | 7,481.6 | |
The diagram below shows part of a city map. The small rectangles represent houses, and the spaces between them represent streets. A student walks daily from point $A$ to point $B$ on the streets shown in the diagram, and can only walk east or south. At each intersection, the student has an equal probability ($\frac{1}{... | $\frac{21}{32}$ | 0 | 7,564.375 | -1 | 7,564.375 | |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha\end{cases}$ ($\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\rho\s... | \sqrt {37}-1 | 0 | 7,443.75 | -1 | 7,443.75 | |
Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?
*Proposed by Michael Tang* | 20 | 0.0625 | 8,117.625 | 7,002 | 8,192 | |
We have that $2a + 1 = 1$ and $b - a = 1.$ What is the value of $b$? | 1 | 1 | 1,702.4375 | 1,702.4375 | -1 | |
Consider license plates consisting of a sequence of four digits followed by two letters. Assume each arrangement is equally likely for these plates. What is the probability that such a license plate contains at least one palindrome sequence (either the four-digit sequence or the two-letter sequence)? Express your resul... | \frac{5}{104} | 0.625 | 5,203.0625 | 3,445.3 | 8,132.666667 | |
Calculate: $\frac{145}{273} \times 2 \frac{173}{245} \div 21 \frac{13}{15}=$ | \frac{7395}{112504} | 0 | 4,392.625 | -1 | 4,392.625 | |
Sophie has collected one of each of the first 25 U.S. state quarters based on the order in which the states joined the union. The following graph indicates the number of states that joined the union in each decade. Determine the fraction of Sophie's 25 coins that represent states that joined the union during the decade... | \frac{2}{5} | 0.3125 | 7,564.0625 | 6,862.8 | 7,882.818182 | |
A fifteen-digit integer is formed by repeating a positive five-digit integer three times. For example, 42563,42563,42563 or 60786,60786,60786 are integers of this form. What is the greatest common divisor of all fifteen-digit integers in this form? | 10000100001 | 0 | 8,192 | -1 | 8,192 | |
For a positive integer $n$, let, $\tau(n)$ be the number of positive integer divisors of $n$. How many integers $1 \leq n \leq 50$ are there such that $\tau(\tau(n))$ is odd? | 17 | Note that $\tau(n)$ is odd if and only if $n$ is a perfect square. Thus, it suffices to find the number of integers $n$ in the given range such that $\tau(n)=k^{2}$ for some positive integer $k$. If $k=1$, then we obtain $n=1$ as our only solution. If $k=2$, we see that $n$ is either in the form $p q$ or $p^{3}$, where... | 0.0625 | 8,024.625 | 8,192 | 8,013.466667 |
In a parallelogram, the lengths of the sides are given as $5$, $10y-2$, $3x+5$, and $12$. Determine the value of $x+y$. | \frac{91}{30} | 0.375 | 5,385.1875 | 2,517.333333 | 7,105.9 | |
Find the sum of $231_5 + 414_5 + 123_5$. Express your answer in base $5$. | 1323_5 | 0.5 | 6,548 | 4,979.5 | 8,116.5 | |
Given an ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) with its left and right foci being $F_1$ and $F_2$ respectively, and a point $$P(1, \frac {3}{2})$$ on the ellipse such that the line connecting $P$ and the right focus of the ellipse is perpendicular to the x-axis.
(1) Find the equation o... | - \frac { \sqrt {3}}{12} | 0 | 7,503.5625 | -1 | 7,503.5625 | |
Kenton watched 2000 adult men and women board a cruise ship. Half of the adults were women. If 20$\%$ of the women and 9$\%$ of the men were wearing sunglasses, what was the total number of men and women wearing sunglasses? | 290 | 1 | 1,254.25 | 1,254.25 | -1 | |
The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers? | 12 | 1 | 1,515.75 | 1,515.75 | -1 | |
Given the function $f(x) = \frac{m}{x} - m + \ln x$ (where $m$ is a constant).
1. Find the monotonic intervals of $f(x)$.
2. For what values of $m$ does $f(x) \geq 0$ hold true? | m = 1 | 0.9375 | 5,399.5 | 5,340.333333 | 6,287 | |
Given $f(x)=x^{3}+ax^{2}+bx+a^{2}$, the extreme value at $x=1$ is $10$. Find the value of $a+b$. | -7 | 1 | 3,908.0625 | 3,908.0625 | -1 | |
Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?
[asy]
size(150);
defaultpen(linewidth(... | 540 | 0.3125 | 6,965.0625 | 5,372.6 | 7,688.909091 | |
Given that the area of $\triangle ABC$ is $S$, and $\overrightarrow{AB} \cdot \overrightarrow{AC} = S$.
(1) Find the values of $\sin A$, $\cos A$, and $\tan 2A$.
