Found problems: 85335
2022 Indonesia Regional, 2
(a) Determine a natural number $n$ such that $n(n+2022)+2$ is a perfect square.
[hide=Spoiler]In case you didn't realize, $n=1$ works lol[/hide]
(b) Determine all natural numbers $a$ such that for every natural number $n$, the number $n(n+a)+2$ is never a perfect square.
2023 Korea National Olympiad, 4
Pentagon $ABCDE$ is inscribed in circle $\Omega$. Line $AD$ meets $CE$ at $F$, and $P (\neq E, F)$ is a point on segment $EF$. The circumcircle of triangle $AFP$ meets $\Omega$ at $Q(\neq A)$ and $AC$ at $R(\neq A)$. Line $AD$ meets $BQ$ at $S$, and the circumcircle of triangle $DES$ meets line $BQ, BD$ at $T(\neq S), U(\neq D)$, respectively. Prove that if $F, P, T, S$ are concyclic, then $P, T, R, U$ are concyclic.
2012 Today's Calculation Of Integral, 790
Define a parabola $C$ by $y=x^2+1$ on the coordinate plane. Let $s,\ t$ be real numbers with $t<0$. Denote by $l_1,\ l_2$ the tangent lines drawn from the point $(s,\ t)$ to the parabola $C$.
(1) Find the equations of the tangents $l_1,\ l_2$.
(2) Let $a$ be positive real number. Find the pairs of $(s,\ t)$ such that the area of the region enclosed by $C,\ l_1,\ l_2$ is $a$.
2023 Malaysian IMO Training Camp, 3
A sequence of reals $a_1, a_2, \cdots$ satisfies for all $m>1$, $$a_{m+1}a_{m-1}=a_m^2-a_1^2$$ Prove that for all $m>n>1$, the sequence satisfies the equation $$a_{m+n}a_{m-n}=a_m^2-a_n^2$$
[i]Proposed by Ivan Chan Kai Chin[/i]
2002 AMC 8, 3
What is the smallest possible average of four distinct positive even integers?
$ \text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 7 $
1997 Akdeniz University MO, 2
If $x$ and $y$ are positive reals, prove that
$$x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}} \geq x^2+y^2$$
2016 NIMO Problems, 7
Let $(a_1,a_2,\ldots, a_{13})$ be a permutation of $(1, 2, \ldots, 13)$. Ayvak takes this permutation and makes a series of [i]moves[/i], each of which consists of choosing an integer $i$ from $1$ to $12$, inclusive, and swapping the positions of $a_i$ and $a_{i+1}$. Define the [i]weight[/i] of a permutation to be the minimum number of moves Ayvak needs to turn it into $(1, 2, \ldots, 13)$.
The arithmetic mean of the weights of all permutations $(a_1, \ldots, a_{13})$ of $(1, 2, \ldots, 13)$ for which $a_5 = 9$ is $\frac{m}{n}$, for coprime positive integers $m$ and $n$. Find $100m+n$.
[i]Proposed by Alex Gu[/i]
1988 IMO Longlists, 16
If $ n$ runs through all the positive integers, then $ f(n) \equal{} \left[n \plus{} \sqrt {\frac {n}{3}} \plus{} \frac {1}{2} \right]$ runs through all positive integers skipping the terms of the sequence $ a_n \equal{} 3 \cdot n^2 \minus{} 2 \cdot n.$
2007 National Olympiad First Round, 17
Let $K$ be the point of intersection of $AB$ and the line touching the circumcircle of $\triangle ABC$ at $C$ where $m(\widehat {A}) > m(\widehat {B})$. Let $L$ be a point on $[BC]$ such that $m(\widehat{ALB})=m(\widehat{CAK})$, $5|LC|=4|BL|$, and $|KC|=12$. What is $|AK|$?
