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37,699 | ## Zadatak A-3.3. (4 boda)
Izračunaj $144^{\log _{5} 1000}: 10^{6 \log _{5} 12}$.
| 1 |
37,712 | Example 7 As shown in Figure 5, given points $A(0,3)$, $B(-2, -1)$, $C(2, -1)$, and $P\left(t, t^{2}\right)$ as a moving point on the parabola $y=x^{2}$ located within $\triangle A B C$ (including the boundary), the line $B P$ intersects $A C$ at point $E$, and the line $C P$ intersects $A B$ at point $F$. Express $\frac{B F}{C E}$ as a function of the variable $t$. | \frac{t^2 + 2t + 5}{t^2 - 2t + 5} |
37,720 | Example 3 In $\triangle A B C$, $\angle A B C=40^{\circ}$, $\angle A C B=30^{\circ}, P$ is a point on the bisector of $\angle A B C$, $\angle P C B=10^{\circ}$. Find the degree measure of $\angle P A B$. | 30^{\circ} |
37,758 | In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$.
[i]Proposed by Michael Tang[/i] | 33 |
37,782 | 14. There is a large square containing two smaller squares. These two smaller squares can move freely within the large square (no part of the smaller squares can move outside the large square, and the sides of the smaller squares must be parallel to the sides of the large square). If the minimum overlapping area of the two smaller squares is 9, and the maximum overlapping area is 25, and the sum of the side lengths of the three squares (one large square and two smaller squares) is 23, then the sum of the areas of the three squares is ( ).
| 189 |
37,798 | One. (20 points) Given that $a$ and $b$ are integers, the equation $a x^{2} + b x + 2 = 0$ has two distinct negative real roots greater than -1. Find the minimum value of $b$.
| 7 |
37,830 | Determine the number of real number $a$, such that for every $a$, equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:
(a) $x_0$ is an even integer;
(b) $|x_0|<1000$. | 999 |
37,864 | Example 5 As shown in Figure 5, in $\triangle A B C$, $A C=7, B C=$ $4, D$ is the midpoint of $A B$, $E$ is a point on side $A C$, and $\angle A E D=$ $90^{\circ}+\frac{1}{2} \angle C$. Find the length of $C E$. | 5.5 |
37,889 | A coin is flipped $20$ times. Let $p$ be the probability that each of the following sequences of flips occur exactly twice:
[list]
[*] one head, two tails, one head
[*] one head, one tails, two heads.
[/list]
Given that $p$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $\gcd (m,n)$.
[i]2021 CCA Math Bonanza Lightning Round #1.3[/i] | 1 |
37,944 | Example 3. As shown in Figure 3, through an internal point $P$ of $\triangle ABC$, three lines parallel to the three sides are drawn, resulting in three triangles $t_{1}, t_{2}$, and $t_{3}$ with areas 4, 9, and 49, respectively. Find the area of $\triangle ABC$.
(2nd American
Mathematical Invitational) | 144 |
37,952 | The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram? | 90 |
37,962 | Determine all four-digit numbers $\overline{abcd}$ which are perfect squares and for which the equality holds:
$\overline{ab}=3 \cdot \overline{cd} + 1$. | 2809 |
37,963 | BdMO National 2016 Higher Secondary
[u][b]Problem 4:[/b][/u]
Consider the set of integers $ \left \{ 1, 2, ......... , 100 \right \} $. Let $ \left \{ x_1, x_2, ......... , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, ......... , 100 \right \}$, where all of the $x_i$ are different. Find the smallest possible value of the sum,
$S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | $. | 198 |
37,964 | 3. In the positive geometric sequence $\left\{a_{n}\right\}$,
$$
a_{5}=\frac{1}{2}, a_{6}+a_{7}=3 \text {. }
$$
Then the maximum positive integer $n$ that satisfies $a_{1}+a_{2}+\cdots+a_{n}>a_{1} a_{2} \cdots a_{n}$ is $\qquad$ | 12 |
37,970 | Example 8 Solve the system of equations in the set of real numbers
$$\left\{\begin{array}{l}
2 x+3 y+z=13 \\
4 x^{2}+9 y^{2}+z^{2}-2 x+15 y+3 z=82
\end{array}\right.$$ | x=3, y=1, z=4 |
37,995 | Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the values of $\sin x$, $\cos x$, and $\tan x$ on three different cards. Then he gave those cards to three students, Malvina, Paulina, and Georgina, one card to each, and asked them to figure out which trigonometric function (sin, cos, or tan) produced their cards. Even after sharing the values on their cards with each other, only Malvina was able to surely identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Malvina's card.
