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 |
|---|---|---|---|---|---|---|
A section is cut out of a circular piece of paper having radius four inches, as shown. Points A and B are then glued together to form a right circular cone. What is the circumference of the base of the resulting cone? Express your answer in terms of $\pi$. (The $270^\circ$ sector forms the cone.)
[asy]import graph;
d... | 6 \pi | 0.625 | 1,444.75 | 1,291.2 | 1,700.666667 | |
Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$? | 2 | 1. **Given Equations**: We are given that $f(x+3) = 3x^2 + 7x + 4$ and $f(x) = ax^2 + bx + c$.
2. **Expression for $f(x+3)$**: Using the expression for $f(x)$, we can write:
\[
f(x+3) = a(x+3)^2 + b(x+3) + c
\]
Expanding $(x+3)^2$ and $(x+3)$, we get:
\[
(x+3)^2 = x^2 + 6x + 9 \quad \text{and} \quad ... | 1 | 2,266.25 | 2,266.25 | -1 |
In the cuboid ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, where AB=3, AD=4, and AA<sub>1</sub>=5, point P is a moving point on the surface A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>. Find the minimum value of |PA|+|PC|. | 5\sqrt{5} | 0.25 | 7,574.0625 | 5,720.25 | 8,192 | |
There are 2021 balls in a crate, numbered from 1 to 2021. Erica calculates the digit sum for each ball. For example, the digit sum of 2021 is 5, since \(2+0+2+1=5\). Balls with equal digit sums have the same color and balls with different digit sums have different colors. How many different colors of balls are there in... | 28 | 0.1875 | 7,818.75 | 6,524 | 8,117.538462 | |
A belt is installed on two pulleys with radii of 14 inches and 4 inches respectively. The belt is taut and does not intersect itself. If the distance between the points where the belt touches the two pulleys is 24 inches, what is the distance (in inches) between the centers of the two pulleys? | 26 | 1 | 1,708.6875 | 1,708.6875 | -1 | |
Use the Horner's method to calculate the value of the polynomial $f(x) = 7x^7 + 6x^6 + 5x^5 + 4x^4 + 3x^3 + 2x^2 + x$ when $x = 3$, and find the value of $v_3$. | 262 | 0.6875 | 4,451.5 | 4,101.090909 | 5,222.4 | |
Calculate the definite integral:
$$
\int_{1}^{8} \frac{5 \sqrt{x+24}}{(x+24)^{2} \cdot \sqrt{x}} \, dx
$$ | \frac{1}{8} | 0.4375 | 6,488.375 | 4,558.142857 | 7,989.666667 | |
On a computer screen is the single character a. The computer has two keys: c (copy) and p (paste), which may be pressed in any sequence. Pressing p increases the number of a's on screen by the number that were there the last time c was pressed. c doesn't change the number of a's on screen. Determine the fewest number o... | 21 | The first keystroke must be c and the last keystroke must be p. If there are $k$ c's pressed in total, let $n_{i}$ denote one more than the number of p's pressed immediately following the $i$ 'th c , for $1 \leq i \leq k$. Then, we have that the total number of keystrokes is $$s:=\sum_{i=1}^{k} n_{i}$$ and the total nu... | 0 | 8,192 | -1 | 8,192 |
The side of a square is increased by $20\%$. To keep the area of the square unchanged, what percentage must the other side be reduced? | 16.67\% | 0.375 | 2,760.1875 | 2,727.333333 | 2,779.9 | |
If $m$ is a root of the equation $4^{x+ \frac {1}{2}}-9\cdot2^{x}+4=0$, then the eccentricity of the conic section $x^{2}+ \frac {y^{2}}{m}=1$ is \_\_\_\_\_\_. | \sqrt {2} | 0 | 7,198.125 | -1 | 7,198.125 | |
Given an arithmetic sequence with a total of $20$ terms, the sum of all terms is $75$, and the sum of the even terms is $25$, determine the common difference $d$. | -2.5 | 0.0625 | 5,482.375 | 6,834 | 5,392.266667 | |
A convex hexagon \( A_{1} A_{2} \ldots A_{6} \) is circumscribed around a circle \( \omega \) with a radius of 1. Consider three segments that connect the midpoints of the opposite sides of the hexagon. What is the greatest \( r \) for which it can be stated that at least one of these segments is not shorter than \( r ... | \sqrt{3} | 0 | 7,920.6875 | -1 | 7,920.6875 | |
Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$. | (3, 2, 6) |
To find all triples \((p, q, r)\) of positive integers, where \(p, q\) are also primes, such that:
\[
\frac{r^2 - 5q^2}{p^2 - 1} = 2,
\]
we start by rearranging the equation:
\[
r^2 - 5q^2 = 2(p^2 - 1).
