The integer $n$, between 10000 and 99999, is $abcde$ when written in decimal notation. The digit $a$ is the remainder when $n$ is divided by 2, the digit $b$ is the remainder when $n$ is divided by 3, the digit $c$ is the remainder when $n$ is divided by 4, the digit $d$ is the remainder when $n$ is divied by 5, and the digit $e$ is the reminader when $n$ is divided by 6. Find $n$.

Let $A, B, C, D, E$ be five points on a circle such that $|AB| = |CD|$ and $|BC| = |DE|$. The segments $AD$ and $BE$ intersect at $F$. Let $M$ denote the midpoint of segment $CD$. Prove that the circle of center $M$ and radius $ME$ passes through the midpoint of segment $AF$.

Prove that $2^{147} - 1$ is divisible by $343.$

Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Let $n>1 \in \mathbb{N}$ and $a_1, a_2, ..., a_n$ be a sequence of $n$ natural integers. Let: $$b_1 = \left[\frac{a_2 + \cdots + a_n}{n-1}\right], b_i = \left[\frac{a_1 + \cdots + a_{i-1} + a_{i+1} + \cdots + a_n}{n-1}\right], b_n = \left[\frac{a_1 + \cdots + a_{n-1}}{n-1}\right]$$ Define a mapping $f$ by $f(a_1,a_2, \cdots a_n) = (b_1,b_2,\cdots,b_n)$. a) Let $g: \mathbb{N} \to \mathbb{N}$ be a function such that $g(1)$ is the number of different elements in $f(a_1,a_2, \cdots a_n)$ and $g(m)$ is the number od different elements in $f^m(a_1,a_2, \cdots a_n) = f(f^{m-1}(a_1,a_2, \cdots a_n)); m>1$. Prove that $\exists k_0 \in \mathbb{N}$ s.t. for $m \ge k_0$ the function $g(m)$ is periodic. b) Prove that $\sum_{m=1}^k \frac{g(m)}{m(m+1)} < C$ for all $k \in \mathbb{N}$, where $C$ is a function that doesn't depend on $k$.

The $x$ and $y$ figures satisfy a condition: for every $n\ge1$ the number $$xx...x6yy...y4$$ ($n$ times $x$ and $n$ times $y$) is a perfect square. Find all possible $x$ and $y$.

How many $4 \times 4$ arrays whose entries are $0$s and $1$s are there such that the row sums (the sum of the entries in each row) are $1,2,3,$ and $4,$ in some order, and the column sums (the sum of the entries in each column) are also $1,2,3,$ and $4,$ in some order? For example, the array $\begin{bmatrix} 1 & 1 & 1 & 0\\ 0 & 1 & 1 & 0\\ 1 & 1 & 1 & 1\\ 0 & 1 & 0 & 0 \end{bmatrix}$ satisfies the condition. $\textbf{(A)}144~\textbf{(B)}240~\textbf{(C)}336~\textbf{(D)}576~\textbf{(E)}624$

In a convex quadrilateral $ABCD$, angles $B$ and $D$ are right angles. On on sides $AB$, $BC$, $CD$, $DA$ points $K$, $L$, $M$, $N$ are taken respectively so that $KN \perp AC$ and $LM \perp AC$. Prove that $KM$, $LN$ and $AC$ intersect at one point.

Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number. (The triangular numbers are the $t_n = n(n+1)/2$ with $n$ in $\{0,1,2,\dots\}$.)

In a family album, there are ten photos. On each of them, three people are pictured: in the middle stands a man, to the right of him stands his brother, and to the left of him stands his son. What is the least possible total number of people pictured, if all ten of the people standing in the middle of the ten pictures are different.

Let $n \ge 2$ be a positive integer. Suppose $a_1, a_2, \dots, a_n$ are distinct integers. For $k = 1, 2, \dots, n$, let \[ s_k := \prod_{\substack{i \not= k, \\ 1 \le i \le n}} |a_k - a_i|, \] i.e. $s_k$ is the product of all terms of the form $|a_k - a_i|$, where $i \in \{ 1, 2, \dots, n \}$ and $i \not= k$. Find the largest positive integer $M$ such that $M$ divides the least common multiple of $s_1, s_2, \dots, s_n$ for any choices of $a_1, a_2, \dots, a_n$.

