For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]

Given is an isosceles trapezoid $ABCD$, such that $AD$ and $BC$ are bases and $AD=2AB$, and it is inscribed in a circle $c$. Points $E$ and $F$ are selected on a circle $c$ so that $AC$ || $DE$ and $BD$ || $AF$. The line $BE$ intersects lines $AC$ and $AF$ at points $X$ and $Y$, respectively. Prove that the circumcircles of triangles $BCX$ and $EFY$ are tangent to each other.

Let $h$ be a semicircle with diameter $AB$. Let $P$ be an arbitrary point inside the diameter $AB$. The perpendicular through $P$ on $AB$ intersects $h$ at point $C$. The line $PC$ divides the semicircular area into two parts. A circle will be inscribed in each of them that touches $AB, PC$ and $h$. The points of contact of the two circles with $AB$ are denoted by $D$ and $E$, where $D$ lies between $A$ and $P$. Prove that the size of the angle $DCE$ does not depend on the choice of $P$. (Walther Janous)

The measures, in degrees, of the angles , $\alpha, \beta$ and $\theta$ are greater than $0$ less than $60$. Find the value of $\theta$ knowing, also, that $\alpha + \beta = 2\theta$ and that $$\sin \alpha \sin \beta \sin \theta = \sin(60 - \alpha ) \sin(60 - \beta) \sin(60 - \theta ).$$

Given $a,b,c>0$ such that $$a+b+c+\frac{1}{abc}=\frac{19}{2}$$ What is the greatest value for $a$?

Find all real numbers $x$ such that $\left\lfloor x^2 \right\rfloor + \left\lceil x^2 \right\rceil = 2003$.

In a triangle $[ABC]$, $\angle C = 2\angle A$. A point $D$ is marked on the side $[AC]$ such that $\angle ABD = \angle DBC$. Knowing that $AB = 10$ and $CD = 3$, what is the length of the side $[BC]$?

Suppose $X, Y, Z$ are collinear points in that order such that $XY = 1$ and $YZ = 3$. Let $W$ be a point such that $YW = 5$, and define $O_1$ and $O_2$ as the circumcenters of triangles $\triangle WXY$ and $\triangle WYZ$, respectively. What is the minimum possible length of segment $\overline{O_1O_2}$?

Find all $n\in\mathbb{Z}$ such that the number $\sqrt{\frac{4n-2}{n+5}}$ is rational.

Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality: \[\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}.\]

A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.

Consider the following two-player game: player $A$ (first mover) and $B$ take turns to write a positive integer less than or equal to $10$ on the blackboard. The integer written at any step cannot be a factor of any existing integer on board. Determine, with proof, who wins.

Point $M$ is chosen in triangle $ABC$ so that the radii of the circumcircles of triangles $AMC, BMC$, and $BMA$ are no smaller than the radius of the circumcircle of $ABC$. Prove that all four radii are equal.

A sleeping rabbit lies in the interior of a convex $2024$-gon. A hunter picks three vertices of the polygon and he lays a trap which covers the interior and the boundary of the triangular region determined by them. Determine the minimum number of times he needs to do this to guarantee that the rabbit will be trapped. [i]Proposed by Anant Mudgal and Rohan Goyal[/i]

Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.

The line $L$ is parallel to the plane $y=z$ and meets the parabola $2x=y^2 ,z=0$ and the parabola $3x=z^2, y=0$. Prove that if $L$ moves freely subject to these constraints then it generates the surface $x=(y-z)\left(\frac{y}{2}-\frac{z}{3}\right)$.

a) A certain locker room contains $n$ lockers numbered $1,2,\ldots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1, T_2 ,\ldots, T_n$, whereby with the operation $T_k$ the state of those lockers whose number is divisible by $k$ is swapped. After all $n$ operations have been performed, it is observed that all lockers whose number is a perfect square (and only those lockers) are open. Prove this. b) Investigate in a meaningful mathematical way a procedure or set of operations similar to those above which will produce the set of cubes, or the set of numbers of the form $2 m^2 $, or the set of numbers of the form $m^2 +1$, or some nontrivial similar set of your own selection.

Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Set $X$ is called [i]fancy[/i] if it satisfies all of the following conditions: [list] [*]The number of elements of $X$ is $2022$. [*]Each element of $X$ is a closed interval contained in $[0, 1]$. [*]For any real number $r \in [0, 1]$, the number of elements of $X$ containing $r$ is less than or equal to $1011$. [/list] For [i]fancy[/i] sets $A, B$, and intervals $I \in A, J \in B$, denote by $n(A, B)$ the number of pairs $(I, J)$ such that $I \cap J \neq \emptyset$. Determine the maximum value of $n(A, B)$.

Let $A$ and $B$ be matrices of size $3\times 2$ and $2\times 3$ respectively. Suppose that $$AB =\begin{pmatrix} 8 & 2 & -2\\ 2 & 5 &4 \\ -2 &4 &5 \end{pmatrix}.$$ Show that the product $BA$ is equal to $\begin{pmatrix} 9 &0\\ 0 &9 \end{pmatrix}.$

Given a circle \( \omega \) and two points \( A \) and \( B \) outside \( \omega \), a quadrilateral \( PQRS \) is defined as[i] "good"[/i] if \( P, Q, R, S \) are four distinct points on \( \omega \) in order, and lines \( PQ \) and \( RS \) intersect at \( A \) and lines \( PS \) and \( QR \) intersect at \( B \). For a quadrilateral \( T \), let \( S_T \) denote its area. If there exists a [i]good[/i] quadrilateral, prove that there exists [i]good[/i] quadrilateral \( T \) such that for any good quadrilateral $T_1 (T_1 \neq T)$, \( S_{T_1} < S_T \).

Two circles $ w_1 $ and $ w_2 $ intersect at two points $ P $ and $ Q $. The common tangent to $ w_1 $ and $ w_2 $, which is closer to the point $ P $ than to $ Q $, touches these circles at $ A $ and $ B $, respectively. The tangent to $ w_1 $ at the point $ P $ intersects $ w_2 $ at the point $ E $ (different from $ P $), and the tangent to $ w_2 $ at the point $ P $ intersects $ w_1 $ at $ F $ (different from $ P $). Let $ H $ and $ K $ be points on the rays $ AF $ and $ BE $, respectively, such that $ AH = AP $ and $ BK = BP $. Prove that the points $ A $, $ H $, $ Q $, $ K $ and $ B $ lie on the same circle.

Find all prime number $p$ such that $p^{2002}+2003^{p-1}-1$ is a multiple of $2003p$.

$a$, $b$, $c$ are three distinct real numbers, given $\lambda >0$. Proof that \[\frac{1+ \lambda ^2a^2b^2}{(a-b)^2}+\frac{1+ \lambda ^2b^2c^2}{(b-c)^2}+\frac{1+ \lambda ^2c^2a^2}{(c-a)^2} \geq \frac 32 \lambda.\] [hide=Remark]Old problem, can be found [url=https://artofproblemsolving.com/community/c6h588854p3487434]here[/url]. Double post to have a cleaner thread for collection (as the original one contains a messy quote)[/hide]

Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.