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36,001 | Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as $\frac{a}{b}$ for positive integers $a,b$ with $\gcd (a,b) = 1$, then find $a + b$.
[i]Proposed by Vismay Sharan[/i] | 831 |
36,016 | The sequence of real numbers $\{a_n\}$, $n \in \mathbb{N}$ satisfies the following condition: $a_{n+1}=a_n(a_n+2)$ for any $n \in \mathbb{N}$. Find all possible values for $a_{2004}$. | [-1, \infty) |
36,018 | Determine all the pairs $(a,b)$ of positive integers, such that all the following three conditions are satisfied:
1- $b>a$ and $b-a$ is a prime number
2- The last digit of the number $a+b$ is $3$
3- The number $ab$ is a square of an integer. | (4, 9) |
36,066 | Example 3 Color each vertex of an $n(n \geqslant 3)$-sided polygon with one color, such that the endpoints of the same edge have different colors. If there are $m(m \geqslant 3)$ colors available, then the total number of different coloring methods is | (m-1)\left[(m-1)^{n-1}+(-1)^{n}\right] |
36,087 | There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.
| 1 + \sqrt{3} |
36,106 | 3. If real numbers $m, n, p, q$ satisfy the conditions
$$
\begin{array}{l}
m+n+p+q=22, \\
m p=n q=100,
\end{array}
$$
then the value of $\sqrt{(m+n)(n+p)(p+q)(q+m)}$ is
$\qquad$ | 220 |
36,117 | A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism?
[i]Proposed by Deyuan Li and Andrew Milas[/i] | 3024 |
36,132 | '1.188 Write the numbers $1,2,3, \cdots, 1986,1987$ on the blackboard. At each step, determine some of the numbers written and replace them with the remainder of their sum divided by 7. After several steps, two numbers remain on the blackboard, one of which is 987. What is the second remaining number? | 0 |
36,144 | Maria and Bilyana play the following game. Maria has $2024$ fair coins and Bilyana has $2023$ fair coins. They toss every coin they have. Maria wins if she has strictly more heads than Bilyana, otherwise Bilyana wins. What is the probability of Maria winning this game? | \frac{1}{2} |
36,151 | A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares. | 27 |
36,177 | Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned in to $3$ subsets with equal sums. Find the number of elements of $E$. | 64 |
36,186 | Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$ | \left\lfloor \frac{n^2}{4} \right\rfloor |
36,192 | 7. Let the moving points be $P(t, 0), Q(1, t)(t \in[0,1])$. Then the area of the plane region swept by the line segment $P Q$ is $\qquad$ . | \frac{1}{6} |
36,210 | Example 12 Solve the equation
$$
5 x^{2}+x-x \sqrt{5 x^{2}-1}-2=0 \text {. }
$$
(1979, National High School Mathematics Competition) | x= \pm \frac{\sqrt{10}}{5} |
36,256 | The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P . | 23 |
36,259 | 4. If a positive integer is equal to 4 times the sum of its digits, then we call this positive integer a quadnumber. The sum of all quadnumbers is $\qquad$ .
| 120 |
36,321 | Find the remainder when $(1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)$ is divided by $43$. Your answer should be an integer between $0$ and $42$. | 4 |
36,342 | 3. Let $x=\sin \alpha+\cos \alpha$, and $\sin ^{3} \alpha+\cos ^{3} \alpha>$ 0. Then the range of $x$ is $\qquad$ | (0, \sqrt{2}] |
36,346 | $6 \cdot 103$ Find the smallest positive integer $n$ (where $n>1$) such that the quadratic mean of the first $n$ natural numbers is an integer. Here, the quadratic mean of $n$ numbers $a_{1}, a_{2}, \cdots, a_{n}$ is given by
$$\left(\frac{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}{n}\right)^{\frac{1}{2}} .$$ | 337 |
36,355 | Let $f(x)$ be a monic cubic polynomial with $f(0)=-64$, and all roots of $f(x)$ are non-negative real numbers. What is the largest possible value of $f(-1)$? (A polynomial is monic if its leading coefficient is 1.) | -125 |
36,374 | Example 4 Given that $a$ is an integer, the equation concerning $x$
$$
\frac{x^{2}}{x^{2}+1}-\frac{4|x|}{\sqrt{x^{2}+1}}+2-a=0
$$
has real roots. Then the possible values of $a$ are $\qquad$
(2008, I Love Mathematics Junior High School Summer Camp Mathematics Competition) | 0, 1, 2 |
36,378 | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy] | 732 |
36,385 | The World Cup, featuring $17$ teams from Europe and South America, as well as $15$ other teams that honestly don’t have a chance, is a soccer tournament that is held once every four years. As we speak, Croatia andMorocco are locked in a battle that has no significance whatsoever on the winner, but if you would like live score updates nonetheless, feel free to ask your proctor, who has no obligation whatsoever to provide them.