| \frac{1 + \sqrt{5}}{2} |
38,011 | What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ?
[i]Posted already on the board I think...[/i] | \binom{n}{4} + \binom{n}{2} + 1 |
38,037 | 5. Place a sphere inside a cone, which is tangent to the cone's side and base. Then the maximum ratio of the sphere's surface area to the cone's surface area is $\qquad$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{1}{2} |
38,072 | Example 6 Solve the system of equations $\left\{\begin{array}{l}x+y+z=3, \\ x^{2}+y^{2}+z^{2}=3,(\text{Example 4 in [1]}) \\ x^{5}+y^{5}+z^{5}=3 .\end{array}\right.$ | x=y=z=1 |
38,085 | There are two equilateral triangles with a vertex at $(0, 1)$, with another vertex on the line $y = x + 1$ and with the final vertex on the parabola $y = x^2 + 1$. Find the area of the larger of the two triangles. | 26\sqrt{3} + 45 |
38,094 | 1. Given that $a$, $b$, $c$, and $d$ are prime numbers, and $a b c d$ is the sum of 77 consecutive positive integers. Then the minimum value of $a+b+c+d$ is $\qquad$ | 32 |
38,100 | Example 3 Let $n$ be a positive integer,
$$
\begin{aligned}
S= & \{(x, y, z) \mid x, y, z \in\{0,1, \cdots, n\}, \\
& x+y+z>0\}
\end{aligned}
$$
is a set of $(n+1)^{3}-1$ points in three-dimensional space. Try to find the minimum number of planes whose union contains $S$ but does not contain $(0,0,0)$. ${ }^{[4]}$ | 3n |
38,105 | Let $\{a_n\}$ be a sequence satisfying: $a_1=1$ and $a_{n+1}=2a_n+n\cdot (1+2^n),(n=1,2,3,\cdots)$. Determine the general term formula of $\{a_n\}$. | a_n = 2^{n-2} (n^2 - n + 6) - n - 1 |
38,140 | For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ (e.g. $d(12)=6$) and let $s(k)$ denote the digit sum of $k$ (e.g. $s(12)=3$). A positive integer $n$ is said to be amusing if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than 1? | 9 |
38,145 | Find all natural numbers $n$ such that $5^{n}+12^{n}$ is a perfect square. | n=2 |
38,155 | 2. For the quadratic function $y=x^{2}+b x+c$, the vertex of its graph is $D$, and it intersects the positive x-axis at points $A$ and $B$ from left to right, and the positive y-axis at point $C$. If $\triangle A B D$ and $\triangle O B C$ are both isosceles right triangles (where $O$ is the origin), then $b+2 c=$ | 2 |
38,166 | 4. If real numbers $x, y, z$ satisfy
$$
x^{2}+y^{2}+z^{2}-(x y+y z+z x)=8 \text {, }
$$
let $A$ denote the maximum value of $|x-y|, |y-z|, |z-x|$, then the maximum value of $A$ is $\qquad$. | \frac{4 \sqrt{6}}{3} |
38,191 | 6. Given that the tangents at two points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ on the parabola $y=4 x$ are perpendicular to each other. Then the equation of the locus of the intersection point of the tangents is $\qquad$ . | x=-1 |
