Let $ABC$ be a triangle with $AB=34,BC=25,$ and $CA=39$. Let $O,H,$ and $ \omega$ be the circumcenter, orthocenter, and circumcircle of $\triangle ABC$, respectively. Let line $AH$ meet $\omega$ a second time at $A_1$ and let the reflection of $H$ over the perpendicular bisector of $BC$ be $H_1$. Suppose the line through $O$ perpendicular to $A_1O$ meets $\omega$ at two points $Q$ and $R$ with $Q$ on minor arc $AC$ and $R$ on minor arc $AB$. Denote $\mathcal H$ as the hyperbola passing through $A,B,C,H,H_1$, and suppose $HO$ meets $\mathcal H$ again at $P$. Let $X,Y$ be points with $XH \parallel AR \parallel YP, XP \parallel AQ \parallel YH$. Let $P_1,P_2$ be points on the tangent to $\mathcal H$ at $P$ with $XP_1 \parallel OH \parallel YP_2$ and let $P_3,P_4$ be points on the tangent to $\mathcal H$ at $H$ with $XP_3 \parallel OH \parallel YP_4$. If $P_1P_4$ and $P_2P_3$ meet at $N$, and $ON$ may be written in the form $\frac{a}{b}$ where $a,b$ are positive coprime integers, find $100a+b$. [i]Proposed by Vincent Huang[/i]

Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy \[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$. Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.

We have erasers, four pencils, two note books and three pens and we want to divide them between two persons so that every one receives at least one of the above stationery. In how many ways is this possible? [Note that the are not distinct.]

Let $X = 123456789$. Find the sum of the tens digits of all integral multiples of $11$ that can be obtained by interchanging two digits of $X$.

For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts: $(1)$ two successive $15\%$ discounts $(2)$ three successive $10\%$ discounts $(3)$ a $25\%$ discount followed by a $5\%$ discount What is the smallest possible positive integer value of $n$? ${ \textbf{(A)}\ \ 27\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}}\ 31\qquad\textbf{(E)}\ 33$

The $ 2007$ AMC $ 10$ will be scored by awarding $ 6$ points for each correct response, $ 0$ points for each incorrect response, and $ 1.5$ points for each problem left unanswered. After looking over the $ 25$ problems, Sarah has decided to attempt the first $ 22$ and leave only the last $ 3$ unanswered. How many of the first $ 22$ problems must she solve correctly in order to score at least $ 100$ points? $ \textbf{(A)}\ 13\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ 16\qquad \textbf{(E)}\ 17$

The quadrilateral $ABCD$ has $AD = DC = CB < AB$ and $AB \parallel CD$. Points $E$ and $F$ lie on the sides $CD$ and $BC$ such that $\angle ADE = \angle AEF$. Prove that: (a) $4CF \le CB$. (b) If $4CF = CB$, then $AE$ is the angle bisector of $\angle DAF$.

In $\Delta ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \dots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum \[ \overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}? \] $ \textbf{(A) }1011 \qquad \textbf{(B) }1012 \qquad \textbf{(C) }2023 \qquad \textbf{(D) }2024 \qquad \textbf{(E) }2025 \qquad $

Given non-constant linear functions $p(x), q(x), r(x)$. Prove that at least one of three trinomials $pq+r, pr+q, qr+p$ has a real root. [b]proposed by S. Berlov[/b]

Let $a_0, a_1, \ldots , a_n$ and $b_0, b_1, \ldots , b_n$ be sequences of real numbers such that $a_0 = b_0 \geqslant 0$, $a_n = b_n > 0$ and \[a_i=\sqrt{\frac{a_{i+1}+a_{i-1}}{2}},\quad b_i=\sqrt{\frac{b_{i+1}+b_{i-1}}{2}},\]for all $i=1,\ldots,n-1$. Prove that $a_1 = b_1$.

Betsy is addicted to chocolate. Every day, she eats $2$ chocolates at breakfast, $3$ chocolates at lunch, $1$ chocolate during her afternoon snack time, and $5$ chocolates at dinner. If she begins eating a bag of $100$ chocolates at breakfast one day, during which meal will she eat the last piece in the bag? $\textbf{(A) }\text{breakfast}\qquad\textbf{(B) }\text{lunch}\qquad\textbf{(C) }\text{snack time}\qquad\textbf{(D) }\text{dinner}\qquad\textbf{(E) }\text{impossible to determine}$

Define an [i]almost-palindrome[/i] as a string of letters that is not a palindrome but can become a palindrome if one of its letters is changed. For example, $TRUST$ is an almost-palindrome because the $R$ can be changed to an $S$ to produce a palindrome, but $TRIVIAL$ is not an almost-palindrome because it cannot be changed into a palindrome by swapping out only one letter (both the $A$ and the $L$ are out of place). How many almost-palindromes contain fewer than $4$ letters.