\]
This can be further rewritten as:
\[
r^2 = 5q^2 + 2(p^2 - 1).
\]
Since \(p\) and \(q\) are primes, we will ... | 0.125 | 8,035.1875 | 6,937.5 | 8,192 |
A school has between 150 and 200 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into six distinct sections of students. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible... | 1575 | 0.8125 | 4,522.5 | 3,675.692308 | 8,192 | |
In an arithmetic sequence $\{a_n\}$ with a non-zero common difference, it is known that $a_1=4$ and $a_7^2=a_1a_{10}$. The sum of the first $n$ terms is $S_n$.
1. Find the general formula for the sequence $\{a_n\}$.
2. Find the maximum value of $S_n$ and the value of $n$ when the maximum is achieved. | 26 | 0.4375 | 5,303 | 5,557 | 5,105.444444 | |
Compute $54 \times 46$ in your head. | 2484 | 1 | 373.25 | 373.25 | -1 | |
For all m and n satisfying \( 1 \leq n \leq m \leq 5 \), the polar equation \( \rho = \frac{1}{1 - C_{m}^{n} \cos \theta} \) represents how many different hyperbolas? | 10 | 0 | 6,843.375 | -1 | 6,843.375 | |
Determine the largest of all integers $n$ with the property that $n$ is divisible by all positive integers that are less than $\sqrt[3]{n}$. | 420 | Observation from that $\operatorname{lcm}(2,3,4,5,6,7)=420$ is divisible by every integer less than or equal to $7=[\sqrt[3]{420}]$ and that $\operatorname{lcm}(2,3,4,5,6,7,8)=840$ is not divisible by $9=[\sqrt[3]{840}]$. One may guess 420 is the required integer. Let $N$ be the required integer and suppose $N>420$. Pu... | 0.25 | 7,811.0625 | 6,668.25 | 8,192 |
Suppose that $\{a_n\}$ is an arithmetic sequence with $$
a_1+a_2+ \cdots +a_{100}=100 \quad \text{and} \quad
a_{101}+a_{102}+ \cdots + a_{200}=200.
$$What is the value of $a_2 - a_1$? Express your answer as a common fraction. | \frac{1}{100} | 0.875 | 4,218.8125 | 3,716.5 | 7,735 | |
A fair coin is flipped 8 times. What is the probability that fewer than 3 of the flips come up heads? | \frac{37}{256} | 0.9375 | 4,438.9375 | 4,188.733333 | 8,192 | |
Let $p(x)$ be a quadratic polynomial such that $[p(x)]^3 - x$ is divisible by $(x - 1)(x + 1)(x - 8).$ Find $p(13).$ | -3 | 0.875 | 4,577.8125 | 4,061.5 | 8,192 | |
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer.
[i]Laurentiu Panaitopol, Romania[/i] | (5, 3) |
We are tasked with finding all pairs of positive integers \( m, n \geq 3 \) such that there exist infinitely many positive integers \( a \) making the expression
\[
\frac{a^m + a - 1}{a^n + a^2 - 1}
\]
an integer. To solve this problem, we aim to explore potential values of \( m \) and \( n \) and identify condition... | 0 | 8,192 | -1 | 8,192 |
If $10$ divides the number $1\cdot2^1+2\cdot2^2+3\cdot2^3+\dots+n\cdot2^n$ , what is the least integer $n\geq 2012$ ? | 2014 | 0 | 7,597.5 | -1 | 7,597.5 | |
In a chess tournament, a team of schoolchildren and a team of students, each consisting of 15 participants, compete against each other. During the tournament, each schoolchild must play with each student exactly once, with the condition that everyone can play at most once per day. Different numbers of games could be pl... | 120 | 0 | 7,859.75 | -1 | 7,859.75 | |
Call a positive integer strictly monotonous if it is a one-digit number or its digits, read from left to right, form a strictly increasing or a strictly decreasing sequence, and no digits are repeated. Determine the total number of strictly monotonous positive integers. | 1013 | 0.125 | 7,661.625 | 7,164 | 7,732.714286 | |
A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015},$ how far from $P_0$ is she, in inches? | 1008 \sqrt{6} + 1008 \sqrt{2} | 0 | 7,903.4375 | -1 | 7,903.4375 | |
Find the greatest constant $N,$ so that
\[\frac{a^2 + b^2 + ab}{c^2} > N\]whenever $a,$ $b,$ and $c$ are the sides of a triangle. | \frac{3}{4} | 0.6875 | 7,409.875 | 7,054.363636 | 8,192 | |
Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x = \frac{y^2}{2}.$ | 2 \sqrt{17} | 0.6875 | 5,364.0625 | 4,078.636364 | 8,192 | |
According to the graph, what is the average monthly balance, in dollars, of David's savings account during the four-month period shown? [asy]
draw((0,0)--(13,0)--(13,8)--(0,8)--cycle,linewidth(1));
draw((0,2)--(13,2),linewidth(1));
draw((0,4)--(13,4),linewidth(1));
draw((0,6)--(13,6),linewidth(1));
draw((1,0)--(1,2)--(... | \$150 | 0.5625 | 3,239.5625 | 2,158.444444 | 4,629.571429 | |
The value of $$\frac {1}{\tan 20^\circ} - \frac {1}{\cos 10^\circ}$$ is equal to \_\_\_\_\_\_. | \sqrt {3} | 0 | 6,553.9375 | -1 | 6,553.9375 | |
Compute the values of $\binom{600}{600}$, $\binom{600}{0}$, and $\binom{600}{1}$. | 600 | 1 | 1,533.0625 | 1,533.0625 | -1 | |
What is the base five product of the numbers $132_{5}$ and $12_{5}$? | 2114_5 | 0 | 5,461.875 | -1 | 5,461.875 | |
A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? | 17 | 1. **Identify the vertices and their coordinates**:
Given that the $y$-coordinates of the vertices of the square are $0$, $1$, $4$, and $5$, we can assume the vertices are $A=(0,0)$, $B=(x_1,1)$, $C=(x_2,5)$, and $D=(x_3,4)$ after a suitable translation.
2. **Calculate the slope of side $AB$**:
The slope of $AB... | 0.0625 | 8,093.3125 | 6,613 | 8,192 |
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, the value of $y$ is | 67.5 | 0 | 8,192 | -1 | 8,192 | |
Given that $a_{1}, a_{2}, b_{1}, b_{2}, \cdots, b_{k}$ are vectors in the plane that are pairwise non-parallel, with $\left|a_{1}-a_{2}\right|=1$, and $\left|a_{i}-b_{j}\right| \in\{1,2,3\} (i=1,2; j=1,2, \cdots, k)$, determine the maximum value of $k$. | 10 | 0 | 8,192 | -1 | 8,192 | |
A circular paper with a radius of 6 inches has a section removed to form a \(240^\circ\) sector. This sector is then used to form a right circular cone by bringing the two radii together. Find the circumference of the base of the cone in terms of \(\pi\). | 8\pi | 0.875 | 1,417.5 | 1,404.5 | 1,508.5 | |
2005^{2} + 2 \times 2005 \times 1995 + 1995^{2} divided by 800. | 20000 | 1 | 2,238.4375 | 2,238.4375 | -1 | |
Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \( a, b, \) and \( c \) are positive real numbers. | \frac{1}{3\sqrt[3]{2}} | 0 | 7,695.5625 | -1 | 7,695.5625 | |
Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s.$
| 380 | 0.75 | 5,561.375 | 5,131.916667 | 6,849.75 | |
Find the number of integer points that satisfy the system of inequalities:
\[
\begin{cases}
y \leqslant 3x \\
y \geqslant \frac{1}{3}x \\
x + y \leqslant 100
\end{cases}
\] | 2551 | 0 | 8,192 | -1 | 8,192 | |
Solve the congruence $11n \equiv 7 \pmod{43}$, as a residue modulo 43. (Give an answer between 0 and 42.) | 28 | 1 | 2,660.875 | 2,660.875 | -1 | |
Let $n$ be a positive integer. A sequence of $n$ positive integers (not necessarily distinct) is called [b]full[/b] if it satisfies the following condition: for each positive integer $k\geq2$, if the number $k$ appears in the sequence then so does the number $k-1$, and moreover the first occurrence of $k-1$ comes befor... | n! |
To solve this problem, we need to determine how many sequences of length \( n \) consisting of positive integers are considered "full" according to the defined condition. The condition implies a hierarchical appearance of integers in the sequence, such that if an integer \( k \) appears, then \( k-1 \) must also appea... | 0 | 8,192 | -1 | 8,192 |
Suppose $a$, $b$, $c$, and $d$ are integers such that:
- $a - b + c = 7$
- $b - c + d = 8$
- $c - d + a = 4$
- $d - a + b = 3$
- $a + b + c - d = 10$
Find the value of $a + b + c + d$. | 16 | 0 | 7,040.4375 | -1 | 7,040.4375 | |
Let the function be $$f(x)= \sqrt {3}\sin 2x+2\cos^{2}x+2$$.
