Compute the $100$th smallest positive multiple of $7$ whose digits in base $10$ are all strictly less than $3.$

Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.

Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that $\bullet$ all of his happy snakes can add $\bullet$ none of his purple snakes can subtract, and $\bullet$ all of his snakes that can’t subtract also can’t add Which of these conclusions can be drawn about Tom’s snakes? $\textbf{(A)}$ Purple snakes can add. $\textbf{(B)}$ Purple snakes are happy. $\textbf{(C)}$ Snakes that can add are purple. $\textbf{(D)}$ Happy snakes are not purple. $\textbf{(E)}$ Happy snakes can't subtract.

A function $f(n)$ is defined on the set of positive integers is said to be multiplicative if $f(mn)=f(m)f(n)$ whenever $m$ and $n$ have no common factors greater than $1$. Are the following functions multiplicative? Justify your answer. (a) $g(n)=5^k$ where $k$ is the number of distinct primes which divide $n$. (b) $h(n)=\begin{cases} 0 & \text{if} \ n \ \text{is divisible by} \ k^2 \ \text{for some integer} \ k>1 \\ 1 & \text{otherwise} \end{cases}$

The diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$ intersect at a point O. It is known that $\angle BAD = 60^o$ and $AO = 3OC$. Prove that the sum of some two sides of a quadrilateral is equal to the sum of the other two sides.

Octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle where $A_1A_2=A_2A_3=A_3A_4=A_6A_7=13$ and $A_4A_5=A_5A_6=A_7A_8=A_8A_1=5. $ The sum of all possible areas of $A_1A_2A_3A_4A_5A_6A_7A_8$ can be expressed as $a+b\sqrt{c}$ where $\gcd{a,b}=1$ and $c$ is squarefree. Find $abc.$ [asy] label("$A_1$",(5,0),E); label("$A_2$",(2.92, -4.05),SE); label("$A_3$",(-2.92,-4.05),SW); label("$A_4$",(-5,0),W); label("$A_5$",(-4.5,2.179),NW); label("$A_6$",(-3,4), NW); label("$A_7$",(3,4), NE); label("$A_8$",(4.5,2.179),NE); draw((5,0)--(2.9289,-4.05235)); draw((2.92898,-4.05325)--(-2.92,-4.05)); draw((-2.92,-4.05)--(-5,0)); draw((-5,0)--(-4.5, 2.179)); draw((-4.5, 2.179)--(-3,4)); draw((-3,4)--(3,4)); draw((3,4)--(4.5,2.179)); draw((4.5,2.179)--(5,0)); dot((0,0)); draw(circle((0,0),5)); [/asy]

$(a)$There are two sequences of numbers, with $2003$ consecutive integers each, and a table of $2$ rows and $2003$ columns $\begin{array}{|c|c|c|c|c|c|} \hline\ \ &\ &\ &\cdots\cdots&\ &\ \\ \hline \ &\ &\ &\cdots\cdots&\ &\ \\ \hline \end{array}$ Is it always possible to arrange the numbers in the first sequence in the first row and the second sequence in the second row, such that the sequence obtained of the $2003$ column-wise sums form a new sequence of $2003$ consecutive integers? $(b)$ What if $2003$ is replaced with $2004$?

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.

Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, where $p$ and $q$ are relatively prime integers, find $p+q$.

Nadia bought a compass and after opening its package realized that the length of the needle leg is $10$ centimeters whereas the length of the pencil leg is $16$ centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least $30$ degrees but the needle leg could be positioned at any angle with respect to the paper. Let $n$ be the difference between the radii of the largest and the smallest circles that Nadia can draw with this compass in centimeters. Which of the following options is closest to $n$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 20$

Find all functions $f:\mathbb R\to\mathbb R$ such that for reals $x,y$, $$f(xf(y)+y)=yf(x)+f(y).$$

Line $l$ that passes right focal point of hyperbola $x^2-\frac{y^2}{2}=1$ intersects the hyperbola at $A,B$. The number of line $l$ that $|AB|=\lambda$ is 3, then $\lambda=$________.

Evaluate $\int_{ 0 }^{ \infty } ( 1 + x^2 )^{-( m + 1 )} \mathrm{d}x$ where $m \in \mathbb{N} $

There are $2024$ points on a circle. A purple elephant labels the points $P_1,P_2,\ldots,P_{2024}$ in some order, and walks along the points from $P_1$ to $P_{2024}$ in this order, while laying some eggs. To ensure the elephant does not step on the eggs it laid, the chords $P_1P_2, P_2P_3, \ldots, P_{2023}P_{2024}$ must not intersect each other except possibly at their endpoints. How many labellings are there? (Note: Two labellings are the same if one is a rotation of the other.) [i](Proposed by Ho Janson)[/i]

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.

Find all positive integers $n$ for which the equation $$x^3 + y^3 = n! + 4$$ has solutions in integers.

Prove:If $ a\equal{}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{n}^{\alpha_n}$ is a perfect number, then $ 2<\prod_{i\equal{}1}^n\frac{p_i}{p_i\minus{}1}<4$ ; if moreover, $ a$ is odd, then the upper bound $ 4$ may be reduced to $ 2\sqrt[3]{2}$.

The figure $ABCDE$ is a convex pentagon. Find the sum $\angle DAC + \angle EBD +\angle ACE +\angle BDA + \angle CEB$?

