Found problems: 85335
Let $\mathbb{N}$ be the set of all positive integers. The function $f: \mathbb{N} \to \mathbb{N}$ satisfies
$f(1)=3, f(mn)=f(m)f(n)-f(m+n)+2$ for all $m,n \in \mathbb{N}$.
Prove that $f$ does not exist.
Comment: The original problem asked for the value of $f(2006)$, which obviously does not exist when $f$ does not. This was probably a mistake by the Olympiad committee. Hence the modified problem.
Let $a,\ b$ be positive constants.
Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]
A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is
[asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((3,0)--(3,9), dashed);
draw((6,0)--(6,9), dashed);[/asy]
$\text{(A)} \ 24 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 64 \qquad \text{(D)} \ 81 \qquad \text{(E)} \ 96$
Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.
One of the two Purple Comet! question writers is an adult whose age is the same as the last two digits of the year he was born. His birthday is in August. What is his age today?
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
If a shrub grows at the rate of $6$ inches per $5$ days, how many feet would it grow in a non-leap year?
$\textbf{(A) } 2\dfrac12\text{ ft}\qquad\textbf{(B) } 36\dfrac12\text{ ft}\qquad\textbf{(C) } 182\dfrac12\text{ ft}\qquad\textbf{(D) } 73\text{ ft}\qquad\textbf{(E) } 365\text{ ft}$
Let $ABCD$ be a random tetrahedron. Let $E$ and $F$ be the midpoints of segments $AB$ and $CD$, respectively. If the angle $a$ is between $AD$ and $BC$, determine $cos a$ in terms of $EF, AD$ and $BC$.
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that
$(\text{i})$ $x$ and $y$ are relatively prime;
$(\text{ii})$ $x$ divides $y^2+m;$
$(\text{iii})$ $y$ divides $x^2+m.$
Let ABC with orthocenter $H$ and circumcenter $O$ be an acute scalene triangle satisfying
$AB = AM$ where $M$ is the midpoint of $BC$. Suppose $Q$ and $K$ are points on $(ABC)$ distinct from
A satisfying $\angle AQH = 90$
and $\angle BAK = \angle CAM$. Let $N$ be the midpoint of $AH$.
• Let $I$ be the intersection of $B\text{-midline}$ and $A\text{-altitude}$ Prove that $IN = IO$.
• Prove that there is point $P$ on the symmedian lying on circle with center $B$ and radius $BM$
such that $(APN)$ is tangent to $AB$.
[i]Proposed by Krutarth Shah[/i]
[b]Q.[/b] Let $ABC$ be a triangle, where $L_A, L_B, L_C$ denote the internal angle bisectors of $\angle BAC, \angle ABC, \angle ACB$ respectively and $\ell_A, \ell_B, \ell_C$, the altitudes from the corresponding vertices. Suppose $ L_A\cap \overline{BC} = \{A_1\}$, $\ell_A \cap \overline{BC} = \{A_2\}$ and the circumcircle of $\triangle AA_1A_2$ meets $AB$ and $AC$ at $S$ and $T$ respectively. If $\overline{ST} \cap \overline{BC} = \{A'\}$, prove that $A',B',C'$ are collinear, where $B'$ and $C'$ are defined in a similar manner.
[i]Proposed by Functional_equation[/i]
([b]8[/b]) Evaluate the integral $ \int_0^1\ln x \ln(1\minus{}x)\ dx$.
$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.
Thirty numbers are arranged on a circle in such a way that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is $2000$, determine the numbers.
Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that
[list]
[*]$m = 1$ and $l = 2k$; or
[*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$.
[/list]
There are relatively prime positive integers $m$ and $n$ so that
\[\frac{m}{n}=\log_4\left(32^{\log_927}\right).\]
Find $m+n$.
Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.
Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed the digit $b$ is registered in the system, and when $\underline{b}$ is pressed the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.
[i]Proposed by Andrew Wen[/i]
About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.
Consider a polynomial $P(x) = ax^2 + bx + c$ with $a > 0$ that has two real roots $x_1, x_2$. Prove that the absolute values of both roots are less than or equal to $1$ if and only if $a + b + c \ge 0, a -b + c \ge 0$, and $a - c \ge 0$.
Given that the expected amount of $1$s in a randomly selected $2021$-digit number is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[i]Proposed by Hannah Shen[/i]
Consider a regular $n$-gon with $n>3$, call a line [i]acceptable[/i] if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons.
(a) Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having $3$ or $4$ sides.
(b) Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have $3$ or $4$ sides.
How many $7$-digit numbers with distinct digits can be made that are divisible by $3$?
Let $m,n$ be natural numbers and let $d = gcd(m,n)$. Let $x = 2^{m} -1$ and $y= 2^n +1$
(a) If $\frac{m}{d}$ is odd, prove that $gcd(x,y) = 1$
(b) If $\frac{m}{d}$ is even, Find $gcd(x,y)$
In a convex hexagon $ABCDEF$ triangles $ABC , CDE , EFA$ are similar. Find conditions on these triangles under which triangle $ACE$ is equilateral if and only if so is $BDF.$