problem
stringclasses 60
values | answer
stringclasses 50
values | reward
float64 0.06
0.44
|
|---|---|---|
The circle is divided by points \(A\), \(B\), \(C\), and \(D\) such that \(AB: BC: CD: DA = 3: 2: 13: 7\). Chords \(AD\) and \(BC\) are extended to intersect at point \(M\).
Find the angle \( \angle AMB \).
|
72
| 0.3125 |
Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$ Find $a-b$ .
|
22
| 0.1875 |
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)?
|
256
| 0.1875 |
What is the largest four-digit negative integer congruent to $1 \pmod{17}$?
|
-1002
| 0.3125 |
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 of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$.
Find $N$.
|
772
| 0.125 |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.
\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular}
|
135
| 0.0625 |
Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$ Find $a-b$ .
|
22
| 0.1875 |
Record the outcome of hitting or missing for 6 consecutive shots in order.
① How many possible outcomes are there?
② How many outcomes are there where exactly 3 shots hit the target?
③ How many outcomes are there where 3 shots hit the target, and exactly two of those hits are consecutive?
|
12
| 0.3125 |
If \( x = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{10^{6}}} \), then the value of \([x]\) is
|
1998
| 0.4375 |
Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
|
71
| 0.3125 |
Convert the quadratic equation $3x=x^{2}-2$ into general form and determine the coefficients of the quadratic term, linear term, and constant term.
|
-2
| 0.3125 |
A banquet has invited 44 guests. There are 15 identical square tables, each of which can seat 1 person per side. By appropriately combining the square tables (to form rectangular or square tables), ensure that all guests are seated with no empty seats. What is the minimum number of tables in the final arrangement?
|
11
| 0.375 |
Máté is always in a hurry. He observed that it takes 1.5 minutes to get to the subway when he stands on the moving escalator, while it takes 1 minute to run down the stationary stairs. How long does it take Máté to get down if he can run down the moving escalator?
|
36
| 0.125 |
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)?
|
256
| 0.1875 |
Find the smallest positive integer $n$ for which $$1!2!\cdots(n-1)!>n!^{2}$$
|
8
| 0.375 |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.
\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular}
|
135
| 0.0625 |
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \).
|
574
| 0.3125 |
A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$ . She also has a box with dimensions $10\times11\times14$ . The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.
Find all possible values of the total number of bricks that she can pack.
|
24
| 0.4375 |
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \).
|
574
| 0.3125 |
What is the largest four-digit negative integer congruent to $1 \pmod{17}$?
|
-1002
| 0.3125 |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?
|
6
| 0.375 |
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \).
|
574
| 0.3125 |
Define an odd function f(x) on ℝ that satisfies f(x+1) is an even function, and when x ∈ [0,1], f(x) = x(3-2x). Evaluate f(31/2).
|
-1
| 0.125 |
Find the smallest positive integer $n$ for which $$1!2!\cdots(n-1)!>n!^{2}$$
|
8
| 0.375 |
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the integer part of $x$, that is, $\lfloor x \rfloor$ is the largest integer not exceeding $x$. Calculate the value of $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \lfloor \log_{2}4 \rfloor + \ldots + \lfloor \log_{2}1024 \rfloor$.
|
8204
| 0.4375 |
1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=
|
0
| 0.4375 |
Given $x= \frac {\pi}{12}$ is a symmetry axis of the function $f(x)= \sqrt {3}\sin(2x+\varphi)+\cos(2x+\varphi)$ $(0<\varphi<\pi)$, after shifting the graph of function $f(x)$ to the right by $\frac {3\pi}{4}$ units, find the minimum value of the resulting function $g(x)$ on the interval $\left[-\frac {\pi}{4}, \frac {\pi}{6}\right]$.
|
-1
| 0.4375 |
What is the value of $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \cdots + [\sqrt{1989 \cdot 1990}] + [-\sqrt{1}] + [-\sqrt{2}] + [-\sqrt{3}] + \cdots + [-\sqrt{1989 \cdot 1990}]$?
