Found problems: 85335
For a point $P$ on an ellipse, let $d$ be the distance from the center of the ellipse to the line tangent to the ellipse at $P.$ Prove that $(PF_1)(PF_2)d^2$ is constant as $P$ varies on the ellipse, where $PF_1$ and $PF_2$ are distances from $P$ to the foci $F_1$ and $F_2$ of the ellipse.
A creative task: propose an original competition problem together with its solution.
a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$
b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd.
(I. Voronovich)
For every real $x$, the polynomial $p(x)$ whose roots are all real satisfies $p(x^2-1)=p(x)p(-x)$. What can the degree of $p(x)$ be at most?
$ \textbf{(A)}\ 0
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \text{There is no upper bound for the degree of } p(x)
\qquad\textbf{(E)}\ \text{None}
$
A positive integer is called [i]shiny[/i] if it can be written as the sum of two not necessarily distinct integers $a$ and $b$ which have the same sum of their digits. For instance, $2012$ is [i]shiny[/i], because $2012 = 2005 + 7$, and both $2005$ and $7$ have the same sum of their digits. Find all positive integers which are [b]not[/b] [i]shiny[/i] (the dark integers).
Solve the following system of equations:
$$x+\frac{1}{x^3}=2y,\quad y+\frac{1}{y^3}=2z,\quad z+\frac{1}{z^3}=2w,\quad w+\frac{1}{w^3}=2x.$$
A set $P$ has the following property: “For any positive integer $k$, if $p$ is a prime factor of $k^3+6$, then $p$ belongs to $P$ ”. Prove that $P$ is infinite.
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.
Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by
permuting these digits leaves the same remainder after the division by $4$.
For all numbers $n\ge 1$ does there exist infinite positive numbers sequence $x_1,x_2,...,x_n$ such that $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_n}$
A $T$-tetromino is a non-convex as well as non-rotationally symmetrical tetromino, which has a maximum number of outside corners (popularly also "Tetris Stone "called). Find all natural numbers $n$ for which, a $n \times n$ chessboard is found that can be covered only with such $T$-tetrominos.
Does there exist a strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ that takes on only irrational values?
Show that for every integer $n \geq 6$, there exists a convex hexagon which can be dissected into exactly $n$ congruent triangles.
As shown in the figure, chess pieces are placed at the intersection points of the $64$ grid lines of the $7\times 7$ grid table. At most $1$ piece is placed at each point, and a total of $k$ left chess pieces are placed. No matter how they are placed, there will always be $4$ chess pieces, and the grid in which they are located the points form the four vertices of a rectangle (the sides of the rectangle are parallel to the grid lines). Try to find the minimum value of $k$.
[img]https://cdn.artofproblemsolving.com/attachments/5/b/23a79f43d3f4c9aade1ba9eaa7a282c3b3b86f.png[/img]
Find the total number of $k$-tuples $(n_1,n_2,...,n_k)$ of positive integers so that $n_{i+1}\ge n_i$ for each $i$, and $k$ regular polygons with numbers of sides $n_1,n_2,...,n_k$ respectively will fit into a tesselation at a point. That is, the sum of one interior angle from each of the polygons is $360^{\circ}$.
Let the trapezoids $A_iB_iC_iD_i$ ($i = 1, 2, 3$) be similar and have the same clockwise direction. Their angles at $A_i$ and $B_i$ are $60^o$ and the sides $A_1B_1$, $B_2C_2$ and $A_3D_3$ are parallel. The lines $B_iD_{i+1}$ and $C_iA_{i+1}$ intersect at the point $P_i$ (the indices are understood cyclically, i.e. $A_4 = A_1$ and $D_4 = D_1$). Prove that the points $P_1$, $P_2$ and $P_3$ lie on a line.
Find with proof all positive $3$ digit integers $\overline{abc}$ satisfying
\[ b\cdot \overline{ac}=c \cdot \overline{ab} +10 \]
Let $n\ge 3$, $d$ be positive integers. For an integer $x$, denote $r(x)$ be the remainder of $x$ when divided by $n$ such that $0\le r(x)\le n-1$. Let $c$ be a positive integer with $1<c<n$ and $\gcd(c,n)=1$, and suppose $a_1, \cdots, a_d$ are positive integers with $a_1+\cdots+a_d\le n-1$. \\
(a) Prove that if $n<2d$, then $\displaystyle\sum_{i=1}^d r(ca_i)\ge n.$ \\
(b) For each $n$, find the smallest $d$ such that $\displaystyle\sum_{i=1}^d r(ca_i)\ge n$ always holds.
[i]Proposed by Yeoh Zi Song & Anzo Teh Zhao Yang[/i]
(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection.
(b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.
Let $A B C D$ be a unit square. Suppose $M$ and $N$ are points on $B C$ and $C D$ respectively such that the perimeter of triangle $M C N$ is 2 . Let $O$ be the circumcentre of triangle $M A N$, and $P$ be the circumcentre of triangle $M O N$. If $\left(\frac{O P}{O A}\right)^2=\frac{m}{n}$ for some relatively prime positive integers $m$ and $n$, find the value of $m+n$.
Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$.
(Vietnam)
Find all positive integers $k$, so that there are only finitely many positive odd numbers $n$ satisfying $n~|~k^n+1$.
Find all non-negative integer solutions of the equation
\[2^x + 3^y = z^2 .\]
Let ABC be a triangle with $\angle A = 90^o$, $\angle B = 75^o$, and $AB = 2$. Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$. Calculate the lenght of $QA$.
Bob has a $12$ foot by $20$ foot garden. He wants to put fencing around it to keep out the neighbor’s dog. Normal fence posts cost $\$2$ each while strong ones cost $\$3$ each. If he needs one fence post for every $2$ feet and has $\$70$ to spend on the fence posts, what is the largest number of strong fence posts he can buy?