(I) Find the smallest positive period and the range of $f(x)$;
(II) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $$A= \frac {\pi }{3}$$ and the area of $\triangle ABC$ is $$\frac { \sqrt {3}}{2}$$,... | \sqrt {3} | 0 | 7,672.75 | -1 | 7,672.75 | |
Given tetrahedron $P-ABC$, if one line is randomly selected from the lines connecting the midpoints of each edge, calculate the probability that this line intersects plane $ABC$. | \frac{3}{5} | 0 | 8,083.375 | -1 | 8,083.375 | |
The function $f$ is linear and satisfies $f(d+1)-f(d) = 3$ for all real numbers $d$. What is $f(3)-f(5)$? | -6 | 1 | 1,378.5625 | 1,378.5625 | -1 | |
Given the function $f(x)=\cos(2x+\varphi), |\varphi| \leqslant \frac{\pi}{2}$, if $f\left( \frac{8\pi}{3}-x \right)=-f(x)$, determine the horizontal shift required to obtain the graph of $y=\sin 2x$ from the graph of $y=f(x)$. | \frac{\pi}{6} | 0.6875 | 6,084.125 | 5,223.636364 | 7,977.2 | |
Let $P R O B L E M Z$ be a regular octagon inscribed in a circle of unit radius. Diagonals $M R, O Z$ meet at $I$. Compute $L I$. | \sqrt{2} | If $W$ is the center of the circle then $I$ is the incenter of $\triangle R W Z$. Moreover, PRIZ is a rhombus. It follows that $P I$ is twice the inradius of a 1-1- $\sqrt{2}$ triangle, hence the answer of $2-\sqrt{2}$. So $L I=\sqrt{2}$. Alternatively, one can show (note, really) that the triangle $O I L$ is isosceles... | 0.5625 | 7,353.75 | 6,701.777778 | 8,192 |
In triangle \(ABC\), \(AC = 8\) and \(BC = 5\). A line parallel to the bisector of the external angle at \(C\) passes through the midpoint of side \(AB\) and point \(E\) on side \(AC\). Find \(AE\). | 1.5 | 0 | 6,783.4375 | -1 | 6,783.4375 | |
A sphere with radius $r$ is inside a cone, whose axial section is an equilateral triangle with the sphere inscribed in it. The ratio of the total surface area of the cone to the surface area of the sphere is \_\_\_\_\_\_. | 9:4 | 0 | 5,184.5 | -1 | 5,184.5 | |
A right square pyramid with base edges of length $12$ units each and slant edges of length $15$ units each is cut by a plane that is parallel to its base and $4$ units above its base. What is the volume, in cubic units, of the top pyramid section that is cut off by this plane? | \frac{1}{3} \times \left(\frac{(144 \cdot (153 - 8\sqrt{153}))}{153}\right) \times (\sqrt{153} - 4) | 0 | 8,192 | -1 | 8,192 | |
For some positive integer $n$ , the sum of all odd positive integers between $n^2-n$ and $n^2+n$ is a number between $9000$ and $10000$ , inclusive. Compute $n$ .
