Found problems: 85335
Integers $x,y,z$ and $a,b,c$ satisfy
$$x^2+y^2=a^2,\enspace y^2+z^2=b^2\enspace z^2+x^2=c^2.$$Prove that the product $xyz$ is divisible by (a) $5$, and (b) $55$.
By [i]centroid[/i] of a quadrilateral $PQRS$ we call a common point of two lines through the midpoints of its opposite sides. Suppose that $ABCDEF$ is a hexagon inscribed into the circle $\Omega$ centered at $O$. Let $AB=DE$, and $BC=EF$. Let $X$, $Y$, and $Z$ be centroids of $ABDE$, $BCEF$; and $CDFA$, respectively. Prove that $O$ is the orthocenter of triangle $XYZ$.
Let $n$ be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves:
(1) select a row and multiply each entry in this row by $n$;
(2) select a column and subtract $n$ from each entry in this column.
Find all possible values of $n$ for which the following statement is true:
Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is $0$.
Suppose that $A, B, C$, and $X$ are any four distinct points in the plane with $$\max \,(BX,CX) \le AX \le BC.$$
Prove that $\angle BAC \le 150^o$.
[b]p1.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. His drink contains $45\%$ of orange juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $60\%$ of orange juice?
[b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm.
[img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img]
[b]p3.[/b] For one particular number $a > 0$ the function f satisfies the equality $f(x + a) =\frac{1 + f(x)}{1 - f(x)}$ for all $x$. Show that $f$ is a periodic function. (A function $f$ is periodic with the period $T$ if $f(x + T) = f(x)$ for any $x$.)
[b]p4.[/b] If $a, b, c, x, y, z$ are numbers so that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1$ and $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}= 0$. Show that $\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1$
[b]p5.[/b] Is it possible that a four-digit number $AABB$ is a perfect square?
(Same letters denote the same digits).
[b]p6.[/b] A finite number of arcs of a circle are painted black (see figure). The total length of these arcs is less than $\frac15$ of the circumference. Show that it is possible to inscribe a square in the circle so that all vertices of the square are in the unpainted portion of the circle.
[img]https://cdn.artofproblemsolving.com/attachments/2/c/bdfa61917a47f3de5dd3684627792a9ebf05d5.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.
If $f(1) = 1$ and $f(n+1) = \frac{2f(n) + 1}{2}$, then find $f(237)$.
$\text{(A) }117\qquad\text{(B) }118\qquad\text{(C) }119\qquad\text{(D) }120\qquad\text{(E) }121$
Denote by $[a, b]$ the least common multiple of $a$ and $b$.
Let $n$ be a positive integer such that $[n, n + 1] > [n, n + 2] >...> [n, n + 35]$. Prove that $[n, n + 35] > [n,n + 36]$.
Five students have the first names Clark, Donald, Jack, Robin and Steve, and have the last names (in a different order) Clarkson, Donaldson, Jackson, Robinson and Stevenson. It is known that Clark is $1$ year older than Clarkson, Donald is $2$ years older than Donaldson, Jack is $3$ years older than Jackson, Robin is $4$ years older than Robinson. Who is older, Steve or Stevenson and what is the difference in their ages?
A ball of mass $m$ is launched into the air. Ignore air resistance, but assume that there is a wind that exerts a constant force $F_0$ in the -$x$ direction. In terms of $F_0$ and the acceleration due to gravity $g$, at what angle above the positive $x$-axis must the ball be launched in order to come back to the point from which it was launched?
$ \textbf{(A)}\ \tan^{-1}(F_0/mg)$
$\textbf{(B)}\ \tan^{-1}(mg/F_0)$
$\textbf{(C)}\ \sin^{-1}(F_0/mg)$
$\textbf{(D)}\ \text{the angle depends on the launch speed}$
$\textbf{(E)}\ \text{no such angle is possible}$
Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]
[i]Proposed by Holden Mui[/i]
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$.
Let $a<b<c<d<e$ be real numbers. Among the $10$ sums of the pairs of these numbers, the least $3$ are $32,36,37$, while the largest two are $48$ and $51$. Find all possible values of $e$
A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures?
