Found problems: 85335
Let $ABC$ be an equilateral triangle with $AB=a$ and $M\in(AB)$ a fixed point. Points $N\in(AC)$ and $P\in(BC)$ are taken such that the perimeter of $MNP$ is minimal. If the ratio between the areas of triangles $MNP$ and $ABC$ is $\textstyle\frac{7}{30},$ find the perimeter of triangle $MNP.$
Find all natural numbers $x,y,z$, such that $7^{x}+13^{y}=2^{z}$.
Find all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integer numbers $n$, such that
$$p^2 = q^2 + r^n$$
(Walther Janous)
Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.
If $(A, <)$ is a partially ordered set, its dimension, $\dim (A, <)$, is the least cardinal $\kappa$ such that there exist $\kappa$ total orderings $\{ <_{\alpha} \colon \alpha < \kappa \}$ on $A$ with $<=\cap_{\alpha < \kappa} <_\alpha$. Show that if $\dim (A, <)>\aleph_0$, then there exist disjoint $A_0, A_1\subseteq A$ with $\dim (A_0, <)$, $\dim (A_1, <)>\aleph_0$. [D. Kelly, A. Hajnal, B. Weiss]
Let $ABC$ be a triangle. Point $M$ and $N$ lie on sides $AC$ and $BC$ respectively such that $MN || AB$. Points $P$ and $Q$ lie on sides $AB$ and $CB$ respectively such that $PQ || AC$. The incircle of triangle $CMN$ touches segment $AC$ at $E$. The incircle of triangle $BPQ$ touches segment $AB$ at $F$. Line $EN$ and $AB$ meet at $R$, and lines $FQ$ and $AC$ meet at $S$. Given that $AE = AF$, prove that the incenter of triangle $AEF$ lies on the incircle of triangle $ARS$.
Point $K$ lies on the circumcircle of a rectangle $ABCD$. Line $CK$ intersects line segment $AD$ at point $M$ so that $AM:MD=2$. $O$ is the center the rectangle. Prove that the centroid of triangle $OKD$ belongs to the circumcircle of triangle $COD$. (Author: V. Shmarov)
Let $n$ and $k$ be two positive integers such that $n>k$. Prove that the equation $x^n+y^n=z^k$ has a solution in positive integers if and only if the equation $x^n+y^n=z^{n-k}$ has a solution in positive integers.
Find the number of the connected graphs with 6 vertices. (Vertices are considered to be different)
There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to
(A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.
From point $D$ of parallelogram $ABCD$ were drawn an arbitrary line $\ell_1$, intersecting the segment $AB$ and the line $BC$ at points $C_1$ and $A_1$, respectively, and an arbitrary line $\ell_2$ intersecting the segment $BC$ and the line $AB$ at the points $A_2$ and $C_2$, respectively. Find the locus of the intersection points of the circles $(A_1BC_2)$ and $(A_2BC_1)$ (other than point $B$).
In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$.
[i]Proposed by Andy Xu[/i]
$ \left(\frac {(x \plus{} 1)^2(x^2 \minus{} x \plus{} 1)^2}{(x^3 \plus{} 1)^2}\right)^2 \cdot \left(\frac {(x \minus{} 1)^2(x^2 \plus{} x \plus{} 1)^2}{(x^3 \minus{} 1)^2}\right)^2$ equals:
$ \textbf{(A)}\ (x \plus{} 1)^4 \qquad\textbf{(B)}\ (x^3 \plus{} 1)^4 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ [(x^3 \plus{} 1)(x^3 \minus{} 1)]^2$
$ \textbf{(E)}\ [(x^3 \minus{} 1)^2]^2$
Find all primes $p,q,r$ such that $qr-1$ is divisible by $p$, $pr-1$ is divisible by $q$, $pq-1$ is divisible by $r$.
If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m\ne0$, $n\ne0$, then the sum of the roots is:
$\textbf{(A)}\ -\dfrac{1}{2} \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ \dfrac{1}{2} \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ \text{Undetermined} $
Suppose that $a_{1},a_{2},\dots,a_{k}\in\mathbb C$ that for each $1\leq i\leq k$ we know that $|a_{k}|=1$. Suppose that \[\lim_{n\to\infty}\sum_{i=1}^{k}a_{i}^{n}=c.\] Prove that $c=k$ and $a_{i}=1$ for each $i$.
A circle overlaps an equilateral triangle of side length $100\sqrt{3}$. The three chords in the circle formed by the three sides of the triangle have lengths 6, 36, and 60, respectively. What is the area of the circle?
Find the minimum value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$ where $-1 \le x \le 1$.
The roots of $64x^3-144x^2+92x-15=0$ are in arithmetic progression. The difference between the largest and smallest roots is:
$\textbf{(A)}\ 2\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ \frac{1}{2}\qquad
\textbf{(D)}\ \frac{3}{8}\qquad
\textbf{(E)}\ \frac{1}{4}$
Let $\omega$ be a circle and let $A,B,C,D,E$ be five points on $\omega$ in this order. Define $F=BC\cap DE$, such that the points $F$ and $A$ are on opposite sides, with regard to the line $BE$ and the line $AE$ is tangent to the circumcircle of the triangle $BFE$.
a) Prove that the lines $AC$ and $DE$ are parallel
b) Prove that $AE=CD$
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.
In a board with $3$ rows and $555$ columns, $3$ squares are colored red, one in each of the $3$ rows.
If the numbers from $1$ to $1665$ are written in the boxes, in row order, from left to right (in the first row from $1$ to $555$, in the second from $556$ to $1110$ and in the third from $1111$ to $1665$) there are $3$ numbers that are written in red squares.
If they are written in the boxes, ordered by columns, from top to bottom, the numbers from $1$ to $1665$ (in the first column from $1$ to $3$, in the second from $4$ to $6$, in the third from $7$ to $9$,... ., and in the last one from $1663$ to $1665$) there are $3$ numbers that are written in red boxes.
We call [i]red[/i] numbers those that in one of the two distributions are written in red boxes.
Indicate which are the $3$ squares that must be colored red so that there are only $3$ red numbers.
Show all the possibilities.
Let $a,b,c $ be the lengths of the three sides of a triangle and $a,b$ be the two roots of the equation $ax^2-bx+c=0 $$ (a<b) . $ Find the value range of $ a+b-c .$
Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is
$\textbf{(A) }42\qquad\textbf{(B) }46\qquad\textbf{(C) }122\qquad\textbf{(D) }290\qquad \textbf{(E) }326$
A duck drew a square $ABCD$, then he reflected $C$ across $B$ to obtain a point $E$. He also drew the center of the square to be $F$. Then, he drew a point $G$ on ray $EF$ beyond $F$ such that $\angle AGC = 135^{\circ}$.
Help the Duck prove that $\angle CGD = 135^{\circ}$ as well.