The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 45$

For all positive integers $n$, find the remainder of $\dfrac{(7n)!}{7^n \cdot n!}$ upon division by 7.

p1. Two integers $m$ and $n$ are said to be [i]coprime [/i] if there are integers $a$ and $ b$ such that $am + bn = 1$. Show that for each integer $p$, the pair of numbers formed by $21p + 4$ and $14p + 3$ are always coprime. p2. Two farmers, Person $A$ and Person $B$ intend to change the boundaries of their land so that it becomes like a straight line, not curvy as in image below. They do not want the area of ​​their origin to be reduced. Try define the boundary line they should agree on, and explain why the new boundary does not reduce the area of ​​their respective origins. [img]https://cdn.artofproblemsolving.com/attachments/4/d/ec771d15716365991487f3705f62e4566d0e41.png[/img] p3. The system of equations of four variables is given: $\left\{\begin{array}{l} 23x + 47y - 3z = 434 \\ 47x - 23y - 4w = 183 \\ 19z + 17w = 91 \end{array} \right. $ where $x, y, z$, and $w$ are positive integers. Determine the value of $(13x - 14y)^3 - (15z + 16w)^3$ p4. A person drives a motorized vehicle so that the material used fuel is obtained at the following graph. [img]https://cdn.artofproblemsolving.com/attachments/6/f/58e9f210fafe18bfb2d9a3f78d90ff50a847b2.png[/img] Initially the vehicle contains $ 3$ liters of fuel. After two hours, in the journey of fuel remains $ 1$ liter. a. If in $ 1$ liter he can cover a distance of $32$ km, what is the distance taken as a whole? Explain why you answered like that? b. After two hours of travel, is there any acceleration or deceleration? Explain your answer. c. Determine what the average speed of the vehicle is. p5. Amir will make a painting of the circles, each circle to be filled with numbers. The circle's painting is arrangement follows the pattern below. [img]https://cdn.artofproblemsolving.com/attachments/8/2/533bed783440ea8621ef21d88a56cdcb337f30.png[/img] He made a rule that the bottom four circles would be filled with positive numbers less than $10$ that can be taken from the numbers on the date of his birth, i.e. $26 \,\, - \,\, 12 \,\, - \,\,1961$ without recurrence. Meanwhile, the circles above will be filled with numbers which is the product of the two numbers on the circles in underneath. a. In how many ways can he place the numbers from left to right, right on the bottom circles in order to get the largest value on the top circle? Explain. b. On another occasion, he planned to put all the numbers on the date of birth so that the number of the lowest circle now, should be as many as $8$ circles. He no longer cares whether the numbers are repeated or not . i. In order to get the smallest value in the top circle, how should the numbers be arranged? ii. How many arrays are worth considering to produce the smallest value?

Let $\{a, b, c, d, e, f, g, h\}$ be a permutation of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that $\overline{abc} +\overline{def}$ is even?

The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time? $\textbf{(A) }5:50\qquad\textbf{(B) }6:00\qquad\textbf{(C) }6:30\qquad\textbf{(D) }6:55\qquad \textbf{(E) }8:10$

Let $ABC$ be an acute triangle such that $AB<AC$. Let $\Omega$ be its circumcircle, $O$ be its circumcenter, and $l$ be the internal angle bisector of $\angle BAC$. Suppose that the tangents to $\Omega$ at $B$ and $C$ intersect at $X$. Let $\omega$ be a circle whose center is $X$ and passes $B$, and $Y$ be the intersection of $l$ and $\omega$ which is chosen inside $\triangle ABC$. Let $D,E$ be the projections of $Y$ onto $AB,AC$, respectively. $OY$ meets $BC$ at $Z$. $ZD,ZE$ meet $l$ at $P,Q$, respectively. Prove that $BQ$ and $CP$ are parallel.

Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$. Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$, where $d(n)$ denotes the number of positive divisors of positive integer $n$.

In acute triangle $ABC$, an arbitrary point $P$ is chosen on altitude $AH$. Points $E$ and $F$ are the midpoints of sides $CA$ and $AB$ respectively. The perpendiculars from $E$ to $CP$ and from $F$ to $BP$ meet at point $K$. Prove that $KB = KC$.