[b]p1.[/b] During the group stage of theWorld Cup, groups of $4$ teams are formed. Every pair of teams in a group play each other once. Each team earns $3$ points for each win and $1$ point for each tie. Find the greatest possible sum of the points of each team in a group.
[b]p2.[/b] In the semi-finals of theWorld Cup, the ref is bad and lets $11^2 = 121$ players per team go on the field at once. For a given team, one player is a goalie, and every other player is either a defender, midfielder, or forward. There is at least one player in each position. The product of the number of defenders, midfielders, and forwards is a mulitple of $121$. Find the number of ordered triples (number of defenders, number of midfielders, number of forwards) that satisfy these conditions.
[b]p3.[/b] Messi is playing in a game during the Round of $16$. On rectangular soccer field $ABCD$ with $AB = 11$, $BC = 8$, points $E$ and $F$ are on segment $BC$ such that $BE = 3$, $EF = 2$, and $FC = 3$. If the distance betweenMessi and segment $EF$ is less than $6$, he can score a goal. The area of the region on the field whereMessi can score a goal is $a\pi +\sqrt{b} +c$, where $a$, $b$, and $c$ are integers. Find $10000a +100b +c$.
[b]p4.[/b] The workers are building theWorld Cup stadium for the $2022$ World Cup in Qatar. It would take 1 worker working alone $4212$ days to build the stadium. Before construction started, there were 256 workers. However, each day after construction, $7$ workers disappear. Find the number of days it will take to finish building the stadium.
[b]p5.[/b] In the penalty kick shootout, $2$ teams each get $5$ attempts to score. The teams alternate shots and the team that scores a greater number of times wins. At any point, if it’s impossible for one team to win, even before both teams have taken all $5$ shots, the shootout ends and nomore shots are taken. If each team does take all $5$ shots and afterwards the score is tied, the shootout enters sudden death, where teams alternate taking shots until one team has a higher score while both teams have taken the same number of shots. If each shot has a $\frac12$ chance of scoring, the expected number of times that any team scores can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. | 18 |
36,388 | Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$. Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$.
[i]2017 CCA Math Bonanza Lightning Round #2.4[/i] | 13 |
36,401 | Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$, $60^\circ$, and $75^\circ$. | 3\sqrt{2} + 2\sqrt{3} - \sqrt{6} |
36,418 | Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$ | (3, 223, 9) |
36,451 | 1. Given positive integers $a$, $b$, $c$, $d$ satisfy $a^{2}=c(d+29)$, $b^{2}=c(d-29)$. Then the value of $d$ is $\qquad$. | 421 |
36,453 | 9. The sequence $1,1,2,1,1,3,1,1,1,4,1,1,1,1$, $5, \cdots, \underbrace{1,1, \cdots, 1}_{n-1 \uparrow}, n, \cdots$ has the sum of its first 2007 terms as | 3898 |
36,461 | Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\triangle GEM$. | 25 |
36,515 | The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$. | 0 |
36,519 | Four. (20 points) Let $x_{1}=a, x_{2}=b$,
$$
x_{n+2}=\frac{x_{n}+x_{n+1}}{2}+c \quad(n=1,2, \cdots) \text {, }
$$
where $a, b, c$ are given real numbers.
(1) Find the expression for $x_{n}$.
