Found problems: 85335
There are $n$ triangles inscribed in a circle and all $3n$ of their vertices are different. Prove that it is possible to put a boy in one of the vertices in each triangle, and a girl in the other, so that boys and girls alternate on a circle.
Find all positive integers $(m,n)$ such that
$$11^n+2^n+6=m^3$$
Given a circle and four points $B,C,X,Y$ on it. Assume $A$ is the midpoint of $BC$, and $Z$ is the midpoint of $XY$. Let $L_1,L_2$ be lines perpendicular to $BC$ and pass through $B,C$ respectively. Let the line pass through $X$ and perpendicular to $AX$ intersects $L_1,L_2$ at $X_1,X_2$ respectively. Similarly, let the line pass through $Y$ and perpendicular to $AY$ intersects $L_1,L_2$ at $Y_1,Y_2$ respectively. Assume $X_1Y_2$ intersects $X_2Y_1$ at $P$. Prove that $\angle AZP=90^o.$
[i]Proposed by William Chao[/i]
Candice starts driving home from work at $5{:}00 \ \text{PM}.$ Starting at exactly $5{:}01 \ \text{PM},$ and every minute after that, Candice encounters a new speed limit sign and slows down by $1$ mph. Candice’s speed, in miles per hour, is always a positive integer. Candice drives for $2/3$ of a mile in total. She drives for a whole number of minutes, and arrives at her house driving slower than when she left. What time is it when she gets home?
Real numbers $x$ and $y$ satisfy the system of equations
$$x^3 + 3x^2 = -3y - 1$$
$$y^3 + 3y^2 = -3x - 1.$$
What is the greatest possible value of $x$?
We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$.
We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!
Two fixed circles are given on the plane, one of them lies inside the other one. From a point $C$ moving arbitrarily on the external circle, draw two chords $CA, CB$ of the larger circle such that they tangent to the smalaler one. Find the locus of the incenter of triangle $ABC$.
Halfway through a $ 100$-shot archery tournament, Chelsea leads by $ 50$ points. For each shot a bullseye scores $ 10$ points, with other possible scores being $ 8, 4, 2, 0$ points. Chelsea always scores at least $ 4$ points on each shot. If Chelsea's next $ n$ shots are bulleyes she will be guaranteed victory. What is the minimum value for n?
$ \textbf{(A)}\ 38\qquad \textbf{(B)}\ 40\qquad \textbf{(C)}\ 42\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 46$
Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$
In a row are 23 boxes such that for $1\le k \le 23$, there is a box containing exactly $k$ balls. In one move, we can double the number of balls in any box by taking balls from another box which has more. Is it always possible to end up with exactly $k$ balls in the $k$-th box for $1\le k\le 23$?
Does there exist a row of Pascal’s Triangle containing four distinct values $a,b,c$ and $d$ such that $b = 2a$ and $d = 2c$?
Recall that Pascal’s triangle is the pattern of numbers that begins as follows
[img]https://cdn.artofproblemsolving.com/attachments/2/1/050e56f0f1f1b2a9c78481f03acd65de50c45b.png[/img]
where the elements of each row are the sums of pairs of adjacent elements of the prior row. For example, $10 =4+6$.
Also note that the last row displayed above contains the four elements $a = 5,b = 10,d = 10,c = 5$, satisfying $b = 2a$ and $d = 2c$, but these four values are NOT distinct.
Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and
$$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$$
Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$
Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$
and $CC_1C_2$ all pass through two common points.
(Note: a scalene triangle is one where no two sides have equal length.)
[i]Proposed by Ankan Bhattacharya, USA[/i]
The base of the quadrangular pyramid $SABCD$ is a quadrilateral $ABCD$, the diagonals of which are perpendicular. The apex of the pyramid is projected at intersection point $O$ of the diagonals of the base. Prove that the feet of the perpendiculars drawn from point $O$ to the side faces of the pyramid lie on one circle.
Let $ABCD$ be a square with side length $4$. Consider points $P$ and $Q$ on segments $AB$ and $BC$, respectively, with $BP=3$ and $BQ=1$. Let $R$ be the intersection of $AQ$ and $DP$. If $BR^2$ can be expressed in the form $\frac{m}{n}$ for coprime positive integers $m,n$, compute $m+n$.
[i]Proposed by Brandon Wang[/i]
Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.
Consider the set $S$ of lattice points with positive coordinates in the plane. For each point $P(a,b)$ from $S$, we draw a segment between it and each of the points in the set \[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\] Show that there is no colouring of the points in $S$ with a finite number of colours such that every two points joined by a segment are coloured with different colours.
[i]Ioan Tomescu[/i]
A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.
Let $n$ be an odd positive integer and let $x_1,x_2,...,x_n$ be n distinct real numbers that satisfy $|x_i -x_j| \le 1$ for $1 \le i < j \le n$. Prove that
$$\sum_{i<j} |x_i -x_j| \le \left[\frac{n}{2} \right] \left(\left[\frac{n}{2} \right]-1 \right)$$
Divide a regular $8960$-gon into non-overlapping parallelograms. Suppose that $R$ of these parallelograms are rectangles. What is the minimum possible value of $R$?
For each positive real number $x$, let $f(x)=\frac{x}{1+x}$ . Prove that if $a$, $b,$ $c$ are the sidelengths of a triangle, then $f(a)$, $f(b),$ $f(c)$ are sidelengths of a triangle.
Let $x, y, z$ be real numbers such that $x+y+z=-2$ and
\[\begin{aligned}
& (x^2+xy+y^2)(y^2+yz+z^2) \\
&+ (y^2+yz+z^2)(z^2+zx+x^2) \\
&+ (z^2+zx+x^2)(x^2+xy+y^2) \\
& = 625+ \tfrac34(xy+yz+zx)^2. \end{aligned}\]
Compute $|xy+yz+zx|$.
[i]Proposed by Michael Tang[/i]
Let $ f$ be a function with the following properties:
(i) $f(1) \equal{} 1$, and
(ii) $ f(2n) \equal{} n\times f(n)$, for any positive integer $ n$.
What is the value of $ f(2^{100})$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2^{99} \qquad \textbf{(C)}\ 2^{100} \qquad \textbf{(D)}\ 2^{4950} \qquad \textbf{(E)}\ 2^{9999}$
ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.
Consider the succession of integers $\{f(n)\}_{n=1}^{\infty}$ defined as:
$\bullet$ $f(1) = 1$.
$\bullet$ $f(n) = f(n/2)$ if $n$ is even.
$\bullet$ If $n > 1$ odd and $f(n-1)$ odd, then $f(n) = f(n-1)-1$.
$\bullet$ If $n > 1$ odd and $f(n-1)$ even, then $f(n) = f(n-1)+1$.
a) Compute $f(2^{2020}-1)$.
b) Prove that $\{f(n)\}_{n=1}^{\infty}$ is not periodical, that is, there do not exist positive integers $t$ and $n_0$ such that $f(n+t) = f(n)$ for all $n \geq n_0$.
Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$.
[img]https://1.bp.blogspot.com/-OWpqpAqR7Zw/Xzj_fyqhbFI/AAAAAAAAMao/5y8vCfC7PegQLIUl9PARquaWypr8_luAgCLcBGAsYHQ/s0/2010%2Boral%2Bmoscow%2Bgeometru%2B8.1.gif[/img]