(The 1st "Hope Cup" Mathematics Contest, 1990)
|
-3956121
| 0.1875 |
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 of the lines $FH$ and $QO$ . Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\dfrac{a + b \sqrt{c}}{d}$ , then find the value of $1000a + 100b + 10c + d$ .
|
1132
| 0.125 |
A positive integer \( n \) cannot be divided by \( 2 \) or \( 3 \), and there do not exist non-negative integers \( a \) and \( b \) such that \( |2^a - 3^b| = n \). Find the smallest value of \( n \).
|
35
| 0.0625 |
How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different?
|
616
| 0.1875 |
Given $x= \frac {\pi}{12}$ is a symmetry axis of the function $f(x)= \sqrt {3}\sin(2x+\varphi)+\cos(2x+\varphi)$ $(0<\varphi<\pi)$, after shifting the graph of function $f(x)$ to the right by $\frac {3\pi}{4}$ units, find the minimum value of the resulting function $g(x)$ on the interval $\left[-\frac {\pi}{4}, \frac {\pi}{6}\right]$.
|
-1
| 0.4375 |
Find the number of ordered pairs $(x,y)$ of real numbers such that
\[16^{x^2 + y} + 16^{x + y^2} = 1.\]
|
1
| 0.1875 |
Máté is always in a hurry. He observed that it takes 1.5 minutes to get to the subway when he stands on the moving escalator, while it takes 1 minute to run down the stationary stairs. How long does it take Máté to get down if he can run down the moving escalator?
|
36
| 0.125 |
An ellipse is defined parametrically by
\[(x,y) = \left( \frac{2 (\sin t - 1)}{2 - \cos t}, \frac{3 (\cos t - 5)}{2 - \cos t} \right).\]Then the equation of the ellipse can be written in the form
\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\]where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are integers, and $\gcd(|A|,|B|,|C|,|D|,|E|,|F|) = 1.$ Find $|A| + |B| + |C| + |D| + |E| + |F|.$
|
1381
| 0.375 |
An ellipse is defined parametrically by
\[(x,y) = \left( \frac{2 (\sin t - 1)}{2 - \cos t}, \frac{3 (\cos t - 5)}{2 - \cos t} \right).\]Then the equation of the ellipse can be written in the form
\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\]where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are integers, and $\gcd(|A|,|B|,|C|,|D|,|E|,|F|) = 1.$ Find $|A| + |B| + |C| + |D| + |E| + |F|.$
|
1381
| 0.375 |
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is
|
13
| 0.0625 |
A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, there are ________ possible duty arrangements for the week.
|
36
| 0.125 |
Alice and Bob are playing a game where Alice declares, "My number is 36." Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in Alice's. What is the smallest possible number Bob can choose?
|
72
| 0.1875 |
Let \( S = \{1, 2, \cdots, 2005\} \). If any \( n \) pairwise coprime numbers in \( S \) always include at least one prime number, find the minimum value of \( n \).
|
16
| 0.125 |
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \).
|
574
| 0.3125 |
Two circles have centers at (1,3) and (4,1) respectively. A line is tangent to the first circle at point (4,6) and to the second circle at point (7,4). Find the slope of the tangent line at these points.
|
-1
| 0.1875 |
What is the value of $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \cdots + [\sqrt{1989 \cdot 1990}] + [-\sqrt{1}] + [-\sqrt{2}] + [-\sqrt{3}] + \cdots + [-\sqrt{1989 \cdot 1990}]$?
(The 1st "Hope Cup" Mathematics Contest, 1990)
|
-3956121
| 0.1875 |
It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calculate the time it takes Mina to ride down the first operating escalator and then ride up the second operating escalator when she just stands on them.
|
69
| 0.3125 |
There are 8 students arranged in two rows, with 4 people in each row. If students A and B must be arranged in the front row, and student C must be arranged in the back row, then the total number of different arrangements is ___ (answer in digits).
|
5760
| 0.4375 |
Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$ Find $a-b$ .
|
22
| 0.1875 |
Alice and Bob are playing a game where Alice declares, "My number is 36." Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in Alice's. What is the smallest possible number Bob can choose?
|
72
| 0.1875 |
We have created a convex polyhedron using pentagons and hexagons where three faces meet at each vertex. Each pentagon shares its edges with 5 hexagons, and each hexagon shares its edges with 3 pentagons. How many faces does the polyhedron have?