(2) If $B = \frac{\pi}{4}, \; |\overrightarrow{CA} - \overrightarrow{CB}| = 6$, find the area $S$ of $\triangle ABC$. | 12 | 0.75 | 6,090.5625 | 5,631 | 7,469.25 | |
A square flag has a green cross of uniform width with a yellow square in the center on a white background. The cross is symmetric with respect to each of the diagonals of the square. If the entire cross (both the green arms and the yellow center) occupies 49% of the area of the flag, what percent of the area of the fla... | 25.14\% | 0 | 8,099.1875 | -1 | 8,099.1875 | |
Josh writes the numbers $2,4,6,\dots,198,200$. He marks out $2$, skips $4$, marks out $6$ and continues this pattern of skipping one number and marking the next until he reaches the end of the list. He then returns to the beginning and repeats this pattern on the new list of remaining numbers, continuing until only one... | 128 | 0.25 | 6,884.5625 | 5,915.75 | 7,207.5 | |
We have learned methods to factorize a polynomial, such as the method of common factors and the application of formulas. In fact, there are other methods for factorization, such as grouping method and splitting method.<br/>① Grouping method:<br/>For example, $x^{2}-2xy+y^{2}-25=(x^{2}-2xy+y^{2})-25=(x-y)^{2}-5^{2}=(x-y... | 14 | 0.875 | 4,621.75 | 4,111.714286 | 8,192 | |
Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$.
(1) Find the smallest positive period and the interval where the function is decreasing;
(2) In triangle $ABC$, the lengths of the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, where $a=7$. If acute angle $A$ satisfies $f(\fra... | 10\sqrt{3} | 0.6875 | 6,809.1875 | 6,180.636364 | 8,192 | |
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$
| 901 | 0.6875 | 5,966.5625 | 5,202.272727 | 7,648 | |
A three-digit natural number $abc$ is termed a "convex number" if and only if the digits $a$, $b$, and $c$ (representing the hundreds, tens, and units place, respectively) satisfy $a < b$ and $c < b$. Given that $a$, $b$, and $c$ belong to the set $\{5, 6, 7, 8, 9\}$ and are distinct from one another, find the probabil... | \frac {1}{3} | 0.6875 | 5,951.3125 | 4,932.818182 | 8,192 | |
Given that vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular and $|\overrightarrow{OA}| = |\overrightarrow{OB}| = 24$. If $t \in [0,1]$, the minimum value of $|t \overrightarrow{AB} - \overrightarrow{AO}| + \left|\frac{5}{12} \overrightarrow{BO} - (1-t) \overrightarrow{BA}\right|$ is: | 26 | 0 | 8,192 | -1 | 8,192 | |
How many even integers are there between $200$ and $700$ whose digits are all different and come from the set $\{1,2,5,7,8,9\}$? | 12 | To find the number of even integers between $200$ and $700$ with all different digits from the set $\{1,2,5,7,8,9\}$, we analyze the conditions:
1. **The number must be even**: This implies that the last digit (units place) of the number must be an even digit. From the given set, the possible even digits are $2$ and $... | 0.125 | 7,631.4375 | 4,661 | 8,055.785714 |
The numbers \( a, b, c, d \) belong to the interval \([-7.5, 7.5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 240 | 0.1875 | 7,690.875 | 5,519.333333 | 8,192 | |
Four people sit around a circular table, and each person will roll a standard six-sided die. What is the probability that no two people sitting next to each other will roll the same number after they each roll the die once? Express your answer as a common fraction. | \frac{35}{72} | 0.125 | 7,948.5625 | 6,244.5 | 8,192 | |
Positive numbers \(a\), \(b\), and \(c\) satisfy the following equations:
\[ a^{2} + a b + b^{2} = 1 \]
\[ b^{2} + b c + c^{2} = 3 \]
\[ c^{2} + c a + a^{2} = 4 \]
Find \(a + b + c\). | \sqrt{7} | 0.8125 | 6,248.0625 | 5,799.461538 | 8,192 | |
Xiaofan checked the step count on the smartwatch app before going out and found that the step count was a two-digit number. After walking downstairs, he found that the tens digit and the units digit had swapped. When he reached the entrance of the residential area, he found that there was an extra $1$ between the two d... | 26 | 0 | 7,734.25 | -1 | 7,734.25 | |
In right triangle $\triangle ABC$ with hypotenuse $\overline{AB}$, $AC = 15$, $BC = 20$, and $\overline{CD}$ is the altitude to $\overline{AB}$. Let $\omega$ be the circle having $\overline{CD}$ as a diameter. Let $I$ be a point outside $\triangle ABC$ such that $\overline{AI}$ and $\overline{BI}$ are both tangent to c... | 97 | 0 | 8,192 | -1 | 8,192 | |
A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ -2 \end{pmatrix} + t \begin{pmatrix} 3 \\ 4 \end{pmatrix}.\]A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -8 \\ 12 \end{pmatrix} + u \begin{pmatrix} 1 \\ 3 \end{pmatrix}.\]If $\theta$ is... | \frac{3}{\sqrt{10}} | 0 | 2,575.5 | -1 | 2,575.5 | |
Given 5 distinct real numbers, any two of which are summed to yield 10 sums. Among these sums, the smallest three are 32, 36, and 37, and the largest two are 48 and 51. What is the largest of these 5 numbers? | 27.5 | 0 | 7,718.9375 | -1 | 7,718.9375 | |
What is the maximum number of diagonals of a regular $12$ -gon which can be selected such that no two of the chosen diagonals are perpendicular?