$
\textbf{(A)}\ 4\sqrt 2
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 9
\qquad\textbf{(E)}\ \text{None of the above}
$
2013 USAMO, 4
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]
1958 AMC 12/AHSME, 6
The arithmetic mean between $ \frac {x \plus{} a}{x}$ and $ \frac {x \minus{} a}{x}$, when $ x \not \equal{} 0$, is:
$ \textbf{(A)}\ {2}\text{, if }{a \not \equal{} 0}\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ {1}\text{, only if }{a \equal{} 0}\qquad \textbf{(D)}\ \frac {a}{x}\qquad \textbf{(E)}\ x$
2009 Today's Calculation Of Integral, 440
For $ a>1$, find $ \lim_{n\to\infty} \int_0^a \frac{e^x}{1\plus{}x^n}dx.$
2012 AMC 10, 2
A square with side length $8$ is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
$ \textbf{(A)}\ 2\text{ by }4
\qquad\textbf{(B)}\ 2\text{ by }6
\qquad\textbf{(C)}\ 2\text{ by }8
\qquad\textbf{(D)}\ 4\text{ by }4
\qquad\textbf{(E)}\ 4\text{ by }8
$
1966 IMO Longlists, 41
Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?
2010 Princeton University Math Competition, 2
On rectangular coordinates, point $A = (1,2)$, $B = (3,4)$. $P = (a, 0)$ is on $x$-axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$.
2004 Czech-Polish-Slovak Match, 4
Solve in real numbers the system of equations: \begin{align*}
\frac{1}{xy}&=\frac{x}{z}+1 \\
\frac{1}{yz}&=\frac{y}{x}+1 \\
\frac{1}{zx}&=\frac{z}{y}+1 \\
\end{align*}
2023 Brazil Team Selection Test, 3
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$.
(For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
2020 Stanford Mathematics Tournament, 1
Pentagon $ABCDE$ has $AB = BC = CD = DE$, $\angle ABC = \angle BCD = 108^o$, and $\angle CDE = 168^o$. Find the measure of angle $\angle BEA$ in degrees.
PEN A Problems, 74
Find an integer $n$, where $100 \leq n \leq 1997$, such that \[\frac{2^{n}+2}{n}\] is also an integer.
2010 Germany Team Selection Test, 2
Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that
\[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\]
[i]Proposed by Dzianis Pirshtuk, Belarus[/i]
1989 Tournament Of Towns, (212) 6
(a) Prove that if 3n stars are placed in $3n$ cells of a $2n \times 2n$ array, then it is possible to remove $n$ rows and $n$ columns in such away that all stars will be removed .
(b) Prove that it is possible to place $3n + 1$ stars in the cells of a $2n \times 2n$ array in such a way that after removing any $n$ rows and $n$ columns at least one star remains.
(K . P. Kohas, Leningrad)
2020 China National Olympiad, 1
Let $a_1,a_2,\cdots,a_{41}\in\mathbb{R},$ such that $a_{41}=a_1, \sum_{i=1}^{40}a_i=0,$ and for any $i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.$ Determine the greatest possible value of
$(1)a_{10}+a_{20}+a_{30}+a_{40};$
$(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.$
1998 Belarus Team Selection Test, 3
For any given triangle $A_0B_0C_0$ consider a sequence of triangles constructed as follows:
a new triangle $A_1B_1C_1$ (if any) has its sides (in cm) that equal to the angles of $A_0B_0C_0$ (in radians). Then for $\vartriangle A_1B_1C_1$ consider a new triangle $A_2B_2C_2$ (if any) constructed in the similar พay, i.e., $\vartriangle A_2B_2C_2$ has its sides (in cm) that equal to the angles of $A_1B_1C_1$ (in radians), and so on.
Determine for which initial triangles $A_0B_0C_0$ the sequence never terminates.
2009 China National Olympiad, 3
Given two integers $ m,n$ satisfying $ 4 < m < n.$ Let $ A_{1}A_{2}\cdots A_{2n \plus{} 1}$ be a regular $ 2n\plus{}1$ polygon. Denote by $ P$ the set of its vertices. Find the number of convex $ m$ polygon whose vertices belongs to $ P$ and exactly has two acute angles.
2012 Kosovo Team Selection Test, 2
Find all three digit numbers, for which the sum of squares of each digit is $90$ .