38,208 | 6. In trapezoid $A B C D$, $A D / / B C, E F$ is the midline, the area ratio of quadrilateral $A E F D$ to quadrilateral $E B C F$ is $\frac{\sqrt{3}+1}{3-\sqrt{3}}$, and the area of $\triangle A B D$ is $\sqrt{3}$. Then the area of trapezoid $A B C D$ is | 2 |
38,211 | 13. If real numbers $x, y$ satisfy $x^{2}+y^{2}=1$, then the minimum value of $\frac{2 x y}{x+y-1}$ is $\qquad$ | 1-\sqrt{2} |
38,215 | Determine the least $n\in\mathbb{N}$ such that $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$ has at least $2010$ positive factors. | 14 |
38,228 | We are given five equal-looking weights of pairwise distinct masses. For any three weights $A$, $B$, $C$, we can check by a measuring if $m(A) < m(B) < m(C)$, where $m(X)$ denotes the mass of a weight $X$ (the answer is [i]yes[/i] or [i]no[/i].) Can we always arrange the masses of the weights in the increasing order with at most nine measurings? | 9 |
38,250 | Let $ABC$ be a triangle whose angles measure $A$, $B$, $C$, respectively. Suppose $\tan A$, $\tan B$, $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$, find the number of possible integer values for $\tan B$. (The values of $\tan A$ and $\tan C$ need not be integers.)
[i] Proposed by Justin Stevens [/i] | 11 |
38,261 | Let $n$ be a positive integer. Mariano divides a rectangle into $n^2$ smaller rectangles by drawing $n-1$ vertical lines and $n-1$ horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer $k$ for which it is possible that Emilio can do $k$ conveniently thought steps in such a way that with the received information, he can determine the area of each one of the $n^2$ smaller rectangles.
| 2n - 1 |
38,275 | 2. As shown in the figure, in $\triangle ABC$, $\angle C=90^{\circ}$, points $P, Q$ are on the hypotenuse $AB$, satisfying the conditions $BP = BC = a$, $AQ = AC = b$, $AB = c$, and $b > a$. Draw $PM \perp BC$ at $M$, $QN \perp AC$ at $N$, and $PM$ and $QN$ intersect at $L$. Given that $\frac{S_{\triangle PQL}}{S_{\triangle ABC}}=\frac{4}{25}$, then $a: b: c=$ | 3: 4: 5 |
38,279 | 3. Real numbers $x, y, z$ satisfy $x^{2}+y^{2}-x+y=1$. Then the range of the function
$$
f(x, y, z)=(x+1) \sin z+(y-1) \cos z
$$
is . $\qquad$ | \left[-\frac{3 \sqrt{2}+\sqrt{6}}{2}, \frac{3 \sqrt{2}+\sqrt{6}}{2}\right] |
38,293 | Let $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$. For all points $(a, b)$, let $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$. Kathy constructs a set $T$ by adding $n$ distinct points from $S$ to $T$ at random. If the expected value of $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$ is 4, then compute $n$.
