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)}$.

Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$. [i]Alex Zhu.[/i]

In their base $ 10$ representation, the integer $ a$ consists of a sequence of $ 1985$ eights and the integer $ b$ consists of a sequence of $ 1985$ fives. What is the sum of the digits of the base 10 representation of $ 9ab$? $ \textbf{(A)}\ 15880 \qquad \textbf{(B)}\ 17856 \qquad \textbf{(C)}\ 17865 \qquad \textbf{(D)}\ 17874 \qquad \textbf{(E)}\ 19851$

The incircle of $\triangle ABC$ touch $AB$,$BC$,$CA$ at $K$,$L$,$M$. The common external tangents to the incircles of $\triangle AMK$,$\triangle BKL$,$\triangle CLM$, distinct from the sides of $\triangle ABC$, are drawn. Show that these three lines are concurrent.

Let $n$ be a positive integer with $k$ digits. A number $m$ is called an $alero$ of $n$ if there exist distinct digits $a_1$, $a_2$, $\dotsb$, $a_k$, all different from each other and from zero, such that $m$ is obtained by adding the digit $a_i$ to the $i$-th digit of $n$, and no sum exceeds 9. For example, if $n$ $=$ $2024$ and we choose $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $3$, then $m$ $=$ $4177$ is an alero of $n$, but if we choose the digits $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $6$, then we don't obtain an alero of $n$, because $4$ $+$ $6$ exceeds $9$. Find the smallest $n$ which is a multiple of $2024$ that has an alero which is also a multiple of $2024$.

Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

The monic polynomial $f$ has rational coefficients and is irreducible over the rational numbers. If $f(\sqrt5 +\sqrt2)= 0$, compute $f(f(\sqrt5 -\sqrt2))$. (A polynomial is [i]monic [/i] if its leading coefficient is $1$. A polynomial is [i]irreducible [/i] over the rational numbers if it cannot be expressed as a product of two polynomials with rational coefficients of positive degree. For example, $x^2 - 2$ is irreducible, but $x^2 - 1 = (x + 1)(x - 1)$ is not.)

Find the maximum number of pairwise disjoint sets of the form $S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$, $a, b \in \mathbb{Z}$.

If nonnegative reals $x_1, x_2, \ldots, x_n$ satisfy \[ \sum_{i=1}^n x_i^2 + 2\sum_{1 \leq k < j \leq n} \sqrt{\frac{k}{j}}x_kx_j = 1 \] what are the minimum and maximum values of $\sum_{i=1}^n x_i$?

In each vertex of a regular $n$-gon, there is a fortress. At the same moment, each fortress shoots one of the two nearest fortresses and hits it. The [i]result of the shooting[/i] is the set of the hit fortresses; we do not distinguish whether a fortress was hit once or twice. Let $P(n)$ be the number of possible results of the shooting. Prove that for every positive integer $k\geqslant 3$, $P(k)$ and $P(k+1)$ are relatively prime. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 3)[/i]

A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are 57, 60, and 91. What is the fourth term of this sequence? $\textbf{(A) }190\qquad\textbf{(B) }194\qquad\textbf{(C) }198\qquad\textbf{(D) }202\qquad\textbf{(E) }206$

The sum of two numbers is $ 10$; their product is $ 20$. The sum of their reciprocals is: $ \textbf{(A)}\ \frac{1}{10}\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 4$

Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle. Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$. [i]Australia[/i]

There are $6$ irrational numbers. Prove that there are always three of them, suppose $a,b,c$ such that $a+b$,$b+c$,$c+a$ are irrational numbers. [i](Erken)[/i]

Let $AB$ be a diameter of a circle with centre $O$, and $CD$ be a chord perpendicular to $AB$. A chord $AE$ intersects $CO$ at $M$, while $DE$ and $BC$ intersect at $N$. Prove that $CM:CO=CN:CB$.

Let $ABC$ be an isosceles triangle right at $C$ and $P$ any point on $CB$. Let also $Q$ be the midpoint of $AB$ and $R, S$ be the points on $AP$ such that $CR$ is perpendicular to $AP$ and $|AS|=|CR|$. Prove that the $|RS| = \sqrt2 |SQ|$.

Prove the inequality [b]a.)[/b] $ \left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers. [b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality \[ a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) }, \] then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

Let $ f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $ f\left(x \plus{} \dfrac{1}{f(y)}\right) \equal{} f\left(y \plus{} \dfrac{1}{f(x)}\right)$ for all $ x$, $ y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $ f$. [i]Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithuania[/i]