Found problems: 85335
Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations:
$a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$.
Before Harry Potter died, he decided to bury his wand in one of eight possible locations (uniformly at random). A squad of Death Eaters decided to go hunting for the wand. They know the eight locations but have poor vision, so even if they're at the correct location they only have a $50\%$ chance of seeing the wand. They also get tired easily, so they can only check three different locations a day. At least they have one thing going for them: they're clever. Assuming they strategize optimally, what is the expected number of days it will take for them to find the wand?
[i]2019 CCA Math Bonanza Individual Round #15[/i]
The Fibonacci numbers are defined by $ F_0 \equal{} 1, F_1 \equal{} 1, F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$. The positive integers $ A, B$ are such that $ A^{19}$ divides $ B^{93}$ and $ B^{19}$ divides $ A^{93}$. Show that if $ h < k$ are consecutive Fibonacci numbers then $ (AB)^h$ divides $ (A^4 \plus{} B^8)^k$
Find the functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$(f(x)-y)f(x+f(y))=f(x^2)-yf(y),$$ for all real numbers $x$ and $y.$
Find all integers $n$ satisfying the following property. There exist nonempty finite integer sets $A$ and $B$ such that for any integer $m$, exactly one of these three statements below is true:
(a) There is $a \in A$ such that $m \equiv a \pmod n$,
(b) There is $b \in B$ such that $m \equiv b \pmod n$, and
(c) There are $a \in A$ and $b \in B$ such that $m \equiv a + b \pmod n$.
Let $S$ be a set of $2022$ lines in the plane, no two parallel, no three concurrent. $S$ divides the plane into finite regions and infinite regions. Is it possible for all the finite regions to have integer area?
[i]Proposed by Tony Wang[/i]
Juhan is touring in Europe. He stands on a highway and watches cars. There are three cars driving along the highway at constant speeds: an Opel and a Trabant in one direction and a Mercedes in the opposite direction. At the moment when the Trabant passes Juhan, the Opel and the Mercedes lie at equal distances from him in opposite directions. At the moment when the Mercedes passes Juhan, the Opel and the Trabant lie at equal distances from him in opposite directions. Prove that at the moment when the Opel passes Juhan, also the Mercedes and the Trabant lie at equal distances from him in opposite directions.
Let $ x, y, z $ be positive real numbers. Prove that
\[ \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1 \]
Find the sum of all positive integers $n$ for which $\mid 2^n + 5^n - 65 \mid$ is a perfect square.
This is a very classical problem.
Let the $ABCD$ be a cyclic quadrilateral with circumcircle $\omega_1$.Lines $AC$ and $BD$ intersect at point $E$,and lines $AD$,$BC$ intersect at point $F$.Circle $\omega_2$ is tangent to segments $EB,EC$ at points $M,N$ respectively,and intersects with circle $\omega_1$ at points $Q,R$.Lines $BC,AD$ intersect line $MN$ at $S,T$ respectively.Show that $Q,R,S,T$ are concyclic.
Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$ such that $OH$ is parallel to $BC$. Let $AH$ intersects again with the circumcircle of $ABC$ at $X$, and let $XB, XC$ intersect with $OH$ at $Y, Z$, respectively. If the projections of $Y,Z$ to $AB,AC$ are $P,Q$, respectively, show that $PQ$ bisects $BC$.
[i]Proposed by usjl[/i]
[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url]
[url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.
[url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that
$$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$
Show that $TB_2AC_2$ is a rectangle.
[url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that
$$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$
When does equality happen?
[url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$
[url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.
Positive real numbers $a,b,c$ satisfy $$2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2.$$
Prove that $$2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2 \geq(ab+bc+ca)^2.$$
In La-La-Land there are 5784 cities. Alpaca chooses for each pair of cities to either build a road or a river between them, and additionally she places a fish in each city to defend it. Subsequently Bear chooses a city to start his trip. At first, he chooses whether to take his trip in a car or in a boat. A boat can sail through rivers but not drive on roads, and a car can drive on roads but not sail through rivers. When Bear enters a city he takes the fish defending it, and consequently the city collapses and he can't return to it again. What is the maximum number of fish Bear can guarantee himself, regardless of the construction of the paths?
Remarks: Bear takes a fish also from the city he begins his trip from (and the city collapses). All roads and rivers are two-way.
For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2y}$. The sum of the solutions to the equation
\[4096f(f(x, x), x) = x^{13}\]
can be written in simplest form as $\tfrac{m}{n}$. Compute $m + n$.
Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD > BC$. Let $E$ be a point on the side $AC$ such that $\frac{AE}{EC} = \frac{BD}{AD-BC}$. Show that $AD > BE$.
For integer $ n,\ a_n$ is difined by $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\cos x)^ndx$.
(1) Find $ a_{\minus{}2},\ a_{\minus{}1}$.
(2) Find the relation of $ a_n$ and $ a_{n\minus{}2}$.
(3) Prove that $ a_{2n}\equal{}b_n\plus{}\pi c_n$ for some rational number $ b_n,\ c_n$, then find $ c_n$ for $ n<0$.
Let $p$ be a prime number. For integers $r, s$ such that $rs(r^2 - s^2)$ is not divisible by $p$, let $f(r, s)$ denote the number of integers $n \in \{1, 2, \ldots, p - 1\}$ such that $\{rn/p\}$ and $\{sn/p\}$ are either both less than $1/2$ or both greater than $1/2$. Prove that there exists $N > 0$ such that for $p \geq N$ and all $r, s$, \[ \left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor. \]
Let $n$ be a positive integer and $p$ be a prime number such that $np+1$ is a perfect square. Prove that $n+1$ can be written as the sum of $p$ perfect squares.
Find all positive integers $n \ge 4$ for which there are $n$ points in general position on the plane such that an arbitrary triangle with vertices belonging to the convex hull of these $n$ points, containing exactly one of $n - 3$ points inside remained.
A paper equilateral triangle of side length $2$ on a table has vertices labeled $A,B,C.$ Let $M$ be the point on the sheet of paper halfway between $A$ and $C.$ Over time, point $M$ is lifted upwards, folding the triangle along segment $BM,$ while $A,B,$ and $C$ on the table. This continues until $A$ and $C$ touch. Find the maximum volume of tetrahedron $ABCM$ at any time during this process.
An integer $n \geq 3$ is called [i]special[/i] if it does not divide $\left ( n-1 \right )!\left ( 1+\frac{1}{2}+\cdot \cdot \cdot +\frac{1}{n-1} \right )$. Find all special numbers $n$ such that $10 \leq n \leq 100$.
Prove that it is impossible for seven distinct straight lines to be situated in the euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines intersect.
Given is a triangle $ABC$. A line $\ell$ passes through reflection wrt $BC$ changes into the line $\ell'$, $\ell'$ changes into $\ell''$ through reflection wrt $AC$ and $\ell''$ through reflection wrt $AB$ changes into $\ell'''$. Construct the line $\ell$ given that $\ell'''$ coincides with $\ell$.
Tim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an angle between $0^\circ$ and $180^\circ$ inclusive. At how many times during that day are the angles on the two clocks equal?