Found problems: 85335
Laurie plays a game called $\textit{bash}$ where she picks two distinct numbers between $1$ and $10$, inclusive, at random. She then finds their sum, product, and non-negative difference. At random, she picks two of these three numbers and tells them to Ali. If the probability that Ali is able to logically deduce the original numbers can be written as $\frac{m}{n}$, with $m$ and $n$ relatively prime, find $m+n$.
[i]Proposed by [b] atmchallenge [/b][/i]
Let $f : R \to R$ be a function from the set $R$ of real numbers to itself. Find all such functions $f$ satisfying the two properties:
(a) $f(x + f(y)) = y + f(x)$ for all $x, y \in R$,
(b) the set $\{ \frac{f(x)}{x} :x$ is a nonzero real number $\}$ is finite
There are $44$ distinct holes in a line and $2017$ ants. Each ant comes out of a hole and crawls along the line with a constant speed into another hole, then comes in. Let $T$ be the set of moments for which the ant comes in or out of the holes. Given that $|T|\leq 45$ and the speeds of the ants are distinct. Prove that there exists two ants that don't collide.
Let $ ABCDE$ be a convex pentagon such that
\[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE.
\]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$.
[i]Proposed by Zuming Feng, USA[/i]
The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square.
[img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]
The triangles $AB'C, CA'B$ and $BC'A$ are constructed on the sides of the equilateral triangle $ABC.$ In the resulting hexagon $AB'CA'BC'$ each of the angles $\angle A'BC',\angle C'AB'$ and $\angle B'CA'$ is greater than $120^\circ$ and the sides satisfy the equalities $AB' = AC',BC' = BA'$ and $CA' = CB'.$ Prove that the segments $AB',BC'$ and $CA'$ can form a triangle.
[i]David Brodsky[/i]
Find the smallest positive integer $n$ such that it is possible to paint each of the $64$ squares of an $8 \times 8$ board of one of $n$ colors so that any four squares that form an $L$ as in the following figure (or congruent figures obtained through rotations and/or reflections) have different colors.
[img]https://cdn.artofproblemsolving.com/attachments/a/2/c8049b1be8f37657c058949e11faf041856da4.png[/img]
A parallelogram $ABCD$ is given. The incircle of triangle $ABC$ with center $I$ touches $AB,BC,CA$ at $R,P,Q$. Ray $DI$ intersects segment $AB$ at $S$. It turned out that $\angle DPR=90$
Prove that the circle with diameter $AS$ is tangent to the circumcircle of triangle $DPQ$
[i]M. Zorka[/i]
$ABC$ is a triangle with $\angle B = 90^o$ and altitude $BH$. The inradii of $ABC, ABH, CBH$ are $r, r_1, r_2$. Find a relation between them.
Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.
Prime numbers $a, b, c$ are bigger that $3$. Show that $(a - b)(b - c)(c - a)$ is divisible by $48$.
The numbers $1,2,...,2n-1,2n$ are divided into two disjoint sets, $a_1 < a_2 < ... < a_n$ and $b_1 > b_2 > ... > b_n$. Prove that $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2.$$
Given is a positive integer $n$. Let $A$ be the set of points $x \in (0;1)$ such that $|x-\frac{p} {q}|>\frac{1}{n^3}$ for each rational fraction $\frac{p} {q}$ with denominator $q \leq n^2$. Prove that $A$ is a union of intervals with total length not exceeding $\frac{100}{n}$.
Proposed by Fedor Petrov
Determine the number of triangles, of any size and shape, in the following figure:
[asy]
size(4cm);
draw(2*dir(0)--dir(120)--dir(240)--cycle);
draw(dir(60)--2*dir(180)--dir(300)--cycle);
[/asy]
[i]Proposed by William Yue[/i]
Find all pairs $\left(a,b\right)$ of posive integers such that $\frac{a^{2}\left(b-a\right)}{b+a}$ is square of prime.
Prove that every polyhedron has at least two faces with the same number of sides.
In the figure of [url]http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view[/url]
$\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$.
The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$,
and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$.
If $DE \parallel O_1A$, prove that $DC \perp CO_2$.
On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on.
If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?
On a large chessboard, there are $4$ puddings that form a square with size $1$. A pudding $A$ could jump over a pudding $B$, or equivalently, $A$ moves to the symmetric point with respect to $B$. Is it possible that after finite times of jumping, the puddings form a square with size $2$?
Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.
For a positive integer $ n $, define $ \phi(n) $ as the number of positive integers less than or equal to $ n $ that are relatively prime to $ n $. Find the sum of all positive integers $ n $ such that $ \phi(n) = 20 $.
Let $x,y$ and $z$ be non-negative real numbers satisfying the equation $x+y+z=xyz$. Prove that $2(x^2+y^2+z^2)\geq3(x+y+z)$.
Write $102$ as the sum of the largest number of distinct primes.
Let $ABC$ be a triangle and let $Q$ be a point such that $\overline{AB} \perp \overline{QB}$ and $\overline{AC} \perp \overline{QC}$. A circle with center $I$ is inscribed in $\triangle ABC$, and is tangent to $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ at points $D$, $E$, and $F$, respectively. If ray $QI$ intersects $\overline{EF}$ at $P$, prove that $\overline{DP} \perp \overline{EF}$.
[i]Proposed by Aaron Lin[/i]
(1) A function is defined $ f(x) \equal{} \ln (x \plus{} \sqrt {1 \plus{} x^2})$ for $ x\geq 0$. Find $ f'(x)$.
(2) Find the arc length of the part $ 0\leq \theta \leq \pi$ for the curve defined by the polar equation: $ r \equal{} \theta\ (\theta \geq 0)$.
Remark:
[color=blue]You may not directly use the integral formula of[/color] $ \frac {1}{\sqrt {1 \plus{} x^2}},\ \sqrt{1 \plus{} x^2}$ here.