[i]Proposed by Lewis Chen[/i] | 2013 |
38,294 | In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is $36^o$. Find all possibilities, how big the angles of this triangle can be. | (36^\circ, 48^\circ, 96^\circ) |
38,315 | 4. As shown in the figure, in $\triangle A B C$, $\angle A=60^{\circ}$, points $D, E$ are on side $A B$, and $F$, $G$ are on side $C A$. Connecting $B F, F E, E G, G D$ divides $\triangle A B C$ into five smaller triangles of equal area, i.e., $S_{\triangle C B F}=S_{\triangle F B E}=S_{\triangle F E C}=S_{\triangle G E D}=S_{\triangle G D A \text { V }}$ and $E B=2 C F$. Find the value of $\frac{B C}{E F}$. | \frac{\sqrt{7}}{2} |
38,317 | 5. In $\triangle A B C$, $A B=3, A C=4, B C=5, I$ is the incenter of $\triangle A B C$, and $P$ is a point within $\triangle I B C$ (including the boundary). If $\overrightarrow{A P}=\lambda \overrightarrow{A B}+\mu \overrightarrow{A C}(\lambda ; \mu \in \mathbf{R})$, then the minimum value of $\lambda+\mu$ is | \frac{7}{12} |
38,319 | 11. Given $\alpha, \beta \in\left[0, \frac{\pi}{4}\right]$. Then the maximum value of $\sin (\alpha-\beta)+$ $2 \sin (\alpha+\beta)$ is $\qquad$ . | \sqrt{5} |
38,333 | Suppose $p < q < r < s$ are prime numbers such that $pqrs + 1 = 4^{p+q}$. Find $r + s$. | 274 |
38,335 | Example 4 Given a unit cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$, $E$ is a moving point on $B C$, and $F$ is the midpoint of $A B$. Determine the position of point $E$ such that $C_{1} F \perp A_{1} E$. | a=\frac{1}{2} |
38,341 | 3. If
$$
\dot{z}=\frac{(1+i)^{2000}(6+2 i)-(1-i)^{1998}(3-i)}{(1+i)^{1996}(23-7 i)+(1-i)^{1994}(10+2 i)} \text {, }
$$
then $|z|=$ . $\qquad$ | 1 |
38,361 | Example 4 Let $x=b y+c z, y=c z+a x, z=a x$ $+b y$. Find the value of $\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}$. | 1 |
38,365 | Example 4—(2005 National High School Mathematics League Additional Question 2) Let positive numbers $a, b, c, x, y, z$ satisfy $c y + b z = a, a z + c x = b, b x + a y = c$, find the minimum value of the function $f(x, y, z) = \frac{x^{2}}{1+x} + \frac{y^{2}}{1+y} + \frac{z^{2}}{1+z}$. | \frac{1}{2} |
38,371 | In triangle $ABC,$ point $D$ is on $\overline{BC}$ with $CD = 2$ and $DB = 5,$ point $E$ is on $\overline{AC}$ with $CE = 1$ and $EA = 3,$ $AB = 8,$ and $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ Points $Q$ and $R$ lie on $\overline{AB}$ so that $\overline{PQ}$ is parallel to $\overline{CA}$ and $\overline{PR}$ is parallel to $\overline{CB}.$ It is given that the ratio of the area of triangle $PQR$ to the area of triangle $ABC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 901 |
38,417 | Zou and Chou are practicing their $100$-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 97 |
38,431 | Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is [i]mediocre[/i] if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$.
Determine the largest mediocre number. | 629 |
38,469 | Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$? | 360 |
38,479 | Let $k$ be a positive integer. $12k$ persons have participated in a party and everyone shake hands with $3k+6$ other persons. We know that the number of persons who shake hands with every two persons is a fixed number. Find $k.$ | k = 3 |
38,480 | Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1.$ Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b.$ Find the sum of all possible values of $(a+b)^2.$ | 85 |
38,520 | 16. Given that one side of the square $A B C D$ lies on the line $y=2 x-17$, and the other two vertices are on the parabola $y=x^{2}$. Then the minimum value of the area of the square is $\qquad$ . | 80 |
38,524 | Faces $ABC^{}_{}$ and $BCD^{}_{}$ of tetrahedron $ABCD^{}_{}$ meet at an angle of $30^\circ$. The area of face $ABC^{}_{}$ is $120^{}_{}$, the area of face $BCD^{}_{}$ is $80^{}_{}$, and $BC=10^{}_{}$. Find the volume of the tetrahedron. | 320 |
38,531 | 3. Let $x, y, z \geqslant 0, x+y+z=1$. Then
$$
\sqrt{2011 x+1}+\sqrt{2011 y+1}+\sqrt{2011 z+1}
$$
The sum of the maximum and minimum values is $\qquad$ L. | \sqrt{6042}+2 \sqrt{503}+2 |
38,559 | 3. If the real number $a$ satisfies $a^{3}+a^{2}-3 a+2=\frac{3}{a}-$ $\frac{1}{a^{2}}-\frac{1}{a^{3}}$, then $a+\frac{1}{a}=$ $\qquad$ | 2 \text{ or } -3 |
38,574 | For a positive integer $n$, define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$. Find the positive integer $k$ for which $7?9?=5?k?$.