Note: sides are not diagonals and diagonals which intersect outside the $12$ -gon at right angles are still considered perpendicular.
*2018 CCA Math Bonanza Tiebreaker Rou... | 24 | 0 | 8,192 | -1 | 8,192 | |
Farmer James has some strange animals. His hens have 2 heads and 8 legs, his peacocks have 3 heads and 9 legs, and his zombie hens have 6 heads and 12 legs. Farmer James counts 800 heads and 2018 legs on his farm. What is the number of animals that Farmer James has on his farm? | 203 | Note that each animal has 6 more legs than heads. Thus, if there are $n$ animals, then there are $6 n$ more legs than heads in total. There are $2018-800=1218$ more legs than heads in total, so there are $\frac{1218}{6}=203$ animals. | 0.9375 | 4,164.125 | 3,895.6 | 8,192 |
The common ratio of the geometric sequence \( a+\log _{2} 3, a+\log _{1} 3, a+\log _{8} 3 \) is ______. | \frac{1}{3} | 0 | 7,972.625 | -1 | 7,972.625 | |
Two rectangles, one measuring \(8 \times 10\) and the other \(12 \times 9\), are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction. | 65 | 0 | 4,131.8125 | -1 | 4,131.8125 | |
Solve
\[\frac{1}{x + 9} + \frac{1}{x + 7} = \frac{1}{x + 10} + \frac{1}{x + 6}.\] | -8 | 0.75 | 3,880.9375 | 2,811.916667 | 7,088 | |
When the municipal government investigated the relationship between changes in citizens' income and tourism demand, a sample of 5000 people was randomly selected using the independence test method. The calculation showed that $K^{2}=6.109$. Based on this data, the municipal government asserts that the credibility of th... | 97.5\% | 0.1875 | 7,027.4375 | 6,719.666667 | 7,098.461538 | |
Determine the maximal size of a set of positive integers with the following properties:
(1) The integers consist of digits from the set \(\{1,2,3,4,5,6\}\).
(2) No digit occurs more than once in the same integer.
(3) The digits in each integer are in increasing order.
(4) Any two integers have at least one digit in... | 32 | 0 | 8,192 | -1 | 8,192 | |
The area of a right triangle is 1, and its hypotenuse is \(\sqrt{5}\). Find the cosine of the acute angle between the medians of the triangle that are drawn to its legs. | \frac{5\sqrt{34}}{34} | 0 | 6,379.5625 | -1 | 6,379.5625 | |
For functions $f(x)$ and $g(x)$, let $m\in \{x|f(x)=0\}$, $n\in \{x|g(x)=0\}$. If there exist $m$ and $n$ such that $|m-n|\leqslant 1$, then $f(x)$ and $g(x)$ are called "zero-point related functions". If the functions $f(x)=e^{x-2}+\ln(x-1)-1$ and $g(x)=x(\ln x-ax)-2$ are "zero-point related functions", then the minim... | -2 | 0.375 | 7,613.0625 | 6,648.166667 | 8,192 | |
Point $P$ is inside equilateral $\triangle ABC$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. Given that $PQ=1$, $PR=2$, and $PS=3$, what is $AB$ in terms of radicals? | 4\sqrt{3} | 0.9375 | 2,586.4375 | 2,212.733333 | 8,192 | |
Two trucks are transporting identical sacks of flour from France to Spain. The first truck carries 118 sacks, and the second one carries only 40. Since the drivers of these trucks lack the pesetas to pay the customs duty, the first driver leaves 10 sacks with the customs officers, after which they only need to pay 800 ... | 1600 | 0 | 8,192 | -1 | 8,192 | |
Given a triangle $ABC$ with angles $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$ , let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$ , and $Q$ be the foot of the perpendicular from $B$ onto $AC$ . Denote by $X$ the intersection point o... | 1132 | 0.125 | 8,095.0625 | 7,416.5 | 8,192 | |
A chess piece called "the four-liner" attacks two vertical and two horizontal squares adjacent to the square on which it stands. What is the maximum number of non-attacking four-liners that can be placed on a $10 \times 10$ board? | 25 | 0 | 8,192 | -1 | 8,192 | |
