Found problems: 85335
Find all continuous functions $f: \mathbb{R}\to\mathbb{R}$ such that for all real $x$ we have
\[f(x)=f\left(x^{2}+\frac{x}{3}+\frac{1}{9}\right). \]
Prove that $\sqrt[3]{5}$ is irrational.
Let the bisectors of the exterior angles at $ B$ and $ C$ of triangle $ ABC$ meet at $ D.$ Then, if all measurements are in degrees, angle $ BDC$ equals:
$ \textbf{(A)}\ \frac {1}{2} (90 \minus{} A) \qquad \textbf{(B)}\ 90 \minus{} A \qquad \textbf{(C)}\ \frac {1}{2} (180 \minus{} A) \qquad \textbf{(D)}\ 180 \minus{} A \qquad \textbf{(E)}\ 180 \minus{} 2A$
If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.
Let a circle $ (O)$ with diameter $ AB$. A point $ M$ move inside $ (O)$. Internal bisector of $ \widehat{AMB}$ cut $ (O)$ at $ N$, external bisector of $ \widehat{AMB}$ cut $ NA,NB$ at $ P,Q$. $ AM,BM$ cut circle with diameter $ NQ,NP$ at $ R,S$.
Prove that: median from $ N$ of triangle $ NRS$ pass over a fix point.
Point $P$ is chosen on the leg $CB$ of right triangle $ABC$ ($\angle ACB = 90^\circ$). The line $AP$ intersects the circumcircle of $ABC$ at point $Q$. Let $L$ be the midpoint of $PB$. Prove that $QL$ is tangent to a fixed circle independent of the choice of point $P$.
Prove that: the polynomial$$(x(x+1)(x+2)(x+3))^{2^{2023}}+1$$is irreducible in $\mathbb{Q}[x].$
Kate bakes a $20$-inch by $18$-inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain?
$
\textbf{(A) }90 \qquad
\textbf{(B) }100 \qquad
\textbf{(C) }180 \qquad
\textbf{(D) }200 \qquad
\textbf{(E) }360 \qquad
$
In triangle $ABC$, $\omega$ is its circumcircle and $O$ is the center of this circle. Points $M$ and $N$ lie on sides $AB$ and $AC$ respectively. $\omega$ and the circumcircle of triangle $AMN$ intersect each other for the second time in $Q$. Let $P$ be the intersection point of $MN$ and $BC$. Prove that $PQ$ is tangent to $\omega$ iff $OM=ON$.
[i]proposed by Mr.Etesami[/i]
[b]p1.[/b] Calculate $\left( \frac12 + \frac13 + \frac14 \right)^2$.
[b]p2.[/b] Find the $2010^{th}$ digit after the decimal point in the expansion of $\frac17$.
[b]p3.[/b] If you add $1$ liter of water to a solution consisting of acid and water, the new solutions will contain of $30\%$ water. If you add another $5$ liters of water to the new solution, it will contain $36\frac{4}{11}\%$ water. Find the number of liters of acid in the original solution.
[b]p4.[/b] John places $5$ indistinguishable blue marbles and $5$ indistinguishable red marbles into two distinguishable buckets such that each bucket has at least one blue marble and one red marble. How many distinguishable marble distributions are possible after the process is completed?
[b]p5.[/b] In quadrilateral $PEAR$, $PE = 21$, $EA = 20$, $AR = 15$, $RE = 25$, and $AP = 29$. Find the area of the quadrilateral.
[b]p6.[/b] Four congruent semicircles are drawn within the boundary of a square with side length $1$. The center of each semicircle is the midpoint of a side of the square. Each semicircle is tangent to two other semicircles. Region $R$ consists of points lying inside the square but outside of the semicircles. The area of $R$ can be written in the form $a - b\pi$, where $a$ and $b$ are positive rational numbers. Compute $a + b$.
[b]p7.[/b] Let $x$ and $y$ be two numbers satisfying the relations $x\ge 0$, $y\ge 0$, and $3x + 5y = 7$. What is the maximum possible value of $9x^2 + 25y^2$?
[b]p8.[/b] In the Senate office in Exie-land, there are $6$ distinguishable senators and $6$ distinguishable interns. Some senators and an equal number of interns will attend a convention. If at least one senator must attend, how many combinations of senators and interns can attend the convention?
[b]p9.[/b] Evaluate $(1^2 - 3^2 + 5^2 - 7^2 + 9^2 - ... + 2009^2) -(2^2 - 4^2 + 6^2 - 8^2 + 10^2- ... + 2010^2)$.
[b]p10.[/b] Segment $EA$ has length $1$. Region $R$ consists of points $P$ in the plane such that $\angle PEA \ge 120^o$ and $PE <\sqrt3$. If point $X$ is picked randomly from the region$ R$, the probability that $AX <\sqrt3$ can be written in the form $a - \frac{\sqrt{b}}{c\pi}$ , where $a$ is a rational number, $b$ and $c$ are positive integers, and $b$ is not divisible by the square of a prime. Find the ordered triple $(a, b, c)$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A space figure is consisted of $n$ vertexes and $l$ lines connecting these vertices, where $n=q^2+q+1, l\geq\frac{1}{2}q(q+1)^2+1,q\geq2,q\in\mathbb{Z}_+$. The figure satisfies: every four vertices are not coplane, every vertex is connected by at least one line, and there is a vertex connected by at least $q+2$ lines. Prove that there exists a space quadrilateral in the figure.