*2020 CCA Math Bonanza Lightning Round #3.1* | 21 | 0.875 | 4,490.6875 | 3,961.928571 | 8,192 | |
Find the remainder when $1^{2}+3^{2}+5^{2}+\cdots+99^{2}$ is divided by 1000. | 650 | We have $S=\sum_{i=0}^{49}(2 i+1)^{2}=\sum_{i=0}^{49} 4 i^{2}+4 i+1=4 \cdot \frac{49 \cdot 50 \cdot 99}{6}+4 \cdot \frac{49 \cdot 50}{2}+50 \equiv 700+900+50(\bmod 1000) \equiv 650(\bmod 1000)$. | 0.6875 | 5,580.0625 | 4,805.636364 | 7,283.8 |
Let $A$ be a point on the parabola $y = x^2 - 9x + 25,$ and let $B$ be a point on the line $y = x - 8.$ Find the shortest possible distance $AB.$ | 4 \sqrt{2} | 0.5625 | 6,871.875 | 5,845.111111 | 8,192 | |
Given an ellipse $C: \frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and the sum of the distances from a point on the ellipse to the two foci is $2\sqrt{2}$. A line $l$ with slope $k(k\neq 0)$ passes through the upper focus of the ellipse and intersects the ellipse at... | \frac{3\sqrt{6}}{16} | 0 | 7,928.625 | -1 | 7,928.625 | |
If $\cos (π+α)=- \frac { \sqrt {10}}{5}$ and $α∈(- \frac {π}{2},0)$, find the value of $\tan ( \frac {3π}{2}+α)$. | - \frac { \sqrt {6}}{3} | 0 | 4,346.875 | -1 | 4,346.875 | |
What is the largest number, with its digits all different and none of them being zero, whose digits add up to 20? | 9821 | 0 | 8,137.6875 | -1 | 8,137.6875 | |
Given the parabola $y=-x^{2}+3$, there exist two distinct points $A$ and $B$ on it that are symmetric about the line $x+y=0$. Find the length of the segment $|AB|$. | 3\sqrt{2} | 0.625 | 5,652.625 | 4,129 | 8,192 | |
Given that there are 4 qualified and 2 defective products, determine the probability of finding the last defective product exactly on the fourth inspection when selectins products one at a time and not returning them after each selection. | \frac{1}{5} | 0.125 | 8,094.125 | 8,192 | 8,080.142857 | |
The equations $2x+7=3$ and $bx-10=-2$ have the same solution $x$. What is the value of $b$? | b = -4 | 1 | 1,243.1875 | 1,243.1875 | -1 | |
The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$ n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n $$ for every positive integer $n$ , where $a_0 = a_1 = 1$ . Calculate the sum $$ \frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}} $$ . | 2009 * 1005 | 0 | 6,267.5625 | -1 | 6,267.5625 | |
Given the sums of the first n terms of two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ denoted as $S_n$ and $T_n$, respectively, if $\frac {S_{n}}{T_{n}} = \frac {2n}{3n+1}$, calculate the value of $\frac {a_{6}}{b_{6}}$. | \frac {11}{17} | 0.875 | 4,429.1875 | 4,161 | 6,306.5 | |
Point \( K \) is the midpoint of edge \( A A_{1} \) of cube \( A B C D A_{1} B_{1} C_{1} D_{1} \), and point \( L \) lies on edge \( B C \). Segment \( K L \) touches the sphere inscribed in the cube. In what ratio does the point of tangency divide segment \( K L \)? | 4/5 | 0.125 | 7,090.6875 | 6,373.5 | 7,193.142857 | |
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values ... | \sqrt{2} |
Given an acute triangle \( ABC \), let \( M \) be the midpoint of \( AC \). A circle \( \omega \) that passes through points \( B \) and \( M \) intersects side \( AB \) at point \( P \) and side \( BC \) at point \( Q \). Point \( T \) is such that \( BPTQ \) forms a parallelogram, and it is given that \( T \) lies o... | 0 | 8,192 | -1 | 8,192 |
Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Then $AK=\tfrac mn,$ where $m$ an... | 11 | 0.25 | 7,774.75 | 6,563.25 | 8,178.583333 | |
A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$ , whose apex $F$ is on the leg $AC$ . Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = ... | 2:3 | 0.25 | 7,273.5 | 6,806.5 | 7,429.166667 | |
A seven-digit number has the following properties: the hundreds digit is twice the ten millions digit, the tens digit is twice the hundred thousands digit, the units digit is twice the ten thousands digit, the thousands digit is 0, and it must be divisible by a five-digit number \( a \). What is \( a \)? | 10002 | 0 | 8,192 | -1 | 8,192 | |
Given a quadratic function $f(x) = ax^2 - 4bx + 1$.
(1) Let set $P = \{-1,1,2,3,4,5\}$ and set $Q = \{-2,-1,1,2,3,4\}$. Randomly select a number from set $P$ as $a$ and from set $Q$ as $b$. Calculate the probability that the function $y = f(x)$ is increasing on the interval $[1,+\infty)$.