Arrange the positive integers into two lines as follows: \begin{align*} 1 \quad 3 \qquad 6 \qquad\qquad\quad 11 \qquad\qquad\qquad\qquad\quad\ 19\qquad\qquad32\qquad\qquad 53\ldots\\ \mbox{\ \ } 2 \quad 4\ \ 5 \quad 7\ \ 8\ \ 9\ \ 10\quad\ 12\ 13\ 14\ 15\ 16\ 17\ 18\quad\ 20 \mbox{ to } 31\quad\ 33 \mbox{ to } 52\quad\ \ldots\end{align*} We start with writing $1$ in the upper line, $2$ in the lower line and $3$ again in the upper line. Afterwards, we alternately write one single integer in the upper line and a block of integers in the lower line. The number of consecutive integers in a block is determined by the first number in the previous block. Let $a_1$, $a_2$, $a_3$, $\ldots$ be the numbers in the upper line. Give an explicit formula for $a_n$.

Characterize all triangles $ABC$ s.t. \[ AI_a : BI_b : CI_c = BC: CA : AB \] where $I_a$ etc. are the corresponding excentres to the vertices $A, B , C$

Let $n$ be a fixed positive integer. An $n$-staircase is a polyomino with $\frac{n(n+1)}{2}$ cells arranged in the shape of a staircase, with arbitrary size. Here are two examples of $5$-staircases: [asy] int s = 5; for(int i = 0; i < s; i=i+1) { draw((0,0)--(0,i+1)--(s-i,i+1)--(s-i,0)--cycle); } for(int i = 0; i < s; i=i+1) { draw((10,.67*s)--(10,.67*(s-i-1))--(.67*(s-i)+10,.67*(s-i-1))--(.67*(s-i)+10,.67*s)--cycle); } [/asy] Prove that an $n$-staircase can be dissected into strictly smaller $n$-staircases. [i]Proposed by James Lin.[/i]

In the triangle $ABC$ it is known that $2AC=AB$ and $\angle A = 2\angle B$. In this triangle draw the angle bisector $AL$, and mark point $M$, the midpoint of the side $AB$. It turned out that $CL = ML$. Prove that $\angle B= 30^o$. (Hilko Danilo)

Let $S$ be an infinite set of real numbers such that $|x_1+x_2+\cdots + x_n | \leq 1 $ for all finite subsets $\{x_1,x_2,\ldots,x_n\} \subset S$. Show that $S$ is countable.

Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from left to right. For example $A_3=A_2A_1=10$, $A_4=A_3A_2=101$, $A_5=A_4A_3=10110$, and so forth. Determine all $n$ such that $11$ divides $A_n$.

For each fixed positiver integer $n$, $n\geq 4$ and $P$ an integer, let $(P)_n \in [1, n]$ be the smallest positive residue of $P$ modulo $n$. Two sequences $a_1, a_2, \dots, a_k$ and $b_1, b_2, \dots, b_k$ with the terms in $[1, n]$ are defined as equivalent, if there is $t$ positive integer, gcd$(t,n)=1$, such that the sequence $(ta_1)_n, \dots, (ta_k)_n$ is a permutation of $b_1, b_2, \dots, b_k$. Let $\alpha$ a sequence of size $n$ and your terms are in $[1, n]$, such that each term appears $h$ times in the sequence $\alpha$ and $2h\geq n$. Show that $\alpha$ is equivalent to some sequence $\beta$ which contains a subsequence such that your size is(at most) equal to $h$ and your sum is exactly equal to $n$.

Let $x,y,n$ be positive integers with $n>1$. How many ordered triples $(x, y, n)$ of solutions are there to the equation $x^n-y^n=2^{100}$?

Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?

$ABCD$ is a cyclic quadrilateral with $\angle BAD=90^\circ$ and $\angle ABC>90^\circ$. $AB$ is extended to a point $E$ such that $\angle AEC=90^\circ$.If $AB=7,BE=9,$ and $EC=12$,calculate $AD$.

Find the last two digits of $2023+202^3+20^{23}$. [i]Proposed by Anthony Yang[/i]

An airplane accelerates at $10$ mph per second and decelerates at $15$ mph/sec. Given that its takeoff speed is $180$ mph, and the pilots want enough runway length to safely decelerate to a stop from any speed below takeoff speed, what’s the shortest length that the runway can be allowed to be? Assume the pilots always use maximum acceleration when accelerating. Please give your answer in miles.

An object moves in two dimensions according to \[\vec{r}(t) = (4.0t^2 - 9.0)\vec{i} + (2.0t - 5.0)\vec{j}\] where $r$ is in meters and $t$ in seconds. When does the object cross the $x$-axis? $ \textbf{(A)}\ 0.0 \text{ s}\qquad\textbf{(B)}\ 0.4 \text{ s}\qquad\textbf{(C)}\ 0.6 \text{ s}\qquad\textbf{(D)}\ 1.5 \text{ s}\qquad\textbf{(E)}\ 2.5 \text{ s}$