(2) For what value of $c$ does the limit $\lim _{n \rightarrow+\infty} x_{n}$ exist? If it exists, find its value. | \frac{a+2 b}{3} |
36,555 | 4. Given $x, y \geqslant 0$, and $x+y \leqslant 2 \pi$. Then the function
$$
f(x, y)=\sin x+\sin y-\sin (x+y)
$$
has a maximum value of | \frac{3 \sqrt{3}}{2} |
36,558 | 4. Let $P(x)=x^{4}+a x^{3}+b x^{2}+c x+d$, where $a, b, c, d$ are constants. If $P(1)=10, P(2)$ $=20, P(3)=30$, then $P(10)+P(-6)=$ $\qquad$ . | 8104 |
36,564 | Since $(36,83)=1$, by Lemma $1-$ there must exist two integers $x$, $y$ such that
$$36 x+83 y=1$$
holds, find $x, y$. | x=30, y=-13 |
36,592 | 8. The unit digit of $\left[\frac{10^{10000}}{10^{100}+9}\right]$ is | 1 |
36,624 | 3. One focus of a hyperbola is $F(0,4)$, the length of the real axis is 6, and point $A(0,3)$ lies on this hyperbola. Then the maximum value of the eccentricity of this hyperbola is $\qquad$ | \frac{4}{3} |
36,627 | Determine all integers $x$ satisfying
\[ \left[\frac{x}{2}\right] \left[\frac{x}{3}\right] \left[\frac{x}{4}\right] = x^2. \]
($[y]$ is the largest integer which is not larger than $y.$) | 0 \text{ and } 24 |
36,629 | 12. If the product of 3 prime numbers is 5 times their sum, then these 3 prime numbers are $\qquad$ . | 7,5,2 |
36,630 | 6. Given the function $f(x)=[x[x]]$, where $[x]$ denotes the greatest integer not exceeding $x$. If $x \in [0, n] (n \in \mathbf{N}_{+})$, the range of $f(x)$ is $A$, and let $a_{n}=\operatorname{card}(A)$, then $a_{n}=$ $\qquad$ | \frac{1}{2}\left(n^{2}-n+4\right) |
36,658 | 3. Among all the eight-digit numbers formed by the digits $1,2, \cdots, 8$ without repetition, the number of those divisible by 11 is. $\qquad$ | 4608 |
36,692 | 4. Arrange the squares of positive integers $1,2, \cdots$ in a sequence: $149162536496481100121144 \cdots$, the digit at the 1st position is 1, the digit at the 5th position is 6, the digit at the 10th position is 4, the digit at the 2008th position is $\qquad$. | 1 |
36,721 | 7. As shown in Figure 3, quadrilateral $ABCD$ is a right trapezoid $\left(\angle B=\angle C=90^{\circ}\right)$, and $AB=BC$. If there exists a point $M$ on side $BC$ such that $\triangle AMD$ is an equilateral triangle, then the value of $\frac{CD}{AB}$ is $\qquad$ | \sqrt{3}-1 |
36,742 | Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ? | \log_2 6 |
36,759 | Example 2 Given that $x$ and $y$ are real numbers, and satisfy $x y + x + y = 17, x^{2} y + x y^{2} = 66$.
Find the value of $x^{4} + x^{3} y + x^{2} y^{2} + x y^{3} + y^{4}$. (2000, Shandong Province Junior High School Mathematics Competition) | 12499 |
36,776 | $$
\begin{aligned}
M= & |2012 x-1|+|2012 x-2|+\cdots+ \\
& |2012 x-2012|
\end{aligned}
$$
The minimum value of the algebraic expression is . $\qquad$ | 1012036 |
36,782 | A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle. | 26 |
36,784 | 5. Define the sequence $\left\{a_{n}\right\}: a_{n}=4+n^{3}, n \in \mathbf{N}_{+}$. Let $d_{n}=\left(a_{n}, a_{n+1}\right)$, i.e., $d_{n}$ is the greatest common divisor of $a_{n}$ and $a_{n+1}$. Then the maximum value of $d_{n}$ is $\qquad$ | 433 |
36,796 | Example 12 Let $a_{1}, a_{2}, \cdots, a_{n}, \cdots$ be a non-decreasing sequence of positive integers. For $m \geqslant 1$, define $b_{m}=\min \left\{n, a_{n} \geqslant m\right\}$, i.e., $b_{m}$ is the smallest value of $n$ such that $a_{n} \geqslant m$. Given that $a_{19}=85$, find
$$a_{1}+a_{2}+\cdots+a_{19}+b_{1}+b_{2}+\cdots+b_{85}$$
the maximum value. | 1700 |
36,801 | Example 7 Given positive real numbers $x, y, z$ satisfy
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=9, \\
y^{2}+y z+z^{2}=16, \\
z^{2}+z x+x^{2}=25 .