[i]Proposed by Tristan Shin[/i] | 10 |
38,599 |
1. Let $A B C D$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B, D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A, C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arcs $A C$ and $B D$ both externally and also touching the side $C D$. Find the radius of the circle $\Gamma$.
| \frac{1}{16} |
38,600 | 6. The axial cross-section of a large round wine glass is the graph of the function $y=x^{4}$. A cherry, a small sphere with radius $r$, is placed into the wine glass. What is the maximum radius $r$ for the cherry to touch the lowest point of the bottom of the glass (in other words, what is the maximum $r$ such that a circle with radius $r$ located in the region $y \geqslant x^{4}$ can pass through the origin)? | \frac{3}{4} \sqrt[3]{2} |
38,611 | For positive numbers $x$, $y$, and $z$ satisfying $x^{2}+y^{2}+z^{2}=1$, find
$$\frac{x}{1-x^{2}}+\frac{y}{1-y^{2}}+\frac{z}{1-z^{2}}$$
the minimum value. | \frac{3 \sqrt{3}}{2} |
38,620 | Let $S_1$ and $S_2$ be sets of points on the coordinate plane $\mathbb{R}^2$ defined as follows
\[S_1={(x,y)\in \mathbb{R}^2:|x+|x||+|y+|y||\le 2}\]
\[S_2={(x,y)\in \mathbb{R}^2:|x-|x||+|y-|y||\le 2}\]
Find the area of the intersection of $S_1$ and $S_2$ | 3 |
38,645 | It's pouring down rain, and the amount of rain hitting point $(x,y)$ is given by
$$f(x,y)=|x^3+2x^2y-5xy^2-6y^3|.$$
If you start at the origin $(0,0)$, find all the possibilities for $m$ such that $y=mx$ is a straight line along which you could walk without any rain falling on you. | -1, \frac{1}{2}, -\frac{1}{3} |
38,650 | A rectangle, $HOMF$, has sides $HO=11$ and $OM=5$. A triangle $\Delta ABC$ has $H$ as orthocentre, $O$ as circumcentre, $M$ be the midpoint of $BC$, $F$ is the feet of altitude from $A$. What is the length of $BC$ ?
[asy]
unitsize(0.3 cm);
pair F, H, M, O;
F = (0,0);
H = (0,5);
O = (11,5);
M = (11,0);
draw(H--O--M--F--cycle);
label("$F$", F, SW);
label("$H$", H, NW);
label("$M$", M, SE);
label("$O$", O, NE);
[/asy] | 28 |
38,667 | 2. Given real numbers $x, y$ satisfy
$$
\begin{array}{l}
\sqrt{125 x-16}+\sqrt{25-8 y}=6, \\
\sqrt{25-80 x}+\sqrt{12.5 y-16}=6 .