(2) Suppose the point $(a, b... | \dfrac{1}{3} | 0.625 | 7,049.375 | 6,435 | 8,073.333333 | |
An eight-sided die, with faces numbered from 1 to 8, is tossed three times. Given that the sum of the first two tosses equals the third, calculate the probability that at least one "2" is tossed. | \frac{11}{28} | 0 | 6,536.75 | -1 | 6,536.75 | |
A game of drawing balls involves a non-transparent paper box containing $6$ identical-sized, differently colored glass balls. Participants pay $1$ unit of fee to play the game once, drawing balls with replacement three times. Participants must specify a color from the box before drawing. If the specified color does not... | 110 | 0.0625 | 4,363.625 | 4,600 | 4,347.866667 | |
Given:
\\((1)y=x+ \\frac {4}{x}\\)
\\((2)y=\\sin x+ \\frac {4}{\\sin x}(0 < x < π)\\)
\\((3)y= \\frac {x^{2}+13}{ \\sqrt {x^{2}+9}}\\)
\\((4)y=4⋅2^{x}+2^{-x}\\)
\\((5)y=\\log \_{3}x+4\\log \_{x}3(0 < x < 1)\\)
Find the function(s) with a minimum value of $4$. (Fill in the correct question number) | (4) | 0 | 6,657.6875 | -1 | 6,657.6875 | |
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 ha... | \frac{1}{4} | 1. **Initial Setup and State Description:**
Each player starts with $1. The possible states of money distribution after each round are $(1-1-1)$ and $(2-1-0)$ in some permutation. The state $(3-0-0)$ is not possible because:
- A player cannot give money to themselves.
- A maximum of $2 is being distributed, an... | 0 | 8,192 | -1 | 8,192 |
Let $A B C$ be a triangle and $\omega$ be its circumcircle. The point $M$ is the midpoint of arc $B C$ not containing $A$ on $\omega$ and $D$ is chosen so that $D M$ is tangent to $\omega$ and is on the same side of $A M$ as $C$. It is given that $A M=A C$ and $\angle D M C=38^{\circ}$. Find the measure of angle $\angl... | 33^{\circ} | By inscribed angles, we know that $\angle B A C=38^{\circ} \cdot 2=76^{\circ}$ which means that $\angle C=104^{\circ}-\angle B$. Since $A M=A C$, we have $\angle A C M=\angle A M C=90^{\circ}-\frac{\angle M A C}{2}=71^{\circ}$. Once again by inscribed angles, this means that $\angle B=71^{\circ}$ which gives $\angle C=... | 0 | 7,829.375 | -1 | 7,829.375 |
The ten smallest positive odd numbers \( 1, 3, \cdots, 19 \) are arranged in a circle. Let \( m \) be the maximum value of the sum of any one of the numbers and its two adjacent numbers. Find the minimum value of \( m \). | 33 | 0 | 8,192 | -1 | 8,192 | |
$-14-(-2)^{3}\times \dfrac{1}{4}-16\times \left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{3}{8}\right)$. | -22 | 0.75 | 618.875 | 615.416667 | 629.25 | |
For positive integers $a$ and $N$, let $r(a, N) \in\{0,1, \ldots, N-1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \leq 1000000$ for which $r(n, 1000)>r(n, 1001)$. | 499500 | Note that $0 \leq r(n, 1000) \leq 999$ and $0 \leq r(n, 1001) \leq 1000$. Consider the $\binom{1000}{2}=499500$ ways to choose pairs $(i, j)$ such that $i>j$. By the Chinese Remainder Theorem, there is exactly one $n$ such that $1 \leq n \leq 1000 \cdot 1001$ such that $n \equiv i(\bmod 1000)$ and $n \equiv j(\bmod 100... | 0 | 8,192 | -1 | 8,192 |
Each of two baskets contains white and black balls such that the total number of balls in both baskets is 25. One ball is randomly drawn from each basket. It is known that the probability that both drawn balls are white is 0.54. Find the probability that both drawn balls are black. | 0.04 | 0 | 7,704.3125 | -1 | 7,704.3125 | |
Given circle M: $(x+1)^2+y^2=1$, and circle N: $(x-1)^2+y^2=9$, a moving circle P is externally tangent to circle M and internally tangent to circle N. The trajectory of the center of circle P is curve C.