We have a machine that has an input and an output. The input is a letter from the finite set $I$ and the output is a lamp that at each moment has one of the colors of the set $C=\{c_1,\dots,c_p\}$. At each moment the machine has an inner state that is one of the $n$ members of finite set $S$. The function $o: S \rightarrow C$ is a surjective function defining that at each state, what color must the lamp be, and the function $t:S \times I \rightarrow S$ is a function defining how does giving each input at each state changes the state. We only shall see the lamp and we have no direct information from the state of the car at current moment. In other words a machine is $M=(S,I,C,o,t)$ such that $S,I,C$ are finite, $t:S \times I \rightarrow S$ , and $o:S \rightarrow C$ is surjective. It is guaranteed that for each two different inner states, there's a sequence of inputs such that the color of the lamp after giving the sequence to the machine at the first state is different from the color of the lamp after giving the sequence to the machine at the second state. (a) The machine $M$ has $n$ different inner states. Prove that for each two different inner states, there's a sequence of inputs of length no more than $n-p$ such that the color of the lamp after giving the sequence to the machine at the first state is different from the color of the lamp after giving the sequence to the machine at the second state. (b) Prove that for a machine $M$ with $n$ different inner states, there exists an algorithm with no more than $n^2$ inputs that starting at any unknown inner state, at the end of the algorithm the state of the machine at that moment is known. Can you prove the above claim for $\frac{n^2}{2}$?

Let $\vartriangle AMO$ be an equilateral triangle. Let $U$ and $G$ lie on side $AM$, and let $S$ and $N$ lie on side $AO$ such that $AU =UG = GM$ and $AS = SN = NO$. Find the value of $\frac{[MONG]}{[U S A]}$

Let $ABC$ be a triangle and let $I$ and $O$ denote its incentre and circumcentre respectively. Let $\omega_A$ be the circle through $B$ and $C$ which is tangent to the incircle of the triangle $ABC$; the circles $\omega_B$ and $\omega_C$ are defined similarly. The circles $\omega_B$ and $\omega_C$ meet at a point $A'$ distinct from $A$; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA',BB'$ and $CC'$ are concurrent at a point on the line $IO$. [i](Russia) Fedor Ivlev[/i]

Prove that any rational $r \in (0, 1)$ can be written uniquely in the form $$r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}$$ where $a_i\text{’s}$ are nonnegative integers with $a_i\leq i-1$ for all $i$.

We call a subset $S$ of vertices of graph $G$, $2$-dominating, if and only if for every vertex $v\notin S,v\in G$, $v$ has at least two neighbors in $S$. prove that every $r$-regular $(r\ge3)$ graph has a $2$-dominating set with size at most $\frac{n(1+\ln(r))}{r}$.(15 points) time of this exam was 3 hours