A triangle $ABC$ has the orthocenter $H$ different from the vertexes and the circumcenter $O$. Let $M, N$ and $P$ be the circumcenters of triangles $HBC, HCA$ and $HAB$. Prove that the lines $AM, BN, CP$ and $OH$ are concurrent.

The angles of an acute triangle $ABC$ at $\alpha , \beta, \gamma$. Let $AD$ be a height, $CF$ a median, and $BE$ the bisector of $\angle B$. Show that $AD,CF$ and $BE$ are concurrent if and only if $\cos \gamma \tan\beta = \sin \alpha$ .

Let $M$ be a point in the plane, distinct from the vertices of $\triangle ABC$. Consider $N,P,Q$ the reflections of $M$ with respect to lines $AB, BC$ and $CA$, in this order. a) Prove that $N, P ,Q$ are collinear if and only if $M$ lies on the circumcircle of $\triangle ABC$. b) If $M$ does not lie on the circumcircle of $\triangle ABC$ and the centroids of triangles $\triangle ABC$ and $\triangle NPQ$ coincide, prove that $\triangle ABC$ is equilateral.

Let $a,b,c,d$ be pairwise distinct positive integers such that $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is an integer. Prove that $a+b+c+d$ is [b]not[/b] a prime number.

The checkered plane (infinitely large house paper) is given. For which pairs (a,, b) one can color each of the squares with one of $a \cdot b$ colors, so that each rectangle of size $ a \times b$ or $b \times a$, placed appropriately in the checkered plane, always contains a unit square with each color ?

What is the smallest value of $k$ for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*} holds for any $8$ real numbers $a,b,c,d,x,y,z,t$? Edit: Fixed a mistake! Thanks @below.

Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. The ratio of the area of the first circle to the sum of areas of all other circles in the sequence, is $\textbf{(A) }(4+3\sqrt{2}):4\qquad\textbf{(B) }9\sqrt{2}:2\qquad\textbf{(C) }(16+12\sqrt{2}):1\qquad$ $\textbf{(D) }(2+2\sqrt{2}):1\qquad \textbf{(E) }3+2\sqrt{2}):1$

Consider the equation $n^2+(n+1)^4=5(n+2)^3$ (a) Show that any integer of the form $3m+1$ or $3m+2$ can not be a solution of this equation. (b) Does the equation have a solution in positive integers?

If the three points of contact of a circle inscribed in a triangle are joined, the angles of the resulting triangle: $ \textbf{(A)}\ \text{are always equal to }60^\circ \\ \textbf{(B)}\ \text{are always one obtuse angle and two unequal acute angles} \\ \textbf{(C)}\ \text{are always one obtuse angle and two equal acute angles} \\ \textbf{(D)}\ \text{are always acute angles} \\ \textbf{(E)}\ \text{are always unequal to each other}$

Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?

Let $p,q$ be two-digit number neither of which are divisible by $10$. Let $r$ be the four-digit number by putting the digits of $p$ followed by the digits of $q$ (in order). As $p,q$ very, a computer prints $r$ on the screen if $\gcd(p,q) = 1$ and $p+q$ divides $r$. Suppose that the largest number that is printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).

$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$

Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Determine all positive integers $n$ for which there exists an infinite subset $A$ of the set $\mathbb{N}$ of positive integers such that for all pairwise distinct $a_1,\ldots , a_n \in A$ the numbers $a_1+\ldots +a_n$ and $a_1a_2\ldots a_n$ are coprime.

What is the units digit of $ 13^{2003}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.

Find the positive integer $N$ for which there exist reals $\alpha, \beta, \gamma, \theta$ which obey \begin{align*} 0.1 &= \sin \gamma \cos \theta \sin \alpha, \\ 0.2 &= \sin \gamma \sin \theta \cos \alpha, \\ 0.3 &= \cos \gamma \cos \theta \sin \beta, \\ 0.4 &= \cos \gamma \sin \theta \cos \beta, \\ 0.5 &\ge \left\lvert N-100 \cos2\theta \right\rvert. \end{align*}[i]Proposed by Evan Chen[/i]