Found problems: 85335
2006 AMC 12/AHSME, 10
For how many real values of $ x$ is $ \sqrt {120 \minus{} \sqrt {x}}$ an integer?
$ \textbf{(A) } 3\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 11$
1983 Canada National Olympiad, 1
Find all positive integers $w$, $x$, $y$ and $z$ which satisfy $w! = x! + y! + z!$.
2015 China Girls Math Olympiad, 7
Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$. Prove that$$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$
1989 National High School Mathematics League, 13
$a_1,a_2,\cdots,a_n$ are positive numbers, satisfying that $a_1a_2\cdots a_n=1$.
Prove that $(2+a_1)(2+a_2)\cdots(2+a_n)\geq3^n$
1989 AMC 8, 22
The letters $\text{A}$, $\text{J}$, $\text{H}$, $\text{S}$, $\text{M}$, $\text{E}$ and the digits $1$, $9$, $8$, $9$ are "cycled" separately as follows and put together in a numbered list:
\[\begin{tabular}[t]{lccc}
& & AJHSME & 1989 \\
& & & \\
1. & & JHSMEA & 9891 \\
2. & & HSMEAJ & 8919 \\
3. & & SMEAJH & 9198 \\
& & ........ &
\end{tabular}\]
What is the number of the line on which $\text{AJHSME 1989}$ will appear for the first time?
$\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24$
Cono Sur Shortlist - geometry, 2020.G1.4
Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.
1990 IMO Longlists, 92
Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$. In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$, satisfying the following conditions:
[b][i]i)[/i][/b] Any two lines are non-parallel.
[b][i]ii)[/i][/b] Any three lines are non-concurrent.
[b][i]iii)[/i][/b] Only $A_1$ does not lies on any line $L_k$, and there are exactly $k + 1$ points $A_j$'s that lie on line $L_k$ $(k = 1, 2, \ldots, n).$
Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$
2012 Mathcenter Contest + Longlist, 6
Let $a,b,c>0$ and $abc=1$. Prove that $$\frac{a}{b^2(c+a)(a+b)}+\frac{b}{c^2(a+b)(b+c)}+\frac{c}{a^2(c+a)(a+b)}\ge \frac{3}{4}.$$
[i](Zhuge Liang)[/i]
2006 Australia National Olympiad, 3
Let $PRUS$ be a trapezium such that $\angle PSR = 2\angle QSU$ and $\angle SPU = 2 \angle UPR$. Let $Q$ and $T$ be on $PR$ and $SU$ respectively such that $SQ$ and $PU$ bisect $\angle PSR$ and $\angle SPU$ respectively. Let $PT$ meet $SQ$ at $E$. The line through $E$ parallel to $SR$ meets $PU$ in $F$ and the line through $E$ parallel to $PU$ meets $SR$ in $G$. Let $FG$ meet $PR$ and $SU$ in $K$ and $L$ respectively. Prove that $KF$ = $FG$ = $GL$.
2012 Saint Petersburg Mathematical Olympiad, 4
Some notzero reals numbers are placed around circle. For every two neighbour numbers $a,b$ it true, that $a+b$ and $\frac{1}{a}+\frac{1}{b}$ are integer. Prove that there are not more than $4$ different numbers.
2024 CMIMC Integration Bee, 5
\[\int_1^e \frac{2x^2+1}{x^3+x\log(x)}\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2024 Thailand TST, 2
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
1996 AMC 12/AHSME, 2
Each day Walter gets $\$3$ for doing his chores or $\$5$ for doing them exceptionally well. After 10 days of doing his chores daily, Walter has received a total of $\$36$. On how many days did Walter do them exceptionally well?
$\textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
2019 Hanoi Open Mathematics Competitions, 4
How many [i]connected subsequences [/i](i.e, consisting of one element or consecutive elements) of the following sequence are there: $1,2,...,100$?