\end{array}
$$
Then $\frac{x}{y}=$ | 0.1 |
38,699 | 3. As shown in Figure 2, quadrilateral $ABCD$ is inscribed in $\odot O$, with $BD$ being the diameter of $\odot O$, and $\overparen{AB}=\overparen{AD}$. If $BC + CD = 4$, then the area of quadrilateral $ABCD$ is $\qquad$ . | 4 |
38,721 | Example 3. As shown in Figure 3, the equilateral triangles $\triangle A B C$ and $\triangle A_{1} B_{1} C_{1}$ have their sides $A C$ and $A_{1} C_{1}$ bisected at $O$. Then $A A_{1}: B B_{1}=$ $\qquad$ | \sqrt{3} : 3 |
38,727 | Example 3 Given in $\triangle A B C$, $A B=A C$, $\angle B A C=100^{\circ}, P$ is a point on the angle bisector of $\angle C$, $\angle P B C=10^{\circ}$. Find the degree measure of $\angle A P B$. | 70^{\circ} |
38,731 | 4. Given a regular tetrahedron $P-ABC$ with a base edge length of 1 and a height of $\sqrt{2}$. Then its inscribed sphere radius is $\qquad$ . | \frac{\sqrt{2}}{6} |
38,772 | Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares. | 0 |
38,774 | Example 4. On the sides $AB$, $BC$, and $CA$ of an equilateral triangle $ABC$, there are moving points $D$, $E$, and $F$ respectively, such that $|AD| + |BE| + |CF| = |AB|$. If $|AB| = 1$, when does the area of $\triangle DEF$ reach its maximum value? What is this maximum value?
---
Translating the text as requested, while preserving the original formatting and line breaks. | \frac{\sqrt{3}}{12} |
38,777 | 4. The largest integer not exceeding $(\sqrt{5}+\sqrt{3})^{6}$ is
$\qquad$ | 3903 |
38,787 | In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.
[asy]
size(250);
defaultpen (linewidth (0.7) + fontsize (10));
pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3);
draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0));
label("$A$", (.5,.5));
label("$M$", (7/6, 1/6));
label("$N$", (1/3, 4/3));[/asy]
[i]Proposed by Aaron Lin[/i] | 36 |
38,795 | Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$. Determine the area of triangle $ AMC$. | \frac{1}{2} |
38,809 | Example 4 Given that $\alpha$ and $\beta$ are acute angles, and $\cos \alpha + \cos \beta - \cos (\alpha + \beta) = \frac{3}{2}$, find the values of $\alpha$ and $\beta$.
| \alpha=\beta=\frac{\pi}{3} |
38,817 | Let $a, b, x, y$ be positive real numbers such that $a+b=1$. Prove that $\frac{1}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$
and find when equality holds. | \frac{1}{\frac{a}{x} + \frac{b}{y}} \leq ax + by |
38,827 | 3. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n+1}=a_{n}^{2}-2\left(n \in \mathbf{N}_{+}\right) \text {, }
$$
and $a_{1}=a, a_{2012}=b(a 、 b>2)$.
Then $a_{1} a_{2} \cdots a_{2011}=$
(express in terms of $a, b$). | \sqrt{\frac{b^{2}-4}{a^{2}-4}} |
38,829 | For which real numbers $ x>1$ is there a triangle with side lengths $ x^4\plus{}x^3\plus{}2x^2\plus{}x\plus{}1, 2x^3\plus{}x^2\plus{}2x\plus{}1,$ and $ x^4\minus{}1$? | x > 1 |
38,841 | The numbers $1,2,\dots,10$ are written on a board. Every minute, one can select three numbers $a$, $b$, $c$ on the board, erase them, and write $\sqrt{a^2+b^2+c^2}$ in their place. This process continues until no more numbers can be erased. What is the largest possible number that can remain on the board at this point?
