Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$. Find all local champions and determine their number. [i]Proposed by Zoran Sunic, USA[/i]

The number of real values of $ x$ satisfying the equality $ (\log_2x)(\log_bx) \equal{} \log_ab$, where $ a > 0$, $ b > 0$, $ a \neq 1$, $ b \neq 1$, is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{a finite integer greater than 2} \qquad \textbf{(E)}\ \text{not finite}$

Let $ABC$ be a triangle with area $1$ (cm$^2$). Points $D,E$ and $F$ lie on the sides $AB, BC$ and CA, respectively. Prove that $min\{$area of $\vartriangle ADF,$ area of $\vartriangle BED,$ area of $\vartriangle CEF\} \le \frac14$ (cm$^2$).

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, such that $CD$ is not a diameter of its circumcircle. The lines $AD$ and $BC$ intersect at point $P$, so that $A$ lies between $D$ and $P$, and $B$ lies between $C$ and $P$. Suppose triangle $PCD$ is acute and let $H$ be its orthocenter. The points $E$ and $F$ on the lines $BC$ and $AD$, respectively, are such that $BD \parallel HE$ and $AC\parallel HF$. The line through $E$, perpendicular to $BC$, intersects $AD$ at $L$, and the line through $F$, perpendicular to $AD$, intersects $BC$ at $K$. Prove that the points $K$, $L$, $O$ are collinear. [i]Amir Parsa Hosseini Nayeri, Iran[/i]

To write all consecutive natural numbers from $1ab$ to $ab2$ inclusive, $1ab1$ digits have been used. Determine how many more digits are needed to write the natural numbers up to $aab$ inclusive. Give all chances. ($a$ and $b$ represent digits)

Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ are tangent to each other externally at point $O$. Points $X$ and $Y$ on $k_1$ and $k_2$ respectively are such that rays $O_1X$ and $O_2Y$ are parallel and codirectional. Prove that two tangents from $X$ to $k_2$ and two tangents from $Y$ to $k_1$ touch the same circle passing through $O$. (V. Yasinsky)

A mink is standing in the center of a field shaped like a regular polygon. The field is surrounded by a fence, and the mink can only exit through the vertices of the polygon. A dog is standing on one of the vertices, and can move along the fence. The mink wants to escape the field, while the dog tries to prevent it. Each of them moves with constant velocity. For what ratio of velocities could the mink escape if: a. The field is a regular triangle? b. The field is a square?

There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?

How many integers $n$ with $10 \le n \le 500$ have the property that the hundreds digit of $17n$ and $17n+17$ are different? [i]Proposed by Evan Chen[/i]

Hannibal and Clarice are still at a barbecue and there are three anticuchos left, each of which it has $10$ pieces. Of the $30$ total pieces, there are $29$ chicken and one meat, the which is at the bottom of one of the anticuchos. To decide who to stay with the piece of meat, they decide to play the following game: they alternately take out a piece of one of the anticuchos (they can take only the outer pieces) and whoever wins the game manages to remove the piece of meat. Clarice decides if she starts or Hannibal starts. What should she decide?

Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$. (a) Prove that $8$ is $100$-discerning. (b) Prove that $9$ is not $100$-discerning. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

Let $\Gamma_B$ and $\Gamma_C$ be excircles of an acute-angled triangle $ABC$ opposite to its vertices $B$ and $C$, respectively. Let $C_1$ and $L$ be the tangent points of $\Gamma_C$ and the side $AB$ and the line $BC$ respectively. Let $B_1$ and $M$ be the tangent points of $\Gamma_B$ and the side $AC$ and the line $BC$, respectively. Let $X$ be the point of intersection of the lines $LC_1$ and $MB_1$. Prove that $AX$ is equal to the inradius of the triangle $ABC$. (A. Voidelevich)

Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$? [asy]unitsize(4cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(1,0), B=(1,1), A=(0,1); pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0]; pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H); draw(A--B--C--D--cycle); draw(D--F); draw(C--E); draw(B--H); draw(A--G); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,NNW); label("$F$",F,ENE); label("$G$",G,SSE); label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$

Let $(X, <)$ be an arbitrary ordered set. Show that the elements of $X$ can be coloured by two colours in such a way that between any two points of the same colour there is a point of the opposite colour. (translated by L. Erdős)

On the grid plane all possible broken lines with the following properties are constructed: each of them starts at the point $(0, 0)$, has all its vertices at integer points, each linear segment goes either up or to the right along the grid lines. For each such broken line consider the corresponding [i]worm[/i], the subset of the plane consisting of all the cells that share at least one point with the broken line. Prove that the number of worms that can be divided into dominoes (rectangles $2\times 1$ and $1\times 2$) in exactly $n > 2$ different ways, is equal to the number of positive integers that are less than n and relatively prime to $n$. (Ilke Chanakchi, Ralf Schiffler)

In triangle $ ABC$, $ AB\equal{}AC$, $ \angle A\equal{}40^\circ$. Point $ O$ is within the triangle with $ \angle OBC \cong \angle OCA$. The number of degrees in angle $ BOC$ is: $ \textbf{(A)}\ 110 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 70$

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$

A semicircle with diameter $AB$ is given. Two non-intersecting circles $k_1$ and $k_2$ with different radii touch the diameter $AB$ and touch the semicircle internally at $C$ and $D$, respectively. An interior common tangent $t$ of $k_1$ and $k_2$ touches $k_1$ at $E$ and $k_2$ at $F$. Prove that the lines $CE$ and $DF$ intersect on the semicircle.

Find the sum of all real values of $A$ such that the equation $Axy+25x^2+25y^2-20x-22y+5=0$ has a unique solution in real numbers $(x,y)$.

Consider a triangle $ABC$ with $AB=AC$, and $D$ the foot of the altitude from the vertex $A$. The point $E$ lies on the side $AB$ such that $\angle ACE= \angle ECB=18^{\circ}$. If $AD=3$, find the length of the segment $CE$.

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

For any positive integer $a$, define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$. Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$.

Prove that $9$-digit number, that contains all the decimal digits except zero and does not ends with $5$ can not be exact square.

If $x_i$ is an integer greater than 1, let $f(x_i)$ be the greatest prime factor of $x_i,x_{i+1} =x_i-f(x_i)$ ($i\ge 0$ and i is an integer). (1) Prove that for any integer $x_0$ greater than 1, there exists a natural number$k(x_0)$, such that $x_{k(x_0)+1}=0$ Grade 10: (2) Let $V_{(x_0)}$ be the number of different numbers in $f(x_0),f(x_1),\cdots,f(x_{k(x_0)})$. Find the largest number in $V(2),V(3),\cdots,V(781)$ and give reasons. Note: Bai Lu Zhou Academy was founded in 1241 and has a history of 781 years. Grade 11: (2) Let $V_{(x_0)}$ be the number of different numbers in $f(x_0),f(x_1),\cdots,f(x_{k(x_0)})$. Find the largest number in $V(2),V(3),\cdots,V(2022)$ and give reasons.

On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types: (a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$. (b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$. Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves. [i]D. Fon-der-Flaas[/i]