$ \text{(A)}\ 2\qquad\text{(B)}\ 3\qquad\text{(C)}\ 4\qquad\text{(D)}\ 5\qquad\text{(E)}\ 6 $
Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$
Consider a triangle whose sidelengths $a$, $b$, $c$ are integers, and which has the property that one of its altitudes equals the sum of the two others.
Then, prove that $a^2+b^2+c^2$ is a perfect square.
The following spiral sequence of squares is drawn on an infinite blackboard: The $1$st square $(1 \times 1)$ has a common vertical side with the $2$nd square (also $1\times 1$) drawn on the right side of it; the 3rd square $(2 \times 2)$ is drawn on the upper side of the $1$st and 2nd ones; the $4$th square $(3 \times 3)$ is drawn on the left side of the $1$st and $3$rd ones; the $5$th square $(5 \times 5)$ is drawn on the bottom side of the $4$th, 1st and $2$nd ones; the $6$th square $(8 \times 8)$ is drawn on the right side, and so on. Each of the squares has a common side with the rectangle consisting of squares constructed earlier. Prove that the centres of all the squares except the $1$st lie on two straight lines.
(A Andjans, Riga)
In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.
Find all functions \(f:\mathbb{R}^{+} \to \mathbb{R}^{+}\) such that
\[
f(x) > x \ \ \text{and} \ \ f(x-y+xy+f(y)) = f(x+y) + xf(y)
\]
for arbitrary positive real numbers \(x\) and \(y\).
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 $
The sequence $a_1, a_2, ... , a_k, ...$ is constructed according to the rules: $$a_{2n} = a_n,a_{4n+1} = 1,a_{4n+3} = 0$$Prove that it is non-periodical sequence.
On the chord $AB$ of the circle $\omega$ points $C$ and $D$ are chosen such that $AC=BD$ and point $C$ lies between $A$ and $D$. On $\omega$ point $E$ and $F$ are marked, they lie on different sides with respect to line $AB$ and lines $EC$ and $FD$ are perpendicular to $AB$. The angle bisector of $AEB$ intersects line $DF$ at $R$
Prove that the circle with center $F$ and radius $FR$ is tangent to $\omega$
[i]V. Kamenetskii, D. Bariev[/i]
There are $5$ ingots weighing $1$, $2$, $3$, $4$ and $5$ kg with an unknown copper content that varies in different ingots. Each ingot must be divided into $5$ parts and $5$ new ingots of the same mass of $1$, $2$, $3$, $4$ and $5$ kg must be made. This requires that the percentage of copper in all pieces be the same, regardless of what it was in the original pieces. What parts should each piece be divided into?
In a tetrahedron $ABCD$ the edges $AB$ and $CD$ are perpendicular and $\angle ACB =\angle ADB$. Prove that the plane through $AB$ and the midpoint of the edge $CD$, is perpendicular to $CD$.
Anne and Bill decide to play a game together. At the beginning, they chose a positive integer $n$; then, starting from a positive integer $\mathcal{N}_0$, Anne subtracts to $\mathcal{N}_0$ an integer $k$-th power (possibly $0$) of $n$ less than or equal to $\mathcal{N}_0$. The resulting number $\mathcal{N}_1=\mathcal{N}_0-n^k$ is then passed to Bill, who repeats the same process starting from $\mathcal{N}_1$: he subtracts to $\mathcal{N}_1$ an integer $j$-th power of $n$ less than or equal to $\mathcal{N}_1$, and he then gives the resulting number $\mathcal{N}_2=\mathcal{N}_1-n^j$ to Anne. The game continues like that until one player gets $0$ as the result of his operation, winning the game. For each $1\leq n \leq 1000$, let $f(n)$ be the number of integers $1\leq \mathcal{N}_0\leq 5000$ such that Anne has a winning strategy starting from them. For how many values of $n$ we have that $f(n)\geq 2520$?
[I]Proposed by [b]FedeX333X[/b][/I]