|
32
| 0.375 |
A positive integer \( n \) cannot be divided by \( 2 \) or \( 3 \), and there do not exist non-negative integers \( a \) and \( b \) such that \( |2^a - 3^b| = n \). Find the smallest value of \( n \).
|
35
| 0.0625 |
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is
|
13
| 0.0625 |
Find the smallest positive integer $n$ for which $$1!2!\cdots(n-1)!>n!^{2}$$
|
8
| 0.375 |
Given $a= \int_{ 0 }^{ \pi }(\sin x-1+2\cos ^{2} \frac {x}{2})dx$, find the constant term in the expansion of $(a \sqrt {x}- \frac {1}{ \sqrt {x}})^{6}\cdot(x^{2}+2)$.
|
-332
| 0.375 |
What is the value of $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \cdots + [\sqrt{1989 \cdot 1990}] + [-\sqrt{1}] + [-\sqrt{2}] + [-\sqrt{3}] + \cdots + [-\sqrt{1989 \cdot 1990}]$?
(The 1st "Hope Cup" Mathematics Contest, 1990)
|
-3956121
| 0.1875 |
Define the sequence \left\{x_{i}\right\}_{i \geq 0} by $x_{0}=x_{1}=x_{2}=1$ and $x_{k}=\frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k>2$. Find $x_{2013}$.
|
9
| 0.3125 |
Suppose that $a$ and $b$ are nonzero integers such that two of the roots of
\[x^3 + ax^2 + bx + 9a\]coincide, and all three roots are integers. Find $|ab|.$
|
1344
| 0.4375 |
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$?
|
15
| 0.125 |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?
|
6
| 0.375 |
Alice and Bob are playing a game where Alice declares, "My number is 36." Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in Alice's. What is the smallest possible number Bob can choose?
|
72
| 0.1875 |
(In the coordinate system and parametric equations optional question) In the polar coordinate system, it is known that the line $l: p(\sin\theta - \cos\theta) = a$ divides the region enclosed by the curve $C: p = 2\cos\theta$ into two parts with equal area. Find the value of the constant $a$.
|
-1
| 0.4375 |
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)?
|
256
| 0.1875 |
A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$ . She also has a box with dimensions $10\times11\times14$ . The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.
Find all possible values of the total number of bricks that she can pack.
|
24
| 0.4375 |
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 of the lines $FH$ and $QO$ . Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\dfrac{a + b \sqrt{c}}{d}$ , then find the value of $1000a + 100b + 10c + d$ .
|
1132
| 0.125 |
Given the set $A=\{x|x=a_0+a_1\times3+a_2\times3^2+a_3\times3^3\}$, where $a_k\in\{0,1,2\}$ ($k=0,1,2,3$), and $a_3\neq0$, calculate the sum of all elements in set $A$.
|
2889
| 0.125 |
(In the coordinate system and parametric equations optional question) In the polar coordinate system, it is known that the line $l: p(\sin\theta - \cos\theta) = a$ divides the region enclosed by the curve $C: p = 2\cos\theta$ into two parts with equal area. Find the value of the constant $a$.
|
-1
| 0.4375 |
Máté is always in a hurry. He observed that it takes 1.5 minutes to get to the subway when he stands on the moving escalator, while it takes 1 minute to run down the stationary stairs. How long does it take Máté to get down if he can run down the moving escalator?
|
36
| 0.125 |
The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?