[i]Proposed by Evan Chen[/i] | 8\sqrt{6} |
38,845 | Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$. | \frac{3\sqrt{3}}{2} |
38,884 | 9. Let $\triangle A B C$ have internal angles $\angle A, \angle B, \angle C$ with opposite sides $a, b, c$ respectively, and $\angle A - \angle C = \frac{\pi}{2}, a, b, c$ form an arithmetic sequence. Then the value of $\cos B$ is . $\qquad$ | \frac{3}{4} |
38,913 | A Geostationary Earth Orbit is situated directly above the equator and has a period equal to the Earth’s rotational period. It is at the precise distance of $22,236$ miles above the Earth that a satellite can maintain an orbit with a period of rotation around the Earth exactly equal to $24$ hours. Be cause the satellites revolve at the same rotational speed of the Earth, they appear stationary from the Earth surface. That is why most station antennas (satellite dishes) do not need to move once they have been properly aimed at a tar get satellite in the sky. In an international project, a total of ten stations were equally spaced on this orbit (at the precise distance of $22,236$ miles above the equator). Given that the radius of the Earth is $3960$ miles, find the exact straight distance between two neighboring stations. Write your answer in the form $a + b\sqrt{c}$, where $a, b, c$ are integers and $c > 0$ is square-free. | -13098 + 13098\sqrt{5} |
38,920 | 12. In a game, two players take turns to "eat squares" from a $5 \times 7$ grid chessboard. To "eat a square," a player selects an uneaten square and moves the piece to that square, then all the squares in the quadrant formed (along the left edge of the square upwards, and along the bottom edge of the square to the right) are eaten. For example, in the right figure, moving the piece to the shaded square results in the shaded square and the four squares marked with $\times$ being eaten (the squares with lines in them were previously eaten). The goal of the game is to make the opponent eat the last square. The figure above shows a situation that can occur during the game. How many different situations can appear at most during the game process? | 792 |
38,961 | 7. In the Cartesian coordinate system $x O y$, the graph of the function $f(x)=$ $a \sin a x+\cos a x(a>0)$ over an interval of the smallest positive period length and the graph of the function $g(x)=\sqrt{a^{2}+1}$ enclose a closed figure whose area is $\qquad$ | \frac{2 \pi}{a} \sqrt{a^{2}+1} |
38,982 | 5. Given that $f(x)$ is a function defined on $\mathbf{R}$. If $f(0)=0$, and for any $x \in \mathbf{R}$, it satisfies
$$
\begin{array}{l}
f(x+4)-f(x) \leqslant x^{2}, \\
f(x+16)-f(x) \geqslant 4 x^{2}+48 x+224,
\end{array}
$$
then $f(64)=$ $\qquad$ | 19840 |
38,984 | Example 2 Let the perfect square $y^{2}$ be the sum of the squares of 11 consecutive integers. Then the minimum value of $|y|$ is $\qquad$ . | 11 |
39,010 | In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$. | 20 |
39,013 | Let $a_1$, $a_2$, $\ldots\,$, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\ell$, and $m$ from 1 through 2019, at least two of the numbers $a_i$, $a_j$, $a_k$, $a_\ell$, and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the given sequence? | 8 |
39,047 | Beto plays the following game with his computer: initially the computer randomly picks $30$ integers from $1$ to $2015$, and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer $k$ and some if the numbers written on the chalkboard, and subtracts $k$ from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all $30$ numbers to $0$, in which case the game ends. Find the minimal number $n$ such that, regardless of which numbers the computer chooses, Beto can end the game in at most $n$ turns. | 11 |
39,053 | Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$). Find the least $m$ such that $m-b_m = 1990$. | 2^{1990} - 1 |
39,069 | Example 2 Given the sets
$$
\begin{array}{l}
M=\{(x, y) \mid x(x-1) \leqslant y(1-y)\}, \\
N=\left\{(x, y) \mid x^{2}+y^{2} \leqslant k\right\} .
\end{array}
$$
If $M \subset N$, then the minimum value of $k$ is $\qquad$ .
(2007, Shanghai Jiao Tong University Independent Admission Examination) | 2 |
39,071 | Example 3 In $\triangle A B C$, $\angle C=90^{\circ}, B C=2$, $P$ is a point inside $\triangle A B C$ such that the minimum value of $P A+P B+P C$ is $2 \sqrt{7}$. Find the degree measure of $\angle A B C$.
