Found problems: 85335
From a square with sides of length $2m$, corners are cut away so as to form a regular octagon. What is the area of the octagon in $m^2$?
$\textbf{(A)}~2\sqrt3$
$\textbf{(B)}~\frac4{\sqrt3}$
$\textbf{(C)}~4\left(\sqrt2-1\right)$
$\textbf{(D)}~\text{None of the above}$
Suppose $ABCD$ is an isosceles trapezoid in which $\overline{AB}\parallel\overline{CD}$. Two mutually externally tangent circles $\omega_1$ and $\omega_2$ are inscribed in $ABCD$ such that $\omega_1$ is tangent to $\overline{AB}$,$\overline{BC}$, and $\overline{CD}$ while $\omega_2$ is tangent to $\overline{AB}$, $\overline{DA}$, and $\overline{CD}$. Given that $AB=1$, $CD=6$, compute the radius of either circle.
How many positive integer solutions are there to $w+x+y+z=20$ where $w+x\ge 5$ and $y+z\ge 5$?
(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$.
(2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition:
For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ .
Sketch the outline of $S$.
(3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition:
For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds.
Find the volume of $V$.
[i]2011 Tokyo University entrance exam/Science, Problem 6[/i]
A set of points is $\emph{convex}$ if the points are the vertices of a convex polygon (that is, a non-self-intersecting polygon with all angles less than or equal to $180^\circ$). Let $S$ be the set of points $(x,y)$ such that $x$ and $y$ are integers and $ 1 \le x, y \le 26 $. Find the number of ways to choose a convex subset of $S$ that contains exactly $98$ points.
Farmer Jim has an $8$ gallon bucket full with water. He has three empty buckets: $3$ gallons, $5$ gallons and $8$ gallons. How can he get two volumes of water, $4$ gallons each, using only the four buckets?
Let $EFGH,ABCD$ and $E_1F_1G_1H_1$ be three convex quadrilaterals satisfying:
i) The points $E,F,G$ and $H$ lie on the sides $AB,BC,CD$ and $DA$ respectively, and $\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1$;
ii) The points $A,B,C$ and $D$ lie on sides $H_1E_1,E_1F_1,F_1,G_1$ and $G_1H_1$ respectively, and $E_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE$.
Suppose that $\frac{E_1A}{AH_1}=\lambda$. Find an expression for $\frac{F_1C}{CG_1}$ in terms of $\lambda$.
[i]Xiong Bin[/i]
Find the maximum possible value of $9\sqrt{x} + 8\sqrt{y} + 5\sqrt{z}$ where $x, y,$ and $z$ are positive real numbers satisfying $9x + 4y + z = 128$.
How many reorderings of $2,3,4,5,6$ have the property that every pair of adjacent numbers are relatively prime?
[i]2021 CCA Math Bonanza Individual Round #3[/i]
The plane is divided into regions by $ n \ge 3$ lines, no two of which are parallel, and no three of which are concurrent. Some regions are coloured , in such a way that no two coloured regions share a common segment or half-line of their borders. Prove that the number of coloured regions is at most $ \frac{n(n\plus{}1)}{3}$
Two integers have a sum of $26$. When two more integers are added to the first two integers the sum is $41$. Finally when two more integers are added to the sum of the previous four integers the sum is $57$. What is the minimum number of even integers among the $6$ integers?
${{ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4}\qquad\textbf{(E)}\ 5} $
For given positive integers $n$ and $N$, let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that:
[list]
(a) $|a_j| \le N$ for $j = 0,1, \cdots ,n$;
(b) The set $\{ j \mid a_j = N\}$ has at most two elements.
[/list]
Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$.
Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by $20$ and increases the larger number by $23,$ only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible sum of the original two numbers on the board.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that:
$$f(xy+f(x))=xf(y)$$
for all $x,y\in\mathbb{R}$.
Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$
Determine $k^2$.
Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$
Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$
[i]Proposed by Ma Zhao Yu
Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.
Find all pairs $(m, n)$ of integers such that $n^2 - 3mn + m - n = 0$.
Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.
(a) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 - 7x + 7 = 0$. Show that there exists a quadratic polynomial $f$ with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$.
(b) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 -7x+4 = 0$. Show that there does not exist a quadratic polynomial $f $with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$.
Which figure's shaded part satisfies the inequality $\log_x(\log_x y^2)>0$?
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi83LzY2ZWE5OGJmZjlhNzI1NDM5ZjdiNjZmYTcyZTFkMzEzZjUzMzk5LnBuZw==&rn=NGRkLnBuZw==[/img]
Let be three real numbers $ a,b,c, $ all in the interval $ (0,\infty ) $ or all in the interval $ (0,1). $ Prove the following inequality:
$$ \sum_{\text{cyc}}\log_a bc\ge 4\cdot\sum_{\text{cyc}} \log_{ab} c . $$
Prove or disprove that there exists a positive real $u$ such that $\lfloor u^n\rfloor-n$ is an even integer for all positive integers $n$.
In a school there are $2021$ doors with the numbers $1,2,\dots,2021$. In a day $2021$ students play the following game: Initially all the doors are closed, and each student receive a card to define the order, there are exactly $2021$ cards. The numbers in the cards are $1,2,\dots,2020,2021$.
The order will be student $1$ first, student $2$ will be the second, and going on. The student $k$ will change the state of the doors $k,2k,4k,8k,\dots,2^pk$ with $2^pk\leq 2021\leq 2^{p+1}k$. Change the state is [b]if the door was close, it will be open and vice versa.[/b]
a) After the round of the student $16$, determine the configuration of the doors $1,2,\dots,16$
b) After the round of the student $2021$, determine how many doors are closed.
In a permutation $(x_1, x_2, \dots , x_n)$ of the set $1, 2, \dots , n$ we call a pair $(x_i, x_j )$ discordant if $i < j$ and $x_i > x_j$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3).$