(1) Find the equation of C:
(2) Let $l$ be a line that is tangent to both circle P and circle M, and $l$ inters... | \frac{18}{7} | 0.1875 | 7,946.1875 | 6,881 | 8,192 | |
The value of the expression \[(3^{1001}+4^{1002})^2-(3^{1001}-4^{1002})^2\]is $k\cdot12^{1001}$ for some positive integer $k$. What is $k$? | 16 | 1 | 3,066.375 | 3,066.375 | -1 | |
A real number $x$ is chosen uniformly at random from the interval $(0,10)$. Compute the probability that $\sqrt{x}, \sqrt{x+7}$, and $\sqrt{10-x}$ are the side lengths of a non-degenerate triangle. | \frac{22}{25} | For any positive reals $a, b, c$, numbers $a, b, c$ is a side length of a triangle if and only if $$(a+b+c)(-a+b+c)(a-b+c)(a+b-c)>0 \Longleftrightarrow \sum_{\text {cyc }}\left(2 a^{2} b^{2}-a^{4}\right)>0$$ (to see why, just note that if $a \geq b+c$, then only the factor $-a+b+c$ is negative). Therefore, $x$ works if... | 0.375 | 7,161.875 | 6,585.666667 | 7,507.6 |
What is the coefficient of \( x^4 \) in the expansion of \( (3x + 4)^8 \)? | 1451520 | 1 | 4,029.8125 | 4,029.8125 | -1 | |
Consider the two points \(A(4,1)\) and \(B(2,5)\). For each point \(C\) with positive integer coordinates, we define \(d_C\) to be the shortest distance needed to travel from \(A\) to \(C\) to \(B\) moving only horizontally and/or vertically. The positive integer \(N\) has the property that there are exactly 2023 point... | 12 | 0 | 8,192 | -1 | 8,192 | |
The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$. | 336 | Let the three integers be $a, b, c$. $N = abc = 6(a + b + c)$ and $c = a + b$. Then $N = ab(a + b) = 6(a + b + a + b) = 12(a + b)$. Since $a$ and $b$ are positive, $ab = 12$ so $\{a, b\}$ is one of $\{1, 12\}, \{2, 6\}, \{3, 4\}$ so $a + b$ is one of $13, 8, 7$ so $N$ is one of $12\cdot 13 = 156, 12\cdot 8 = 96, 12\cdo... | 1 | 3,967.375 | 3,967.375 | -1 |
Assume the function $f(x) = 2\sin x \cos^2\left(\frac{\varphi}{2}\right) + \cos x \sin\varphi - \sin x$, where $(0 < \varphi < \pi)$, takes its minimum value at $x = \pi$.
(i) Find the value of $\varphi$ and simplify $f(x)$.
(ii) In triangle $ABC$, $a$, $b$, and $c$ are the lengths of the sides opposite to angles $... | \frac{7\pi}{12} | 0.6875 | 6,225.0625 | 5,466.727273 | 7,893.4 | |
A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with a radius of 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$ | \sqrt{59} | 0.875 | 3,527.5625 | 2,861.214286 | 8,192 | |
If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is: | -13\frac{1}{2} | 1. **Identify the nature of the sequence**: Given that $x, 2x+2, 3x+3, \dots$ are in geometric progression, the ratio between consecutive terms must be constant. Let's denote this common ratio by $r$.
2. **Set up the equation for the common ratio**:
\[
\frac{2x+2}{x} = \frac{3x+3}{2x+2}
\]
This equation ar... | 0 | 6,066.9375 | -1 | 6,066.9375 |
Given the function $f(x)=-\frac{1}{2}x^{2}+x$ with a domain that contains an interval $[m,n]$, and its range on this interval is $[3m,3n]$. Find the value of $m+n$. | -4 | 0.1875 | 7,668.1875 | 5,398.333333 | 8,192 | |
A thief on a bus gets off at a bus stop and walks in the direction opposite to the bus’s travel direction. The bus continues its journey, and a passenger realizes they have been robbed. The passenger gets off at the next stop and starts chasing the thief. If the passenger's speed is twice that of the thief, the bus's s... | 440 | 0 | 5,926.875 | -1 | 5,926.875 | |
A random simulation method is used to estimate the probability of a shooter hitting the target at least 3 times out of 4 shots. A calculator generates random integers between 0 and 9, where 0 and 1 represent missing the target, and 2 through 9 represent hitting the target. Groups of 4 random numbers represent the resul... | 0.75 | 0.1875 | 4,909.125 | 4,075 | 5,101.615385 | |
The fraction $\frac{1}{2015}$ has a unique "(restricted) partial fraction decomposition" of the form $\frac{1}{2015}=\frac{a}{5}+\frac{b}{13}+\frac{c}{31}$ where $a, b, c$ are integers with $0 \leq a<5$ and $0 \leq b<13$. Find $a+b$. | 14 | This is equivalent to $1=13 \cdot 31 a+5 \cdot 31 b+5 \cdot 13 c$. Taking modulo 5 gives $1 \equiv 3 \cdot 1 a (\bmod 5)$, so $a \equiv 2(\bmod 5)$. Taking modulo 13 gives $1 \equiv 5 \cdot 5 b=25 b \equiv-b(\bmod 13)$, so $b \equiv 12 (\bmod 13)$. The size constraints on $a, b$ give $a=2, b=12$, so $a+b=14$. | 0.75 | 5,466.5625 | 4,558.083333 | 8,192 |
Consider a larger grid extending from point $A$ to point $B$, now divided into a 3x2 grid. You can still only move right or down along the drawn segments. How many different routes are there from point $A$ to point $B$?