[b]p1.[/b] How many lines of symmetry does a square have? [b]p2.[/b] Compute$ 1/2 + 1/6 + 1/12 + 1/4$. [b]p3.[/b] What is the maximum possible area of a rectangle with integer side lengths and perimeter $8$? [b]p4.[/b] Given that $1$ printer weighs $400000$ pennies, and $80$ pennies weighs $2$ books, what is the weight of a printer expressed in books? [b]p5.[/b] Given that two sides of a triangle are $28$ and $3$ and all three sides are integers, what is the sum of the possible lengths of the remaining side? [b]p6.[/b] What is half the sum of all positive integers between $1$ and $15$, inclusive, that have an even number of positive divisors? [b]p7.[/b] Austin the Snowman has a very big brain. His head has radius $3$, and the volume of his torso is one third of his head, and the volume of his legs combined is one third of his torso. If Austin's total volume is $a\pi$ where $a$ is an integer, what is $a$? [b]p8.[/b] Neethine the Kiwi says that she is the eye of the tiger, a fighter, and that everyone is gonna hear her roar. She is standing at point $(3, 3)$. Neeton the Cat is standing at $(11,18)$, the farthest he can stand from Neethine such that he can still hear her roar. Let the total area of the region that Neeton can stand in where he can hear Neethine's roar be $a\pi$ where $a$ is an integer. What is $a$? [b]p9.[/b] Consider $2018$ identical kiwis. These are to be divided between $5$ people, such that the first person gets $a_1$ kiwis, the second gets $a_2$ kiwis, and so forth, with $a_1 \le a_2 \le a_3 \le a_4 \le a_5$. How many tuples $(a_1, a_2, a_3, a_4, a_5)$ can be chosen such that they form an arithmetic sequence? [b]p10.[/b] On the standard $12$ hour clock, each number from $1$ to $12$ is replaced by the sum of its divisors. On this new clock, what is the number of degrees in the measure of the non-reflex angle between the hands of the clock at the time when the hour hand is between $7$ and $6$ while the minute hand is pointing at $15$? [b]p11.[/b] In equiangular hexagon $ABCDEF$, $AB = 7$, $BC = 3$, $CD = 8$, and $DE = 5$. The area of the hexagon is in the form $\frac{a\sqrt{b}}{c}$ with $b$ square free and $a$ and $c$ relatively prime. Find $a+b+c$ where $a, b,$ and $c$ are integers. [b]p12.[/b] Let $\frac{p}{q} = \frac15 + \frac{2}{5^2} + \frac{3}{5^3} + ...$ . Find $p + q$, where $p$ and $q$ are relatively prime positive integers. [b]p13.[/b] Two circles $F$ and $G$ with radius $10$ and $4$ respectively are externally tangent. A square $ABMC$ is inscribed in circle $F$ and equilateral triangle $MOP$ is inscribed in circle $G$ (they share vertex $M$). If the area of pentagon $ABOPC$ is equal to $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers $c$ is square free, then find $a + b + c$. [b]p14.[/b] Consider the polynomial $P(x) = x^3 + 3x^2 + ax + 8$. Find the sum of all integer $a$ such that the sum of the squares of the roots of $P(x)$ divides the sum of the coecients of $P(x)$. [b]p15.[/b] Nithin and Antonio play a number game. At the beginning of the game, Nithin picks a prime $p$ that is less than $100$. Antonio then tries to find an integer $n$ such that $n^6 + 2n^5 + 2n^4 + n^3 + (n^2 + n + 1)^2$ is a multiple of $p$. If Antonio can find such a number n, then he wins, otherwise, he loses. Nithin doesn't know what he is doing, and he always picks his prime randomly while Antonio always plays optimally. The probability of Antonio winning is $a/b$ where $a$ and $b$ are relatively prime positive integers. Find$a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be the set of all positive integers $n$ satisfying the following two conditions: $\bullet$ $n$ is relatively prime to all positive integers less than or equal to $\frac{n}{6}$ $\bullet$ $2^n \equiv 4$ mod $n$ What is the sum of all numbers in $S$?

Let $A$, $B$, and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $<PBC=60$. Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$, and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$. Let $X$ be the point of intersection of the lines $BQ$ and $PC$, and let $Y$ be the point of intersection of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.

Consider the succession of integers $\{f(n)\}_{n=1}^{\infty}$ defined as: $\bullet$ $f(1) = 1$. $\bullet$ $f(n) = f(n/2)$ if $n$ is even. $\bullet$ If $n > 1$ odd and $f(n-1)$ odd, then $f(n) = f(n-1)-1$. $\bullet$ If $n > 1$ odd and $f(n-1)$ even, then $f(n) = f(n-1)+1$. a) Compute $f(2^{2020}-1)$. b) Prove that $\{f(n)\}_{n=1}^{\infty}$ is not periodical, that is, there do not exist positive integers $t$ and $n_0$ such that $f(n+t) = f(n)$ for all $n \geq n_0$.

Jon and Steve ride their bicycles on a path that parallels two side-by-side train tracks running in the east/west direction. Jon rides east at 20 miles per hour, and Steve rides west at 20 miles per hour. Two trains of equal length traveling in opposite directions at constant but different speeds, each pass the two riders. Each train takes exactly 1 minute to go past Jon. The westbound train takes 10 times as long as the eastbound train to go past Steve. The length of each train is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.