[asy]
draw((0,0)--(0,7)--(24,7)--(24,0)--cycle);
label("KEY:",(3,5));
fill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..cycle);
label("- one(1) baseball player",(14,2));
[/asy]
[asy]
draw((18,0)--(0,0)--(0,18));
label("6",(3,-1));
label("7",(6,-1));
label("8",(9,-1));
label("9",(12,-1));
label("10",(15,-1));
fill((3,.5)..(3.5,1)..(3,1.5)..(2.5,1)..cycle);
fill((3,2)..(3.5,2.5)..(3,3)..(2.5,2.5)..cycle);
fill((3,3.5)..(3.5,4)..(3,4.5)..(2.5,4)..cycle);
fill((3,5)..(3.5,5.5)..(3,6)..(2.5,5.5)..cycle);
fill((3,6.5)..(3.5,7)..(3,7.5)..(2.5,7)..cycle);
fill((3,8)..(3.5,8.5)..(3,9)..(2.5,8.5)..cycle);
fill((6,.5)..(6.5,1)..(6,1.5)..(5.5,1)..cycle);
fill((6,2)..(6.5,2.5)..(6,3)..(5.5,2.5)..cycle);
fill((6,3.5)..(6.5,4)..(6,4.5)..(5.5,4)..cycle);
fill((6,5)..(6.5,5.5)..(6,6)..(5.5,5.5)..cycle);
fill((9,.5)..(9.5,1)..(9,1.5)..(8.5,1)..cycle);
fill((9,2)..(9.5,2.5)..(9,3)..(8.5,2.5)..cycle);
fill((9,3.5)..(9.5,4)..(9,4.5)..(8.5,4)..cycle);
fill((15,.5)..(15.5,1)..(15,1.5)..(14.5,1)..cycle);
label("Number of Home Runs",(9,-3));
picture perpLabel;
label(perpLabel,"Number of Top Hitters");
add(rotate(90)*perpLabel,(-1,9));
[/asy]
|
7
| 0.0625 |
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.
|
76
| 0.4375 |
How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different?
|
616
| 0.1875 |
Find $\frac{a^{8}-256}{16 a^{4}} \cdot \frac{2 a}{a^{2}+4}$, if $\frac{a}{2}-\frac{2}{a}=3$.
|
33
| 0.375 |
In triangle $ABC,\,$ angle $C$ is a right angle and the altitude from $C\,$ meets $\overline{AB}\,$ at $D.\,$ The lengths of the sides of $\triangle ABC\,$ are integers, $BD=29^3,\,$ and $\cos B=m/n\,$, where $m\,$ and $n\,$ are relatively prime positive integers. Find $m+n.\,$
|
450
| 0.3125 |
Define the sequence \left\{x_{i}\right\}_{i \geq 0} by $x_{0}=x_{1}=x_{2}=1$ and $x_{k}=\frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k>2$. Find $x_{2013}$.
|
9
| 0.3125 |
(In the coordinate system and parametric equations optional question) In the polar coordinate system, it is known that the line $l: p(\sin\theta - \cos\theta) = a$ divides the region enclosed by the curve $C: p = 2\cos\theta$ into two parts with equal area. Find the value of the constant $a$.
|
-1
| 0.4375 |
How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different?
|
616
| 0.1875 |
Convert the quadratic equation $3x=x^{2}-2$ into general form and determine the coefficients of the quadratic term, linear term, and constant term.
|
-2
| 0.3125 |
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy = 1$ and both branches of the hyperbola $xy = -1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting them is contained in $S$.)
|
4
| 0.0625 |
A banquet has invited 44 guests. There are 15 identical square tables, each of which can seat 1 person per side. By appropriately combining the square tables (to form rectangular or square tables), ensure that all guests are seated with no empty seats. What is the minimum number of tables in the final arrangement?
|
11
| 0.375 |
Record the outcome of hitting or missing for 6 consecutive shots in order.
① How many possible outcomes are there?
② How many outcomes are there where exactly 3 shots hit the target?
③ How many outcomes are there where 3 shots hit the target, and exactly two of those hits are consecutive?
|
12
| 0.3125 |
In the diagram, \(ABCD\) is a rectangle with \(AD = 13\), \(DE = 5\), and \(EA = 12\). The area of \(ABCD\) is
|
60
| 0.375 |
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.
|
76
| 0.4375 |
Define an odd function f(x) on ℝ that satisfies f(x+1) is an even function, and when x ∈ [0,1], f(x) = x(3-2x). Evaluate f(31/2).
|
-1
| 0.125 |
What is the largest four-digit negative integer congruent to $1 \pmod{17}$?
|
-1002
| 0.3125 |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.