\end{array}\right.
$$
Find the value of $x y+y z+z x$. | 8 \sqrt{3} |
36,819 | II. (50 points) Define a "Hope Set" (Hope Set) abbreviated as HS as follows: HS is a non-empty set that satisfies the condition "if $x \in \mathrm{HS}$, then $2 x \notin \mathrm{HS}$". How many "Hope Subsets" are there in the set $\{1,2, \cdots, 30\}$? Please explain your reasoning. | 26956799 |
36,823 | 8. Given $a, b \in R$. Try to find $s=\max \{|a+b|,|a-b|, |1-a|, |1-b|\}$, the minimum value. | \frac{2}{3} |
36,849 | 1. Let the sum of the digits of the natural number $x$ be $S(x)$. Then the solution set of the equation $x+S(x)+S(S(x))+S(S(S(x)))=2016$ is $\qquad$ | 1980 |
36,857 |
Problem G2. Consider a triangle $A B C$ and let $M$ be the midpoint of the side $B C$. Suppose $\angle M A C=\angle A B C$ and $\angle B A M=105^{\circ}$. Find the measure of $\angle A B C$.
| 30 |
36,866 | Example 2 Find a set $S$ composed of positive integers with at least two elements, such that the sum of all elements in $S$ equals the product of all elements in $S$.
(2006, Tsinghua University Independent Admission Examination) | S=\{1,2,3\} |
36,873 | Example 4 Let $z$ be a complex number with modulus 2. Then the sum of the maximum and minimum values of $\left|z-\frac{1}{z}\right|$ is $\qquad$ [2] | 4 |
36,877 | II. (20 points) Find the positive integer $n$ such that
$$
\left[\log _{3} 1\right]+\left[\log _{3} 2\right]+\cdots+\left[\log _{3} n\right]=2007 \text {, }
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$. | 473 |
36,887 | A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use? | n(n+1) |
36,907 | 5. Color each vertex of a square pyramid with one color, and make the endpoints of the same edge have different colors. If only 5 colors are available, the total number of different coloring methods is $\qquad$ | 420 |
36,913 | Six, let a large cube of $4 \times 4 \times 4$ be composed of 64 unit cubes. Select 16 of these unit cubes to be painted red, such that in the large cube, each $1 \times 1 \times 4$ small rectangular prism composed of 4 unit cubes contains exactly 1 red cube. How many different ways are there to select the 16 red cubes? Explain your reasoning. | 576 |
36,922 | Example 2 In $\triangle A B C$, $A B=A C, \angle B A C=$ $80^{\circ}, O$ is a point inside the triangle, $\angle O B C=10^{\circ}, \angle O C B=$ $30^{\circ}$. Find the degree measure of $\angle B A O$. | 70^{\circ} |
36,975 | 2. Let $x_{1}=\frac{1}{4}, x_{2}=\frac{3}{16}, x_{n+2}=\frac{3}{16}-\frac{1}{4}\left(x_{1}\right.$ $\left.+x_{2}+\cdots+x_{n}\right)(n=1,2, \cdots)$. Then $x_{n}=$ | \frac{n+1}{2^{n+2}} |
36,985 | 11. Given real numbers $a, b$ satisfy
$$
\left\{\begin{array}{l}
a+b-2 \geqslant 0, \\
b-a-1 \leqslant 0, \\
a \leqslant 1 .
\end{array}\right.
