Found problems: 85335
Let $a,b\in\mathbb{N}$ such that :
\[ ab(a-b)\mid a^3+b^3+ab \]
Then show that $\operatorname{lcm}(a,b)$ is a perfect square.
Let $N\geq 4$ be a fixed positive integer. Two players, $A$ and $B$ are forming an ordered set $\{x_1,x_2,...\},$ adding elements alternatively. $A$ chooses $x_1$ to be $1$ or $-1,$ then $B$ chooses $x_2$ to be $2$ or $-2,$ then $A$ chooses $x_3$ to be $3$ or $-3,$ and so on. (at the $k^{th}$ step, the chosen number must always be $k$ or $-k$)
The winner is the first player to make the sequence sum up to a multiple of $N.$ Depending on $N,$ find out, with proof, which player has a winning strategy.
Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal connecting them. Let $d$ be the number of diagonals Alice drew. Find the least possible value that $d$ can obtain.
The quadrilateral $ABCD$ has an incircle of diameter $d$ which touches $BC$ at $K$ and touches $DA$ at $L$. Is it always true that the harmonic mean of $AB$ and $CD$ is equal to $KL$ if and only if the geometric mean of $AB$ and $CD$ is equal to $d$?
Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.
Find all functions $f : \mathbb{Z}\to \mathbb{Z}$ that satisfy:
$i) f(f(x))=x, \forall x\in\mathbb{Z}$
$ii)$ For any integer $x$ and $y$ such that $x + y$ is odd, it holds that $f(x) + f(y) \ge x + y.$
How many ordered pairs $(x, y)$ of real numbers $x$ and $y$ are there such that $-100 \pi \le x \le 100 \pi$, $-100 \pi \le y \le 100 \pi$, $x + y = 20.19$, and $\tan x + \tan y = 20.19$?
Merlin summons the $n$ knights of Camelot for a conference. Each day, he assigns them to the $n$ seats at the Round Table. From the second day on, any two neighbours may interchange their seats if they were not neighbours on the first day. The knights try to sit in some cyclic order which has already occurred before on an earlier day. If they succeed, then the conference comes to an end when the day is over. What is the maximum number of days for which Merlin can guarantee that the conference will last?
Let $a,b,c$ be reals and
\[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\]
Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$
Let $\omega$ be the circumcircle of an acute angled tirangle $ABC.$ The line tangent to $\omega$ at $A$ intersects the line $BC$ at the point $T.$ Let the midpoint of segment $AT$ be $N,$ and the centroid of $\triangle ABC$ be the point $G.$ The other tangent line drawn from $N$ to $\omega$ intersects $\omega$ at the point $L.$ The line $LG$ meets $\omega$ at $S\neq L.$
Prove that $AS\parallel BC.$
Find all $f\colon\mathbf{R}\rightarrow\mathbf{R}$ such that
\[f(f(x)f(y)) = f(x + y) + f(xy)\]
for all $x,y\in\mathbf{R}$.
Let $n$ be the number of points $P$ interior to the region bounded by a circle with radius $1$, such that the sum of the squares of the distances from $P$ to the endpoints of a given diameter is $3$. Then $n$ is:
$\textbf{(A) }0\qquad
\textbf{(B) }1\qquad
\textbf{(C) }2\qquad
\textbf{(D) }4\qquad
\textbf{(E) }\text{infinite}$
While running from an unrealistically rendered zombie, Willy Smithers runs into a vacant lot in the shape of a square, $100$ meters on a side. Call the four corners of the lot corners $1$, $2$, $3$, and $4$, in clockwise order. For $k = 1, 2, 3, 4$, let $d_k$ be the distance between Willy and corner $k$. Let
(a) $d_1<d_2<d_4<d_3$,
(b) $d_2$ is the arithmetic mean of $d_1$ and $d_3$, and
(c) $d_4$ is the geometric mean of $d_2$ and $d_3$.
If $d_1^2$ can be written in the form $\dfrac{a-b\sqrt c}d$, where $a,b,c,$ and $d$ are positive integers, $c$ is square-free, and the greatest common divisor of $a$, $b$, and $d$ is $1,$ find the remainder when $a+b+c+d$ is divided by $2008$.
For a positive integer $n$ write down all its positive divisors in increasing order: $1=d_1<d_2<\ldots<d_k=n$.
Find all positive integers $n$ divisible by $2019$ such that $n=d_{19}\cdot d_{20}$.
[i](I. Gorodnin)[/i]
In an acute-angled triangle $ABC, CD$ is the altitude. A line through the midpoint $M$ of side $AB$ meets the rays $CA$ and $CB$ at $K$ and $L$ respectively such that $CK = CL.$ Point $S$ is the circumcenter of the triangle $CKL.$ Prove that $SD = SM.$
If $k$ and $n$ are positive integers, prove the inequality
$$\frac{1}{kn} +\frac{1}{kn+1} +...+\frac{1}{(k+1)n-1} \ge n \left(\sqrt[n]{\frac{k+1}{k}}-1\right)$$
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.
The area of a circle (in square inches) is numerically larger than its circumference (in inches). What is the smallest possible integral area of the circle, in square inches?
[i]Proposed by James Lin[/i]
Prove that there exist infinitely many positive integers $n$ such that $3^n+2$ and $5^n+2$ are all composite numbers.
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments.
A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i].
a) Prove that there are infinite [i]non-charrua[/i] pairs.
b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].
Thirty numbers are placed on a circle. For every number $A$ we have: $A$ equals the absolute value of $(B- C)$, where $B$ and $C$ follow $A$ clockwise. The total sum of the numbers equals $1$. Find all the numbers.
(Folklore)
Determine all numbers $x, y$ and $z$ satisfying the system of equations
$$\begin{cases} x^2 + yz = 1 \\ y^2 - xz = 0 \\ z^2 + xy = 1\end{cases}$$
Cut the $10$ cm $x 20$ cm rectangle into two pieces with one straight cut so that they can be placed inside the $19.4$ cm diameter circle without intersecting.