[b]A.[/b] $1010$ [b]B.[/b] $2020$ [b]C.[/b] $3030$ [b]D.[/b] $4040$ [b]E.[/b] $5050$
PEN A Problems, 118
Determine the highest power of $1980$ which divides \[\frac{(1980n)!}{(n!)^{1980}}.\]
1969 Czech and Slovak Olympiad III A, 6
A sphere with unit radius is given. Furthermore, circles $k_0,k_1,\ldots,k_n\ (n\ge3)$ of the same radius $r$ are given on the sphere. The circle $k_0$ is tangent to all other circles $k_i$ and every two circles $k_i,k_{i+1}$ are tangent for $i=1,\ldots,n$ (assuming $k_{n+1}=k_1$).
a) Find relation between numbers $n,r.$
b) Determine for which $n$ the described situation can occur and compute the corresponding radius $r.$
(We say non-planar circles are tangent if they have only a single common point and their tangent lines in this point coincide.)
1997 AMC 12/AHSME, 12
If $ m$ and $ b$ are real numbers and $ mb > 0$, then the line whose equation is $ y \equal{} mx \plus{} b$ [u]cannot[/u] contain the point
$ \textbf{(A)}\ (0,1997)\qquad
\textbf{(B)}\ (0,\minus{}1997)\qquad
\textbf{(C)}\ (19,97)\qquad
\textbf{(D)}\ (19,\minus{}97)\qquad
\textbf{(E)}\ (1997,0)$
2007 National Olympiad First Round, 6
How many positive integers $n$ are there such that $n!(2n+1)$ and $221$ are relatively prime?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 13
\qquad\textbf{(E)}\ \text{None of the above}
$
2016 Romania National Olympiad, 1
The orthocenter $ H $ of a triangle $ ABC $ is distinct from its vertices and its circumcenter $ O. $ $ M,N,P $ are the circumcenters of the triangles $ HBC,HCA, $ respectively, $ HAB. $ Prove that $ AM,BN,CP $ and $ OH $ are concurrent.
2010 APMO, 4
Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.
2021 Czech-Polish-Slovak Junior Match, 3
A [i]cross [/i] is the figure composed of $6$ unit squares shown below (and any figure made of it by rotation).
[img]https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png[/img]
Find the greatest number of crosses that can be cut from a $6 \times 11$ divided sheet of paper into unit squares (in such a way that each cross consists of six such squares).
2003 Oral Moscow Geometry Olympiad, 3
Inside the segment $AC$, an arbitrary point $B$ is selected and circles with diameters $AB$ and $BC$ are constructed. Points $M$ and $L$ are chosen on the circles (in one half-plane with respect to $AC$), respectively, so that $\angle MBA = \angle LBC$. Points $K$ and $F$ are marked, respectively, on rays $BM$ and $BL$ so that $BK = BC$ and $BF = AB$. Prove that points $M, K, F$ and $L$ lie on the same circle.
2025 Kosovo National Mathematical Olympiad`, P3
Let $g_a$, $g_b$ and $g_c$ be the medians of a triangle $\triangle ABC$ erected from the vertices $A$, $B$ and $C$, respectively.
Similarly, let $g_x$, $g_y$ and $g_z$ be the medians of an another triangle $\triangle XYZ$. Show that if
$$g_a : g_b : g_c = g_x : g_y : g_z, $$
then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.
2018 District Olympiad, 4
Let $a < b$ be real numbers and let $f : (a, b) \to \mathbb{R}$ be a function such that the functions $g : (a, b) \to \mathbb{R}$, $g(x) = (x - a) f(x)$ and $h : (a, b) \to \mathbb{R}$, $h(x) = (x - b) f(x)$ are increasing. Show that the function $f$ is continuous on $(a, b)$.
2024 ELMO Shortlist, A6
Let $\mathbb R^+$ denote the set of positive real numbers. Find all functions $f:\mathbb R^+\to\mathbb R$ and $g:\mathbb R^+\to\mathbb R$ such that for all $x,y\in\mathbb R^+$, $g(x)-g(y)=(x-y)f(xy)$.
[i]Linus Tang[/i]