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Ben was trying to solve for $x$ in the equation $6 + x = 1$. Unfortunately, he was reading upside-down and misread the equation as $1 = x + 9$. What is the positive difference between Ben's answer and the correct answer? [b]p2.[/b] Anjali and Meili each have a chocolate bar shaped like a rectangular box. Meili's bar is four times as long as Anjali's, while Anjali's is three times as wide and twice as thick as Meili's. What is the ratio of the volume of Anjali's chocolate to the volume of Meili's chocolate? [b]p3.[/b] For any two nonnegative integers $m, n$, not both zero, define $m?n = m^n + n^m$. Compute the value of $((2?0)?1)?7$. [b]p4.[/b] Eliza is making an in-scale model of the Phillips Exeter Academy library, and her prototype is a cube with side length $6$ inches. The real library is shaped like a cube with side length $120$ feet, and it contains an entrance chamber in the front. If the chamber in Eliza's model is $0.8$ inches wide, how wide is the real chamber, in feet? [b]p5.[/b] One day, Isaac begins sailing from Marseille to New York City. On the exact same day, Evan begins sailing from New York City to Marseille along the exact same route as Isaac. If Marseille and New York are exactly $3000$ miles apart, and Evan sails exactly 40 miles per day, how many miles must Isaac sail each day to meet Evan's ship in $30$ days? [b]p6.[/b] The conversion from Celsius temperature C to Fahrenheit temperature F is: $$F = 1.8C + 32.$$ If the lowest temperature at Exeter one day was $20^o$ F, and the next day the lowest temperature was $5^o$ C higher, what would be the lowest temperature that day, in degrees Fahrenheit? [b]p7.[/b] In a school, $60\%$ of the students are boys and $40\%$ are girls. Given that $40\%$ of the boys like math and $50\%$ of the people who like math are girls, what percentage of girls like math? [b]p8.[/b] Adam and Victor go to an ice cream shop. There are four sizes available (kiddie, small, medium, large) and seventeen different flavors, including three that contain chocolate. If Victor insists on getting a size at least as large as Adam's, and Adam refuses to eat anything with chocolate, how many different ways are there for the two of them to order ice cream? [b]p9.[/b] There are $10$ (not necessarily distinct) positive integers with arithmetic mean $10$. Determine the maximum possible range of the integers. (The range is defined to be the nonnegative difference between the largest and smallest number within a list of numbers.) [b]p10.[/b] Find the sum of all distinct prime factors of $11! - 10! + 9!$. [b]p11.[/b] Inside regular hexagon $ZUMING$, construct square $FENG$. What fraction of the area of the hexagon is occupied by rectangle $FUME$? [b]p12.[/b] How many ordered pairs $(x, y)$ of nonnegative integers satisfy the equation $4^x \cdot 8^y = 16^{10}$? [b]p13.[/b] In triangle $ABC$ with $BC = 5$, $CA = 13$, and $AB = 12$, Points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, such that $EF \parallel BC$. Given that triangle $AEF$ and trapezoid $EFBC$ have the same perimeter, find the length of $EF$. [b]p14.[/b] Find the number of two-digit positive integers with exactly $6$ positive divisors. (Note that $1$ and $n$ are both counted among the divisors of a number $n$.) [b]p15.[/b] How many ways are there to put two identical red marbles, two identical green marbles, and two identical blue marbles in a row such that no red marble is next to a green marble? [b]p16.[/b] Every day, Yannick submits $8$ more problems to the EMCC problem database than he did the previous day. Every day, Vinjai submits twice as many problems to the EMCC problem database as he did the previous day. If Yannick and Vinjai initially both submit one problem to the database on a Monday, on what day of the week will the total number of Vinjai's problems first exceed the total number of Yannick's problems? [b]p17.[/b] The tiny island nation of Konistan is a cone with height twelve meters and base radius nine meters, with the base of the cone at sea level. If the sea level rises four meters, what is the surface area of Konistan that is still above water, in square meters? [b]p18.[/b] Nicky likes to doodle. On a convex octagon, he starts from a random vertex and doodles a path, which consists of seven line segments between vertices. At each step, he chooses a vertex randomly among all unvisited vertices to visit, such that the path goes through all eight vertices and does not visit the same vertex twice. What is the probability that this path does not cross itself? [b]p19.[/b] In a right-angled trapezoid $ABCD$, $\angle B = \angle C = 90^o$, $AB = 20$, $CD = 17$, and $BC = 37$. A line perpendicular to $DA$ intersects segment $BC$ and $DA$ at $P$ and $Q$ respectively and separates the trapezoid into two quadrilaterals with equal area. Determine the length of $BP$. [b]p20.[/b] A sequence of integers $a_i$ is defined by $a_1 = 1$ and $a_{i+1} = 3i - 2a_i$ for all integers $i \ge 1$. Given that $a_{15} = 5476$, compute the sum $a_1 + a_2 + a_3 + ...+ a_{15}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The root(s) of $ \frac {15}{x^2 \minus{} 4} \minus{} \frac {2}{x \minus{} 2} \equal{} 1$ is (are): $ \textbf{(A)}\ \minus{} 5\text{ and }3 \qquad\textbf{(B)}\ \pm 2 \qquad\textbf{(C)}\ 2\text{ only} \qquad\textbf{(D)}\ \minus{} 3\text{ and }5 \qquad\textbf{(E)}\ 3\text{ only}$

Evaluate: $ \sum\limits_{n\equal{}1}^\infty \arctan{\left(\frac{1}{n^2\minus{}n\plus{}1}\right)}$

Points $ P,Q,R$ are given in the plane. It is known that there is a triangle $ ABC$ such that $ P$ is the midpoint of $ BC$, $ Q$ the point on side $ CA$ with $ \frac{CQ}{QA}\equal{}2$, and $ R$ the point on side $ AB$ with $ \frac{AR}{RB}\equal{}2$. Determine with proof how the triangle $ ABC$ may be reconstructed from $ P,Q,R$.

Consider the set: $A = \{1, 2,..., 100\}$ Prove that if we take $11$ different elements from $A$, there are $x, y$ such that $x \neq y$ and $0 < |\sqrt{x} - \sqrt{y}| < 1$

Given a triangle $ABC$ where $|BC| = a, |CA| = b$ and $|AB| = c$, prove that the equality $\frac{1}{a + b}+\frac{1}{b + c}=\frac{3}{a + b + c}$ holds if and only if $\angle ABC = 60^o$.

Let \( \triangle ABC \) be an equilateral triangle, and let \( M \) be the midpoint of \( BC \). Let \( C_1 \) be the circumcircle of triangle \( \triangle ABC \) and \( C_2 \) the circumcircle of triangle \( \triangle ABM \). Determine the ratio between the areas of the circles \( C_1 \) and \( C_2 \).

Suppose $S$ cans of soda can be purchased from a vending machine for $Q$ quarters. Which of the following expressions describes the number of cans of soda that can be purchased for $D$ dollars, where $1$ dollar is worth $4$ quarters? $\textbf{(A) }\frac{4DQ}S\qquad\textbf{(B) }\frac{4DS}Q\qquad\textbf{(C) }\frac{4Q}{DS}\qquad\textbf{(D) }\frac{DQ}{4S}\qquad\textbf{(E) }\frac{DS}{4Q}$