For how many positive integers $ m$ does there exist at least one positive integer $ n$ such that $ m\cdot n \le m \plus{} n$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}$ infinitely many

Ana choses two real numbers $ y>0,x $ and Bogdan repeatedly tries to guess these in the following manner: at step $ j $ he choses a real number $ b_j, $ asks her if $ b_j=x+jy, $ and she tells him the truth. [b]a)[/b] If $ x=0, $ can Bogdan find Ana's numbers in a finite number of steps? [b]b)[/b] If $ x\neq 0, $ can Bogdan find Ana's numbers in a finite number of steps?

Let $O$ be an interior point of an acute triangle $ABC$. The circles with centers the midpoints of its sides and passing through $O$ mutually intersect the second time at the points $K$, $L$ and $M$ different from $O$. Prove that $O$ is the incenter of the triangle $KLM$ if and only if $O$ is the circumcenter of the triangle $ABC$.

Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$ [asy] // diagram by SirCalcsALot draw(arc((0,0),17,180,0)); draw((-17,0)--(17,0)); fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey); draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle); dot("$A$",(8,0), 1.25*S); dot("$B$",(8,15), 1.25*N); dot("$C$",(-8,15), 1.25*N); dot("$D$",(-8,0), 1.25*S); dot("$E$",(17,0), 1.25*S); dot("$F$",(-17,0), 1.25*S); label("$16$",(0,0),N); label("$9$",(12.5,0),N); label("$9$",(-12.5,0),N); [/asy] $\textbf{(A) }240 \qquad \textbf{(B) }248 \qquad \textbf{(C) }256 \qquad \textbf{(D) }264 \qquad \textbf{(E) }272$

We will call the ordered pair $(a,b)$ “parallel”, where $a,b\in \mathbb{N}$, if $\sqrt{ab}\in \mathbb{N}$. Prove that the number of “parallel” pairs $(a,b)$, for which $1\leq a,b\leq 10^6$ is at least $3.10^6(ln\, 10-1)$.

A graph $ G$ with $ n$ vertices is given. Some $ x$ of its edges are colored red so that each triangle has at most one red edge. The maximum number of vertices in $ G$ that induce a bipartite graph equals $ y.$ Prove that $ n\ge 4x/y.$

Let $\left< F_n\right>$ be the Fibonacci sequence, that is, $F_0=0$, $F_1=1$, and $F_{n+2}=F_{n+1}+F_{n}$ holds for all nonnegative integers $n$. Find all pairs $(a,b)$ of positive integers with $a < b$ such that $F_n-2na^n$ is divisible by $b$ for all positive integers $n$.

The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]

If $ (x^2+1)(y^2+1)+9=6(x+y)$ where $x,y$ are real numbers, what is $x^2+y^2$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3 $

An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of length $1$ unit either up or to the right. How many up-right paths from $(0, 0)$ to $(7, 7),$ when drawn in the plane with the line $y = x - 2.021$, enclose exactly one bounded region below that line?

$7662$ chairs are placed in a circle around the city of Chiang Mai. They are also marked with a label for either $1$st, $2$nd, or $3$rd grade students, so that there are $2554$ chairs labeled with each label. The following situations happen, in order [list=i] [*] $2554$ students each from the $1$st, $2$nd, and $3$rd grades are given a ball as follows: $1$st grade students receive footballs, $2$nd grade students receive basketballs, and $3$rd grade students receive volleyballs. [*] The students go sit in chairs labeled for their grade [*] The students simultaneously send their balls to the student to their left, and this happens some positive number of times. [/list] A labelling of the chairs is called [i]lin-ping[/i] if it is possible for all $1$st, $2$nd, and $3$rd grade students to now hold volleyballs, footballs, and basketballs respectively. Compute the number of [i]lin-ping[/i] labellings

A $10$ digit number is called interesting if its digits are distinct and is divisible by $11111$. Then find the number of interesting numbers.

Shen, Ling, and Ru each place four slips of paper with their name on it into a bucket. They then play the following game: slips are removed one at a time, and whoever has all of their slips removed first wins. Shen cheats, however, and adds an extra slip of paper into the bucket, and will win when four of his are drawn. Given that the probability that Shen wins can be expressed as simplified fraction $\tfrac{m}{n}$, compute $m+n$.

Quadrilateral $ABCD$ is cyclic. Line through point $D$ parallel with line $BC$ intersects $CA$ in point $P$, line $AB$ in point $Q$ and circumcircle of $ABCD$ in point $R$. Line through point $D$ parallel with line $AB$ intersects $AC$ in point $S$, line $BC$ in point $T$ and circumcircle of $ABCD$ in point $U$. If $PQ=QR$, prove that $ST=TU$

Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $ Show that $ f $ and $ g $ are the same function.

Evaluate \[\int_0^{\pi} \frac{x^2\cos ^ 2 x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx\]

In a $3n \times 3n$ grid, each square is either black or red. Each red square not on the edge of the grid has exactly five black squares among its eight neighbor squares.. On every black square that not at the edge of the grid, there are exactly four reds in the adjacent squares box. How many black and how many red squares are in the grid?

Let $a,b,c$ be positive real numbers with $a \leq c$ and $b \leq c$. Prove that $$ (a +10b)(b +22c)(c +7a) \geq 2024 abc.$$

[b]Problem 4[/b] : A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent line to both circles touches the circle with $AC$ as diameter at $P \neq C$ and the circle with $CB$ as diameter at $Q \neq C.$ Prove that lines $AP, BQ$ and the common tangent line to both circles at $C$ all meet at a single point which lies on the circle with $AB$ as diameter.

[b]1.[/b] Some proper partitions $P_1, \dots , P_n$ of a finite set $S$ (that is, partitions containing at least two parts) are called [i]independent[/i] if no matter how we choose one class from each partition, the intersection of the chosen classes is nonempty. Show that if the inequality $\frac{\left | S \right | }{2} < \left |P_1 \right | \dots \left |P_n \right |\qquad \quad (*)$ holds for some independent partitions, then $P_1, \dots , P_n$ is maximal in the sense that there is no partition $P$ such that $P,P_1, \dots , P_n$ are independent. On the other hand, show that inequality $(*)$ is not necessary for this maximality. ([b]C.20[/b]) [E. Gesztelyi]

Compute the maximum area of a triangle having a median of length 1 and a median of length 2.

Given a cyclic quadrilateral $ABCD$ with circumcenter $O$, let the circle $(AOD)$ intersect the segments $AB$, $AC$, $DB$, $DC$ at $P$, $Q$, $R$, $S$ respectively. Suppose $X$ is the reflection of $D$ about $PQ$ and $Y$ is the reflection of $A$ about $RS$. Prove that the circles $(AOD)$, $(BPX)$, $(CSY)$ meet at a common point. [i]Proposed by Leia Mayssa & Ivan Chan Kai Chin[/i]

There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $. Find the sum of the sum of the coordinates of all such points.

Jack writes whole numbers starting from $ 1$ and skips all numbers that contain either a $2$ or $9$. What is the $100$th number that Jack writes down?