[asy]
unitsize(0.09inch);
draw((0,0)--(15,0)--(15,10)--(0,10)--cycle);
draw((5,0)--(5,10));
draw((1... | 10 | 0.9375 | 4,088.5 | 3,938.333333 | 6,341 | |
Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$. | {(a,b)\in \{(-2,1), (-1,1), (0,1), (1,1), (2,1), (-1,-1), (0,-1), (1,-1)\}} |
To solve this problem, we need to determine all integer pairs \((a, b)\) such that there exists a polynomial \( P(x) \in \mathbb{Z}[X] \) with the product \((x^2 + ax + b) \cdot P(x)\) having all coefficients either \(1\) or \(-1\).
Assume \( P(x) = c_m x^m + c_{m-1} x^{m-1} + \ldots + c_1 x + c_0 \) with \( c_i \in ... | 0 | 8,192 | -1 | 8,192 |
Given the sequence $\{v_n\}$ defined by $v_1 = 7$ and the relationship $v_{n+1} - v_n = 2 + 5(n-1)$ for $n=1,2,3,\ldots$, express $v_n$ as a polynomial in $n$ and find the sum of its coefficients. | 4.5 | 0 | 5,254.25 | -1 | 5,254.25 | |
Given $$\sqrt {2 \frac {2}{3}}=2 \sqrt { \frac {2}{3}}$$, $$\sqrt {3 \frac {3}{8}}=3 \sqrt { \frac {3}{8}}$$, $$\sqrt {4 \frac {4}{15}}=4 \sqrt { \frac {4}{15}}$$, ..., if $$\sqrt {6 \frac {a}{t}}=6 \sqrt { \frac {a}{t}}$$ (where $a$, $t$∈$R^*$), then $a=$ \_\_\_\_\_\_ , $t=$ \_\_\_\_\_\_ . | 35 | 0.5 | 6,488.125 | 4,784.25 | 8,192 | |
A regular octagon is inscribed in a circle of radius 2. Alice and Bob play a game in which they take turns claiming vertices of the octagon, with Alice going first. A player wins as soon as they have selected three points that form a right angle. If all points are selected without either player winning, the game ends i... | 2 \sqrt{2}, 4+2 \sqrt{2} | A player ends up with a right angle iff they own two diametrically opposed vertices. Under optimal play, the game ends in a draw: on each of Bob's turns he is forced to choose the diametrically opposed vertex of Alice's most recent choice, making it impossible for either player to win. At the end, the two possibilities... | 0 | 8,176.3125 | -1 | 8,176.3125 |
On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are 101, 102, 103, 104, 105, and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?
A) 75%
B) 80%
C) 90%
D) 95%
E) 100% | 80\% | 0 | 7,780.8125 | -1 | 7,780.8125 | |
What is the reciprocal of $\frac{3}{4} + \frac{4}{5}$?
A) $\frac{31}{20}$
B) $\frac{20}{31}$
C) $\frac{19}{20}$
D) $\frac{20}{19}$ | \frac{20}{31} | 0 | 1,419.5 | -1 | 1,419.5 | |
Calculate the sum of the series:
\[
\sum_{n=1}^\infty \frac{3^n}{1 + 3^n + 3^{n+1} + 3^{2n+1}}.
\] | \frac{1}{4} | 0.0625 | 5,498.9375 | 3,115 | 5,657.866667 | |
In triangle $\triangle ABC$, side $a$ is 2 units longer than side $b$, and side $b$ is 2 units longer than side $c$. If the sine of the largest angle is $\frac {\sqrt {3}}{2}$, then the area of triangle $\triangle ABC$ is \_\_\_\_\_\_. | \frac {15 \sqrt {3}}{4} | 0 | 5,691.0625 | -1 | 5,691.0625 | |
How many ways can one fill a $3 \times 3$ square grid with nonnegative integers such that no nonzero integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7 ? | 216 | In what ways could we potentially fill a single row? The only possibilities are if it contains the numbers $(0,0,7)$ or $(0,1,6)$ or $(0,2,5)$ or $(0,3,4)$ or $(1,2,4)$. Notice that if we write these numbers in binary, in any choices for how to fill the row, there will be exactly one number with a 1 in its rightmost di... | 0 | 8,192 | -1 | 8,192 |
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