$$
Then the maximum value of $\frac{a+2 b}{2 a+b}$ is $\qquad$ | \frac{7}{5} |
37,011 | 4. As shown in Figure 1, let $A B C D-A_{1} B_{1} C_{1} D_{1}$ be a cube, and $M$ be the midpoint of edge $A A_{1}$. Then the size of the dihedral angle $C-M D_{1}-B_{1}$ is equal to $\qquad$ (express using radians or the arccosine function). | \arccos \frac{\sqrt{6}}{6} |
37,014 | 2. Let the sum of the squares of the first 101 positive integers starting from a positive integer $k$ be equal to the sum of the squares of the next 100 positive integers. Then the value of $k$ is $\qquad$ . | 20100 |
37,064 | 4. Group all positive integers that are coprime with 2012 in ascending order, with the $n$-th group containing $2n-1$ numbers:
$$
\{1\},\{3,5,7\},\{9,11,13,15,17\}, \cdots \text {. }
$$
Then 2013 is in the $\qquad$ group. | 32 |
37,067 | 2. Given that $a$ is a natural number, there exists a linear polynomial with integer coefficients and $a$ as the leading coefficient, which has two distinct positive roots less than 1. Then, the minimum value of $a$ is $\qquad$ . | 5 |
37,083 | Example 3. Find the value of $\cos \frac{\pi}{7}+\cos \frac{3 \pi}{7}+\cos \frac{5 \pi}{7}$. | \frac{1}{2} |
37,112 | Example 6 In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=2, A A_{1}=A D=1$. Find the angle formed by $A B$ and the plane $A B_{1} C$. | \arcsin \frac{1}{3} |
37,118 | II. (16 points) Find all four-digit numbers that satisfy the following conditions: they are divisible by 111, and the quotient obtained is equal to the sum of the digits of the four-digit number. | 2997 |
37,122 | Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$ | 43 |
37,132 | 5. Let $\{x\}$ denote the fractional part of the real number $x$. If $a=$ $(5 \sqrt{13}+18)^{2005}$, then $a \cdot\{a\}=$ $\qquad$ | 1 |
37,165 | Example 8 Let $x, y, z, w$ be real numbers, not all zero, find $\frac{x y+2 y z+z w}{x^{2}+y^{2}+z^{2}+w^{2}}$ | \frac{\sqrt{2}+1}{2} |
37,186 | Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009. | 42 |
37,206 | Derek fills a square $10$ by $10$ grid with $50$ $1$s and $50$ $2$s. He takes the product of the numbers in each of the $10$ rows. He takes the product of the numbers in each of the $10$ columns. He then sums these $20$ products up to get an integer $N.$ Find the minimum possible value of $N.$ | 640 |
37,219 | The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy
\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]
where $a,b$, and $c$ are (not necessarily distinct) digits. Find the three-digit number $abc$. | 447 |
37,233 | 8. Given the sequence $\left\{a_{n}\right\}$ with the first term being 2, and satisfying
$$
6 S_{n}=3 a_{n+1}+4^{n}-1 \text {. }
$$
Then the maximum value of $S_{n}$ is $\qquad$. | 35 |
37,248 | 8. Let the lines $l_{1} / / l_{2}$, and take 10 points $A_{1}, A_{2}, \cdots, A_{10}$ and $B_{1}, B_{2}, \cdots, B_{10}$ on $l_{1}$ and $l_{2}$ respectively. Then the line segments $A_{1} B_{1}, A_{2} B_{2}, \cdots, A_{10} B_{10}$ can divide the strip region between $l_{1}$ and $l_{2}$ into at most $\qquad$ non-overlapping parts. | 56 |
37,256 | Willy Wonka has $n$ distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is $p(n)$, then we have
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline
$n$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ \\\hline
$p(n)$ & $1$ & $2$ & $5$ & $15$ & $52$ & $203$ & $877$ & $4140$ & $21147$ & $115975$ \\\hline
\end{tabular}
Define a [i]splitting[/i] of the $n$ distinguishable pieces of candy to be a way of splitting them into groups. If Willy Wonka has $8$ candies, what is the sum of the number of groups over all splittings he can use?
