Found problems: 85335
1997 AIME Problems, 6
Point $B$ is in the exterior of the regular $n$-sided polygon $A_1A_2\cdots A_n,$ and $A_1A_2B$ is an equilateral triangle. What is the largest value of $n$ for which $A_n, A_1,$ and $B$ are consecutive vertices of a regular polygon?
2004 National High School Mathematics League, 5
For a 3-digit-number $n=\overline{abc}$, if $a,b,c$ can be three sides of an isosceles triangle (regular triangle included), then the number of such numbers is
$\text{(A)}45\qquad\text{(B)}81\qquad\text{(C)}165\qquad\text{(D)}216$
2009 National Olympiad First Round, 8
$ S \equal{} \{1,2,\dots,n\}$ is divided into two subsets. How the set is divided, if there exist two elements whose sum is a perfect square, then $ n$ is at least ?
$\textbf{(A)}\ 13 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ 17$
II Soros Olympiad 1995 - 96 (Russia), 11.8
The following is known about the quadrilateral $ABCD$: triangles $ABC$ and $CDA$ are equal in area, the area of triangle $BCD$ is $k$ times greater than the area of triangle $DAB$, the bisectors of angles $ABC$ and $CDA$ intersect on the diagonal $AC$, straight lines $AC$ and $BD$ are not perpendicular. Find the ratio $AC/BD$.
1981 AMC 12/AHSME, 18
The number of real solutions to the equation \[ \frac{x}{100} = \sin x \] is
$\text{(A)} \ 61 \qquad \text{(B)} \ 62 \qquad \text{(C)} \ 63 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$
2011 Tuymaada Olympiad, 2
In a word of more than $10$ letters, any two consecutive letters are different. Prove that one can change places of two consecutive letters so that the resulting word is not [i]periodic[/i], that is, cannot be divided into equal subwords.
2009 Indonesia TST, 3
In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.
1961 IMO Shortlist, 3
Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.
2003 Vietnam Team Selection Test, 3
Let $n$ be a positive integer. Prove that the number $2^n + 1$ has no prime divisor of the form $8 \cdot k - 1$, where $k$ is a positive integer.
1975 Vietnam National Olympiad, 2
Solve this equation
$\frac{y^{3}+m^{3}}{\left ( y+m \right )^{3}}+\frac{y^{3}+n^{3}}{\left ( y+n \right )^{3}}+\frac{y^{3}+p^{3}}{\left ( y+p \right )^{3}}-\frac{3}{2}+\frac{3}{2}.\frac{y-m}{y+m}.\frac{y-n}{y+n}.\frac{y-p}{y+p}=0$
2011 Purple Comet Problems, 2
The diagram below shows a $12$-sided figure made up of three congruent squares. The figure has total perimeter $60$. Find its area.
[asy]
size(150);
defaultpen(linewidth(0.8));
path square=unitsquare;
draw(rotate(360-135)*square^^rotate(345)*square^^rotate(105)*square);
[/asy]
2008 Polish MO Finals, 2
A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition:
\[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\]
Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds:
\[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]
2008 Brazil Team Selection Test, 3
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
2006 Junior Balkan Team Selection Tests - Romania, 3
For any positive integer $n$ let $s(n)$ be the sum of its digits in decimal representation. Find all numbers $n$ for which $s(n)$ is the largest proper divisor of $n$.
1981 National High School Mathematics League, 7
The equation $x|x|+px+q=0$ is given. Which of the following is not true?
$\text{(A)}$It has at most three real roots.
$\text{(B)}$It has at least one real root.
$\text{(C)}$Only if $p^2-4q\geq0 $,it has real roots.
$\text{(D)}$If $p<0$ and $q>0$, it has three real roots.
2019 AMC 12/AHSME, 20
Points $A(6,13)$ and $B(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$?
$\textbf{(A) }\frac{83\pi}{8}\qquad\textbf{(B) }\frac{21\pi}{2}\qquad\textbf{(C) }
\frac{85\pi}{8}\qquad\textbf{(D) }\frac{43\pi}{4}\qquad\textbf{(E) }\frac{87\pi}{8}$
2021 Silk Road, 2
For every positive integer $m$ prove the inquality
$|\{\sqrt{m}\} - \frac{1}{2}| \geq \frac{1}{8(\sqrt m+1)} $
(The integer part $[x]$ of the number $x$ is the largest integer not exceeding $x$. The fractional part of the number $x$ is a number $\{x\}$ such that $[x]+\{x\}=x$.)
A. Golovanov
2016 Thailand TSTST, 1
Find all polynomials $P\in\mathbb{Z}[x]$ such that $$|P(x)-x|\leq x^2+1$$ for all real numbers $x$.
2000 IMO Shortlist, 3
Find all pairs of functions $ f : \mathbb R \to \mathbb R$, $g : \mathbb R \to \mathbb R$ such that \[f \left( x + g(y) \right) = xf(y) - y f(x) + g(x) \quad\text{for all } x, y\in\mathbb{R}.\]
2017 CMIMC Number Theory, 10
For each positive integer $n$, define \[g(n) = \gcd\left\{0! n!, 1! (n-1)!, 2 (n-2)!, \ldots, k!(n-k)!, \ldots, n! 0!\right\}.\] Find the sum of all $n \leq 25$ for which $g(n) = g(n+1)$.
1995 Vietnam National Olympiad, 1
Let a tetrahedron $ ABCD$ and $ A',B',C',D'$ be the circumcenters of triangles $ BCD,CDA,DAB,ABC$ respectively. Denote planes $ (P_A),(P_B),(P_C),(P_D)$ be the planes which pass through $ A,B,C,D$ and perpendicular to $ C'D',D'A',A'B',B'C'$ respectively. Prove that these planes have a common point called $ I.$ If $ P$ is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
2021 South East Mathematical Olympiad, 4
For positive integer $k,$ we say that it is a [i]Taurus integer[/i] if we can delete one element from the set $M_k=\{1,2,\cdots,k\},$ such that the sum of remaining $k-1$ elements is a positive perfect square. For example, $7$ is a Taurus integer, because if we delete $3$ from $M_7=\{1,2,3,4,5,6,7\},$ the sum of remaining $6$ elements is $25,$ which is a positive perfect square.
$(1)$ Determine whether $2021$ is a Taurus integer.
$(2)$ For positive integer $n,$ determine the number of Taurus integers in $\{1,2,\cdots,n\}.$
1962 All-Soviet Union Olympiad, 14
Given are two sets of positive numbers with the same sum. The first set has $m$ numbers and the second $n$. Prove that you can find a set of less than $m+n$ positive numbers which can be arranged to part fill an $m \times n$ array, so that the row and column sums are the two given sets.
2022 SG Originals, Q3
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$.
[i]Proposed by DVDthe1st[/i]
2019 China Team Selection Test, 3
Find all positive integer $n$, such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for
\\a) $k=2018$
\\b) $k=2019$.