\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular}
|
135
| 0.0625 |
1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=
|
0
| 0.4375 |
A pyramid \( S A B C D \) has a trapezoid \( A B C D \) as its base, with bases \( B C \) and \( A D \). Points \( P_1, P_2, P_3 \) lie on side \( B C \) such that \( B P_1 < B P_2 < B P_3 < B C \). Points \( Q_1, Q_2, Q_3 \) lie on side \( A D \) such that \( A Q_1 < A Q_2 < A Q_3 < A D \). Let \( R_1, R_2, R_3, \) and \( R_4 \) be the intersection points of \( B Q_1 \) with \( A P_1 \); \( P_2 Q_1 \) with \( P_1 Q_2 \); \( P_3 Q_2 \) with \( P_2 Q_3 \); and \( C Q_3 \) with \( P_3 D \) respectively. It is known that the sum of the volumes of the pyramids \( S R_1 P_1 R_2 Q_1 \) and \( S R_3 P_3 R_4 Q_3 \) equals 78. Find the minimum value of
\[ V_{S A B R_1}^2 + V_{S R_2 P_2 R_3 Q_2}^2 + V_{S C D R_4}^2 \]
and give the closest integer to this value.
|
2028
| 0.1875 |
In triangle $ABC,\,$ angle $C$ is a right angle and the altitude from $C\,$ meets $\overline{AB}\,$ at $D.\,$ The lengths of the sides of $\triangle ABC\,$ are integers, $BD=29^3,\,$ and $\cos B=m/n\,$, where $m\,$ and $n\,$ are relatively prime positive integers. Find $m+n.\,$
|
450
| 0.3125 |
There are 8 students arranged in two rows, with 4 people in each row. If students A and B must be arranged in the front row, and student C must be arranged in the back row, then the total number of different arrangements is ___ (answer in digits).
|
5760
| 0.4375 |
Given the sets
$$
\begin{array}{l}
M=\{x, x y, \lg (x y)\} \\
N=\{0,|x|, y\},
\end{array}
$$
and $M=N$, determine the value of:
$$\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right).$$
|
-2
| 0.1875 |
Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
|
71
| 0.3125 |
A positive integer \( n \) cannot be divided by \( 2 \) or \( 3 \), and there do not exist non-negative integers \( a \) and \( b \) such that \( |2^a - 3^b| = n \). Find the smallest value of \( n \).
|
35
| 0.0625 |
Máté is always in a hurry. He observed that it takes 1.5 minutes to get to the subway when he stands on the moving escalator, while it takes 1 minute to run down the stationary stairs. How long does it take Máté to get down if he can run down the moving escalator?
|
36
| 0.125 |
What is the value of $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \cdots + [\sqrt{1989 \cdot 1990}] + [-\sqrt{1}] + [-\sqrt{2}] + [-\sqrt{3}] + \cdots + [-\sqrt{1989 \cdot 1990}]$?
(The 1st "Hope Cup" Mathematics Contest, 1990)
|
-3956121
| 0.1875 |
We have created a convex polyhedron using pentagons and hexagons where three faces meet at each vertex. Each pentagon shares its edges with 5 hexagons, and each hexagon shares its edges with 3 pentagons. How many faces does the polyhedron have?
|
32
| 0.375 |
It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calculate the time it takes Mina to ride down the first operating escalator and then ride up the second operating escalator when she just stands on them.
|
69
| 0.3125 |
What is the value of $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \cdots + [\sqrt{1989 \cdot 1990}] + [-\sqrt{1}] + [-\sqrt{2}] + [-\sqrt{3}] + \cdots + [-\sqrt{1989 \cdot 1990}]$?
(The 1st "Hope Cup" Mathematics Contest, 1990)
|
-3956121
| 0.1875 |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?
|
12
| 0.0625 |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?
|
6
| 0.375 |
Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$ Find $a-b$ .
|
22
| 0.1875 |
The circular region of the sign now has an area of 50 square inches. To decorate the edge with a ribbon, Vanessa plans to purchase 5 inches more than the circle’s circumference. How many inches of ribbon should she buy if she estimates \(\pi = \frac{22}{7}\)?
|
30
| 0.3125 |
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the integer part of $x$, that is, $\lfloor x \rfloor$ is the largest integer not exceeding $x$. Calculate the value of $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \lfloor \log_{2}4 \rfloor + \ldots + \lfloor \log_{2}1024 \rfloor$.
|
8204
| 0.4375 |
Alice and Bob are playing a game where Alice declares, "My number is 36." Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in Alice's. What is the smallest possible number Bob can choose?
|
72
| 0.1875 |
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