The perimeter of triangle $APM$ is $152,$ and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}.$ Given that $OP=m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Find the least positive integer $ a$ such that $ 2001$ divides $ 55^n\plus{}a \cdot 32^n$ for some odd $ n$.

Let $f(z) = az^4+ bz^3+ cz^2+ dz + e = a(z -r_1)(z -r_2)(z -r_3)(z -r_4)$ where $a, b, c, d, e$ are integers, $a \not= 0$. Show that if $r_1 + r_2$ is a rational number, and if $r_1 + r_2 \neq r_3 + r_4$, then $r_1r_2$ is a rational number.

[1] Tokyo Univ. of Science: $\int \frac{\ln x}{(x+1)^2}dx$ [2] Saitama Univ.: $\int \frac{5}{3\sin x+4\cos x}dx$ [3] Yokohama City Univ.: $\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx$ [4] Daido Institute of Technology: $\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx$ [5] Gunma Univ.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

Let $n$ be the least positive integer for which $149^n - 2^n$ is divisible by $3^3 \cdot 5^5 \cdot 7^7$. Find the number of positive divisors of $n$.

Let $ ABC$ be a triangle, let $ E, F$ be the tangency points of the incircle $ \Gamma(I)$ to the sides $ AC$, respectively $ AB$, and let $ M$ be the midpoint of the side $ BC$. Let $ N \equal{} AM \cap EF$, let $ \gamma(M)$ be the circle of diameter $ BC$, and let $ X, Y$ be the other (than $ B, C$) intersection points of $ BI$, respectively $ CI$, with $ \gamma$. Prove that \[ \frac {NX} {NY} \equal{} \frac {AC} {AB}. \] [i]Cosmin Pohoata[/i]

Find $\lim_{t\rightarrow 1^-} (1-t) \sum_{n=1}^{\infty}\frac{t^n}{1+t^n}$.

A fisherman has caught a number of fish. The three heaviest together make up $35\%$ of the total weight of the catch. He sells them. After that, the three lightest make up together $5/13$ of the weight of the rest. How many fish did he catch?

On the sides $AB$,$BC$, and $CA$ of $\triangle ABC$ are chosen points $C_1$, $A_1$, and $B_1$ respectively, in such way that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $X$. If $\angle A_1C_1B = \angle B_1C_1A$, prove that $CC_1$ is perpendicular to $AB$.

Leticia has a $9\times 9$ board. She says that two squares are [i]friends[/i] is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules: - If the square is green, write the number of red friends plus twice the number of blue friends. - If the square is red, write the number of blue friends plus twice the number of green friends. - If the square is blue, write the number of green friends plus twice the number of red friends. Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

Knowing that $23$ October $1948$ was a Saturday, which is more frequent for New Year’s Day, Sunday or Monday?

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $R$ be the radius of circumcircle of $\triangle ABC$. Let $A',B',C'$ be the points on $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ respectively such that $AH.AA'=R^2,BH.BB'=R^2,CH.CC'=R^2$. Prove that $O$ is incenter of $\triangle A'B'C'$.

Boris is driving on a remote highway. His car’s odometer reads $24942\text{ km}$, which Boris notices is a palindromic number, meaning it is not changed when it is reversed. “Hm,” he thinks, “it should be a long time before I see that again.” But it takes only $1$ hour for the odometer to once again show a palindromic number! How fast is Boris driving in $\text{km/h}$?

Let $ABC$ be an equilateral triangle. The points $M$ and $K$ are located in different half-planes with respect to line $BC$, so that the point $M \in (AB)$ ¸and the triangle $MKC$ is equilateral. Prove that the lines $AC$ and $BK$ are parallel.

A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$. For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$, then there are at least $3$ people that got the right answer of a certain problem, among the $n$ people. Find the least possible $n$.

Let $S$ be the set of natural numbers whose digits are different and belong to the set $\{1, 3, 5, 7\}$. Calculate the sum of the elements of $S$.

Let $m$ be a positive integer and $a$ a positive real number. Find the greatest value of $a^{m_1}+a^{m_2}+\ldots+a^{m_p}$ where $m_1+m_2+\ldots+m_p=m, m_i\in\mathbb{N},i=1,2,\ldots,p;$ $1\leq p\leq m, p\in\mathbb{N}$.

Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions: \[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\] Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$

There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.

Clarabelle wants to travel from $(0,0)$ to $(6,2)$ in the coordinate plane. She is able to move in one-unit steps up, down, or right, must stay between $y=0$ and $y=2$ (inclusive), and is not allowed to visit the same point twice. How many paths can she take? [i]Proposed by Connor Gordon[/i]

The shape of a dog kennel from above is an equilateral triangle with side length $1$ m and its corners in points $A, B$ and $C$, as shown in the picture. The chain of the dog is of length $6$ m and its end is fixed to the corner in point $A$. The dog himself is in point $K$ in a way that the chain is tight and points $K, A$ and $B$ are on the same straight line. The dog starts to move clockwise around the kennel, holding the chain tight all the time. How long is the walk of the dog until the moment when the chain is tied round the kennel at full? [img]https://cdn.artofproblemsolving.com/attachments/9/5/616f8adfe66e2eb60f1a6c3f26e652c45f3e27.png[/img]

Let $ f$ be a function defined on the set of non-negative integers and taking values in the same set. Given that (a) $ \displaystyle x \minus{} f(x) \equal{} 19\left[\frac{x}{19}\right] \minus{} 90\left[\frac{f(x)}{90}\right]$ for all non-negative integers $ x$; (b) $ 1900 < f(1990) < 2000$, find the possible values that $ f(1990)$ can take. (Notation : here $ [z]$ refers to largest integer that is $ \leq z$, e.g. $ [3.1415] \equal{} 3$).

Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that \[ (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,\] then either all these numbers are nonnegative or all these numbers are nonpositive.