Two real numbers $a$ and $b$ satisfy $a+b=8$ and $a^3+b^3=172$. Compute the value of $ab$. | \frac{85}{6} | 1 | 2,576.4375 | 2,576.4375 | -1 | |
In trapezoid \( KLMN \), the lengths of the bases are \( KN = 25 \), \( LM = 15 \), and the lengths of the legs are \( KL = 6 \), \( MN = 8 \). Find the length of the segment connecting the midpoints of the bases. | 20 | 0 | 4,846.375 | -1 | 4,846.375 | |
Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is ... | 1116 | 0.875 | 5,186.4375 | 4,757.071429 | 8,192 | |
Let $a$ and $b$ be integers such that $ab = 100.$ Find the minimum value of $a + b.$ | -101 | 0.875 | 4,596.8125 | 4,580 | 4,714.5 | |
A point $A(-2,-4)$ outside the parabola $y^{2}=2px (p > 0)$ is connected to a line $l$: $\begin{cases} x=-2+ \frac{\sqrt{2}}{2}t \\ y=-4+ \frac{\sqrt{2}}{2}t \end{cases} (t$ is a parameter, $t \in \mathbb{R})$ intersecting the parabola at points $M_{1}$ and $M_{2}$. The distances $|AM_{1}|$, $|M_{1}M_{2}|$, and $|AM_{2... | 2\sqrt{10} | 0.6875 | 6,240.5 | 5,870.454545 | 7,054.6 | |
In triangle $ABC$, $3 \sin A + 4 \cos B = 6$ and $4 \sin B + 3 \cos A = 1$. Find all possible values of $\angle C,$ in degrees. Enter all the possible values, separated by commas. | 30^\circ | 0.75 | 6,779.875 | 6,309.166667 | 8,192 | |
Find the minimum value of
\[\frac{x^2}{x - 8}\]for $x > 8.$ | 32 | 1 | 3,269.875 | 3,269.875 | -1 | |
Milan has a bag of 2020 red balls and 2021 green balls. He repeatedly draws 2 balls out of the bag uniformly at random. If they are the same color, he changes them both to the opposite color and returns them to the bag. If they are different colors, he discards them. Eventually the bag has 1 ball left. Let $p$ be the p... | 2021 | The difference between the number of green balls and red balls in the bag is always 1 modulo 4. Thus the last ball must be green and $p=1$. | 0.3125 | 7,809.1875 | 7,115.6 | 8,124.454545 |
Find the sum of all integers $n$ not less than $3$ such that the measure, in degrees, of an interior angle of a regular $n$ -gon is an integer.
*2016 CCA Math Bonanza Team #3* | 1167 | 0.75 | 5,411.875 | 4,485.166667 | 8,192 | |
Place 6 balls, labeled from 1 to 6, into 3 different boxes. If each box is to contain 2 balls, and the balls labeled 1 and 2 are to be placed in the same box, calculate the total number of different ways to do this. | 18 | 0.5 | 7,407.5 | 6,623 | 8,192 | |
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal? | 18 | 0.5 | 5,645.5 | 5,930 | 5,361 | |
Given that $r$ and $s$ are relatively prime positive integers such that $\frac{r}{s} = \frac{2(\sqrt{2} + \sqrt{10})}{5(\sqrt{3 + \sqrt{5}})}$, find $r$ and $s$. | r = 4, s = 5 | Squaring both sides of the given equation yields $\frac{r^{2}}{s^{2}} = \frac{4(12 + 4 \sqrt{5})}{25(3 + \sqrt{5})} = \frac{16(3 + \sqrt{5})}{25(3 + \sqrt{5})} = \frac{16}{25}$. Because $r$ and $s$ are positive and relatively prime, then by inspection, $r = 4$ and $s = 5$. | 0 | 3,952.5 | -1 | 3,952.5 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\vec{m}=(a,c)$ and $\vec{n}=(\cos C,\cos A)$.
1. If $\vec{m}\parallel \vec{n}$ and $a= \sqrt {3}c$, find angle $A$;
2. If $\vec{m}\cdot \vec{n}=3b\sin B$ and $\cos A= \frac {3}{5}$, find the value of $\... | \frac {4-6 \sqrt {2}}{15} | 0 | 7,847.75 | -1 | 7,847.75 | |
What is the least common multiple of 220 and 504? | 27720 | 0.9375 | 2,840.5 | 2,483.733333 | 8,192 | |
Let point G be the centroid of triangle ABC. If $\angle A=120^\circ$ and $\overrightarrow {AB} \cdot \overrightarrow {AC}=-1$, find the minimum value of $|\overrightarrow {AG}|$. | \frac{\sqrt{2}}{3} | 0 | 5,404.0625 | -1 | 5,404.0625 |
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