[i]2020 CCA Math Bonanza Lightning Round #3.4[/i] | 17007 |
37,259 | Let $ n $ be a natural number. How many numbers of the form $ \pm 1\pm 2\pm 3\pm\cdots\pm n $ are there? | \frac{n(n+1)}{2} + 1 |
37,270 | 6. Two differently sized cubic building blocks are glued together, forming the three-dimensional shape shown in the right figure. Among them, the four vertices of the small block's glued face are the one-third points on each side of the large block's glued face. If the edge length of the large block is 3, then the surface area of this three-dimensional shape is $\qquad$
| 74 |
37,288 | 7. Let $A$ be the set of all positive integers not exceeding 2009, i.e., $A=\{1,2, \cdots, 2009\}$, and let $L \subseteq A$, where the difference between any two distinct elements of $L$ is not equal to 4. Then the maximum possible number of elements in the set $L$ is | 1005 |
37,292 | 6. Given $a \cdot b \neq 1$, and $5 a^{2}+1995 a+8=0$ as well as $8 b^{2}$ $+1995 b+5=0$. Then $\frac{a}{b}=$ $\qquad$ . | \frac{8}{5} |
37,297 | 8. Square $A B C D$ has a side length of $10 \mathrm{~cm}$, point $E$ is on the extension of side $C B$, and $E B=10$ $\mathrm{cm}$, point $P$ moves on side $D C$, and the intersection of $E P$ and $A B$ is point $F$. Let $D P=x \mathrm{~cm}$, and the sum of the areas of $\triangle E F B$ and quadrilateral $A F P D$ is $y \mathrm{~cm}^{2}$. Then, the functional relationship between $y$ and $x$ is $\qquad$ $(0<x<10)$. | y=5x+50 |
37,308 | There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group. | \frac{3}{455} |
37,325 | Find the positive integer $n\,$ for which
\[\lfloor\log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994\]
(For real $x\,$, $\lfloor x\rfloor\,$ is the greatest integer $\le x.\,$) | 312 |
37,333 | 4. Let $a$, $b$, $c$ be positive integers, and satisfy
$$
a^{2}+b^{2}+c^{2}-a b-b c-c a=19 \text {. }
$$
Then the minimum value of $a+b+c$ is $\qquad$ | 10 |
37,342 | Milly chooses a positive integer $n$ and then Uriel colors each integer between $1$ and $n$ inclusive red or blue. Then Milly chooses four numbers $a, b, c, d$ of the same color (there may be repeated numbers). If $a+b+c= d$ then Milly wins. Determine the smallest $n$ Milly can choose to ensure victory, no matter how Uriel colors. | 11 |
37,356 | 14. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=0, a_{n+1}=\frac{n+2}{n} a_{n}+\frac{1}{n}$. Find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{(n-1)(n+2)}{4} |
37,387 | 6. Let $a, b$ be positive integers, and $a+b \sqrt{2}$ $=(1+\sqrt{2})^{100}$. Then the units digit of $a b$ is $\qquad$ | 4 |
37,460 | Find all positive integers $m$ for which $2001\cdot S (m) = m$ where $S(m)$ denotes the sum of the digits of $m$. | 36018 |
37,516 | A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below $250$. How many bricks remain in the end? | 240 |
37,518 | 7. If $m$ and $n$ are positive integers, and $m \leqslant 1996, r=2-\frac{m}{n}$ $>0$, then the minimum value of $r$ is | \frac{1}{998} |
37,520 | Cyclic pentagon $ ABCDE$ has a right angle $ \angle{ABC} \equal{} 90^{\circ}$ and side lengths $ AB \equal{} 15$ and $ BC \equal{} 20$. Supposing that $ AB \equal{} DE \equal{} EA$, find $ CD$. | 7 |
37,522 | 5. Find the smallest positive integer $n$ such that the equation $\left[\frac{10^{n}}{x}\right]=1989$ has an integer solution $x$.
(1989, Soviet Union Mathematical Olympiad) | 7 |
37,565 | $101$ numbers are written on a blackboard: $1^2, 2^2, 3^2, \cdots, 101^2$. Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number. | 1 |
37,593 | In a rectangle $ABCD$, two segments $EG$ and $FH$ divide it into four smaller rectangles. $BH$ intersects $EG$ at $X$, $CX$ intersects $HF$ and $Y$, $DY$ intersects $EG$ at $Z$. Given that $AH=4$, $HD=6$, $AE=4$, and $EB=5$, find the area of quadrilateral $HXYZ$. | 8 |
37,625 | Example 7 Solve the equation $\sqrt{x-\frac{1}{x}}+\sqrt{\frac{x-1}{x}}=x$. | x=\frac{1+\sqrt{5}}{2} |
37,628 | 1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer.
2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$. | \frac{k^m}{n^{m-1}} |
37,675 | In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.
| (0, \frac{197}{4}) |
37,678 | In a [rectangular](https://artofproblemsolving.com/wiki/index.php/Rectangle) array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1 = y_2,$ $x_2 = y_1,$ $x_3 = y_4,$ $x_4 = y_5,$ and $x_5 = y_3.$ Find the smallest possible value of $N.$ | 149 |
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