Note: a space quadrilateral is figure with four vertices $A, B, C, D$ and four lines $ AB, BC, CD, DA$.
Quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$, both of which have real roots, are called friendly if for all $t \in [0,1]$ quadratic polynomial $tP(x)+(1-t)Q(x)$ also has real roots.
a) Provide an example of quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ and which have real roots, that are not friendly.
b) Prove that for any two quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ that have real roots, there is a quadratic polynomial $R(x)$ which has a leading coefficient $1$ and which is friendly to both $P$ and $Q$
Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ , circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.
On a circle are given the points $A_1, B_1, A_2, B_2, \cdots, A_9, B_9$ in this order. All the segments $A_iB_j (i, j=1, 2, \cdots, 9$ must be colored in one of $k$ colors, so that no two segments from the same color intersect (inside the circle) and for every $i$ there is a color, such that no segments with an end $A_i$, nor $B_i$ is colored such. What is the least possible $k$?
Three flies are crawling along the perimeter of the triangle $ABC$ in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the triangle $ABC$ . (The centre of masses for the triangle is the point of medians intersection.
Four students take an exam. Three of their scores are $70, 80,$ and $90$. If the average of their four scores is $70$, then what is the remaining score?
$\textbf{(A) }40\qquad\textbf{(B) }50\qquad\textbf{(C) }55\qquad\textbf{(D) }60\qquad \textbf{(E) }70$
Let $b$ be a positive integer such that 2032 has 3 digits when expressed in base $b$. Define the function $S_k(n)$ as the sum of the digits of the base $k$ representation of $n$. Given that $S_b(2032)+S_{b^2}(2032) = 14$, find $b$.
[i]Proposed by Anthony Yang[/i]
Acute triangle $ABC$ with $AB>BC$ is inscribed in circle $\Omega$. Points $D$ and $E$, that lie on $(BC)$ and $(AB)$ are the feet of altitudes from $A$ and $C$ in triangle $ABC$, and $M$ is the midpoint of the segment $DE$. Half-line $(AM$ intersects the circle $\Omega$ for the second time in $N$. Show that the circumcenter of triangle $MDN$ lies on the line $BC$.
How many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ has an area of $p$, where $p$ is a prime number less than $100$?
[i]Proposed by Joshua Hsieh[/i]
Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a [i]saddle pair[/i] if the following two conditions are satisfied:
[list]
[*] $(i)$ For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geqslant a\left(r^{\prime}, c\right)$ for all $c \in C$;
[*] $(ii)$ For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leqslant a\left(r, c^{\prime}\right)$ for all $r \in R$.
[/list]
A saddle pair $(R, C)$ is called a [i]minimal pair[/i] if for each saddle pair $\left(R^{\prime}, C^{\prime}\right)$ with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows.
Show that for every integer $n \ge 3$ there exists positive integers $x_1, x_2, . . . , x_n$, pairwise different, so that $\{2, n\} \subset \{x_1, x_2, . . . , x_n\}$ and
$$\frac{1}{x_1}+\frac{1}{x_2}+.. +\frac{1}{x_n}= 1.$$
Let $P$ be a point inside a tetrahedron $ABCD$ and let $S_A,S_B,S_C,S_D$ be the centroids (i.e. centers of gravity) of the tetrahedra $PBCD,PCDA,PDAB,PABC$. Show that the volume of the tetrahedron $S_AS_BS_CS_D$ equals $1/64$ the volume of $ABCD$.
A subset of the integers $1, 2, ..., 100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
$ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 66 \qquad\textbf{(C)}\ 67 \qquad\textbf{(D)}\ 76 \qquad\textbf{(E)}\ 78 $
There is a set of 1000 switches, each of which has four positions, called $A, B, C,$ and $D.$ When the position of any switch changes, it is only from $A$ to $B,$ from $B$ to $C,$ from $C$ to $D,$ or from $D$ to $A.$ Initially each switch is in position $A.$ The switches are labeled with the 1000 different integers $2^x3^y5^z,$ where $x, y,$ and $z$ take on the values $0, 1, \ldots, 9.$ At step $i$ of a 1000-step process, the $i$th switch is advanced one step, and so are all the other switches whose labels divide the label on the $i$th switch. After step 1000 has been completed, how many switches will be in position $A$?
Let $\omega$ be a circle with radius $1$. Equilateral triangle $\vartriangle ABC$ is tangent to $\omega$ at the midpoint of side $BC$ and $\omega$ lies outside $\vartriangle ABC$. If line $AB$ is tangent to $\omega$ , compute the side length of $\vartriangle ABC$.