Analysis: The key to this problem is to find the line segment when $P A+P B+P C$ reaches its minimum value. From the analysis of Example 2, if we construct an equilateral $\triangle A C D$ outside $\triangle A B C$ with $A C$ as one side, and connect $B D$, then the length of $B D$ is the minimum value of $P A+P B+P C$. | 60^{\circ} |
39,093 | 9. (16 points) Let the range of the function $f(x)$ be $[1,2]$, and let $g(x)=f(x)+\frac{a}{f(x)}$. Find the minimum value of $p(a)=\max g(x)-\min g(x)$. | 3-2\sqrt{2} |
39,101 | A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10} = .4$ and $a_n\le.4$ for all $n$ such that $1\le n\le9$ is given to be $p^aq^br/\left(s^c\right)$ where $p$, $q$, $r$, and $s$ are primes, and $a$, $b$, and $c$ are positive integers. Find $\left(p+q+r+s\right)\left(a+b+c\right)$. | 660 |
39,102 | The diagram below shows some small squares each with area $3$ enclosed inside a larger square. Squares that touch each other do so with the corner of one square coinciding with the midpoint of a side of the other square. Find integer $n$ such that the area of the shaded region inside the larger square but outside the smaller squares is $\sqrt{n}$.
[asy]
size(150);
real r=1/(2sqrt(2)+1);
path square=(0,1)--(r,1)--(r,1-r)--(0,1-r)--cycle;
path square2=(0,.5)--(r/sqrt(2),.5+r/sqrt(2))--(r*sqrt(2),.5)--(r/sqrt(2),.5-r/sqrt(2))--cycle;
defaultpen(linewidth(0.8));
filldraw(unitsquare,gray);
filldraw(square2,white);
filldraw(shift((0.5-r/sqrt(2),0.5-r/sqrt(2)))*square2,white);
filldraw(shift(1-r*sqrt(2),0)*square2,white);
filldraw(shift((0.5-r/sqrt(2),-0.5+r/sqrt(2)))*square2,white);
filldraw(shift(0.5-r/sqrt(2)-r,-(0.5-r/sqrt(2)-r))*square,white);
filldraw(shift(0.5-r/sqrt(2)-r,-(0.5+r/sqrt(2)))*square,white);
filldraw(shift(0.5+r/sqrt(2),-(0.5+r/sqrt(2)))*square,white);
filldraw(shift(0.5+r/sqrt(2),-(0.5-r/sqrt(2)-r))*square,white);
filldraw(shift(0.5-r/2,-0.5+r/2)*square,white);
[/asy] | 288 |
39,109 | Example 1 Let $a, b, c$ be the lengths of the three sides of a right-angled triangle, where $c$ is the length of the hypotenuse. Find the maximum value of $k$ such that $\frac{a^{3}+b^{3}+c^{3}}{a b c} \geqslant k$ holds. ${ }^{[2]}$ | 2+\sqrt{2} |
39,112 | II. (50 points) Let positive numbers $a, b, c, x, y, z$ satisfy $c y + b z = a$, $a z + c x = b$, $b x + a y = c$. Find the minimum value of the function
$$
f(x, y, z)=\frac{x^{2}}{1+x}+\frac{y^{2}}{1+y}+\frac{z^{2}}{1+z}
$$ | \frac{1}{2} |
39,122 | Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*} | 383 |
39,129 | Let $P$ be a point outside a circle $\Gamma$ centered at point $O$, and let $PA$ and $PB$ be tangent lines to circle $\Gamma$. Let segment $PO$ intersect circle $\Gamma$ at $C$. A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$, respectively. Given that $EF=8$ and $\angle{APB}=60^\circ$, compute the area of $\triangle{AOC}$.
[i]2020 CCA Math Bonanza Individual Round #6[/i] | 12 \sqrt{3} |
39,144 | One, (20 points) As shown in the figure, given that $AB, CD$ are perpendicular chords in a circle $\odot O$ with radius 5, intersecting at point $P$. $E$ is the midpoint of $AB$, $PD=AB$, and $OE=3$. Try to find the value of $CP + CE$.
---
The translation is provided as requested, maintaining the original text's format and line breaks. | 4 |
39,145 | Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2-bx+c=0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$. | 64 |
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