Found problems: 85335
2012 Sharygin Geometry Olympiad, 15
Given triangle $ABC$. Consider lines $l$ with the next property: the reflections of $l$ in the sidelines of the triangle concur. Prove that all these lines have a common point.
1955 AMC 12/AHSME, 23
In checking the petty cash a clerk counts $ q$ quarters, $ d$ dimes, $ n$ nickels, and $ c$ cents. Later he discovers that $ x$ of the nickels were counted as quarters and $ x$ of the dimes were counted as cents. To correct the total obtained the clerk must:
$ \textbf{(A)}\ \text{make no correction} \qquad
\textbf{(B)}\ \text{subtract 11 cents} \qquad
\textbf{(C)}\ \text{subtract 11}x\text{ cents} \\
\textbf{(D)}\ \text{add 11}x\text{ cents} \qquad
\textbf{(E)}\ \text{add }x\text{ cents}$
2016 IberoAmerican, 1
Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$
2022 VIASM Summer Challenge, Problem 2
Give $P(x) = {x^{2022}} + {a_{2021}}{x^{2021}} + ... + {a_1}x + 1$ is a polynomial with real coefficents.
a) Assume that $2021a_{2021}^2 - 4044{a_{2020}} < 0.$ Prove that: $P(x)$ can't have $2022$ real roots.
b) Assume that $a_1^2 + a_2^2 + ... + a_{2021}^2 \le \frac{4}{{2021}}.$ Prove that: $P(x)\ge 0$, for all $x\in \mathbb{R}.$
2016 Thailand Mathematical Olympiad, 10
A [i]Pattano coin[/i] is a coin which has a blue side and a yellow side. A positive integer not exceeding $100$ is written on each side of every coin (the sides may have different integers).
Two Pattano coins are [i]identical [/i] if the number on the blue side of both coins are equal and the number on the yellow side of both coins are equal.
Two Pattano coins are [i]pairable [/i] if the number on the blue side of both coins are equal or the number on the yellow side of both coins are equal.
Given $2559$ Pattano coins such that no two coins are identical. Show that at least one Pattano coin is pairable with at least $50$ other coins
2012 Moldova Team Selection Test, 5
Find all pairs $(m, n)$ of integers for which $$\sqrt{m^2-6}<2\sqrt{n}-m<\sqrt{m^2-2}.$$
2022 Serbia Team Selection Test, P2
Given is a triangle $ABC$ with circumcircle $\gamma$. Points $E, F$ lie on $AB, AC$ such that $BE=CF$. Let $(AEF)$ meet $\gamma$ at $D$. The perpendicular from $D$ to $EF$ meets $\gamma$ at $G$ and $AD$ meets $EF$ at $P$. If $PG$ meets $\gamma$ at $J$, prove that $\frac {JE} {JF}=\frac{AE} {AF}$.
2010 AIME Problems, 7
Define an ordered triple $ (A, B, C)$ of sets to be minimally intersecting if $ |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1$ and $ A \cap B \cap C \equal{} \emptyset$. For example, $ (\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $ N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $ \{1,2,3,4,5,6,7\}$. Find the remainder when $ N$ is divided by $ 1000$.
[b]Note[/b]: $ |S|$ represents the number of elements in the set $ S$.
1927 Eotvos Mathematical Competition, 1
Let the integers $a, b, c, d$ be relatively prime to $$m = ad - bc.$$
Prove that the pairs of integers $(x,y)$ for which $ax+by$ is a multiple of $m$ are identical with those for which $cx + dy$ is a multiple of $m$.
1992 National High School Mathematics League, 5
Points on complex plane that complex numbers $z_1,z_2$ corresponding to are $A,B$, and $|z_1|=4,4z_1^2-2z_1z_2+z_2^2=0$. $O$ is original point, then the area of $\triangle OAB$ is
$\text{(A)}8\sqrt3\qquad\text{(B)}4\sqrt3\qquad\text{(C)}6\sqrt3\qquad\text{(D)}12\sqrt3$
2006 Kurschak Competition, 1
Is there a set $S\subset\mathbb{R}^3$ of $2006$ points such that not all its points are coplanar, no three of the points are collinear, and for any $A,B\in S$ we can find points $C,D\in S$ for which $AB||CD$?
1992 Romania Team Selection Test, 3
Let $\pi$ be the set of points in a plane and $f : \pi \to \pi$ be a mapping such that the image of any triangle (as its polygonal line) is a square. Show that $f(\pi)$ is a square.
2021 CMIMC, 1
You place $n^2$ indistinguishable pieces on an $n\times n$ chessboard, where $n=2^{90}\approx 1.27\times10^{27}$. Of these pieces, $n$ of them are slightly lighter than usual, while the rest are all the same standard weight, but you are unable to discern this simply by feeling the pieces.\\
However, beneath each row and column of the chessboard, you have set up a scale, which, when turned on, will tell you [i]only[/i] whether the average weight of all the pieces on that row or column is the standard weight, or lighter than standard. On a given step, you are allowed to rearrange every piece on the chessboard, and then turn on all the scales simultaneously, and record their outputs, before turning them all off again. (Note that you can only turn on the scales if all $n^2$ pieces are placed in different squares on the board.)
Find an algorithm that, in at most $k$ steps, will always allow you to rearrange the pieces in such a way that every row and column measures lighter than standard on the final step.
An algorithm that completes in at most $k$ steps will be awarded:
1 pt for $k>10^{55}$
10 pts for $k=10^{55}$
30 pts for $k=10^{30}$
50 pts for $k=10^{28}$
65 pts for $k=10^{20}$
80 pts for $k=10^5$
90 pts for $k=2021$
100 pts for $k=500$
1972 Yugoslav Team Selection Test, Problem 4
Determine the largest integer $k(n)$ with the following properties: There exist $k(n)$ different subsets of a given set with $n$ elements such that each two of them have a non-empty intersection.
2015 IFYM, Sozopol, 3
Find all functions $f:\mathbb R^{+} \longrightarrow \mathbb R^{+}$ so that
$f(xy + f(x^y)) = x^y + xf(y)$ for all positive reals $x,y$.
Kyiv City MO Juniors 2003+ geometry, 2019.8.3
In the triangle $ABC$ it is known that $2AC=AB$ and $\angle A = 2\angle B$. In this triangle draw the angle bisector $AL$, and mark point $M$, the midpoint of the side $AB$. It turned out that $CL = ML$. Prove that $\angle B= 30^o$.
(Hilko Danilo)
2011 Bogdan Stan, 3
Find all Riemann integrable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ which have the property that, for all nonconstant and continuous functions $ g:\mathbb{R}\longrightarrow\mathbb{R}, $ and all real numbers $ a,b $ such that $ a<b, $ the following equality holds.
$$ \int_a^b \left( f\circ g \right) (x)dx=\int_a^b \left( g\circ f \right) (x)dx $$
[i]Cosmin Nițu[/i]
2009 Tournament Of Towns, 4
Several zeros and ones are written down in a row. Consider all pairs of digits (not necessarily adjacent) such that the left digit is $1$ while the right digit is $0$. Let $M$ be the number of the pairs in which $1$ and $0$ are separated by an even number of digits (possibly zero), and let $N$ be the number of the pairs in which $1$ and $0$ are separated by an odd number of digits. Prove that $M \ge N$.
1989 All Soviet Union Mathematical Olympiad, 507
Find the least possible value of $(x + y)(y + z)$ for positive reals satisfying $(x + y + z) xyz = 1$.
2010 LMT, 9
Given a triangle $XYZ$ with $\angle Y = 90^{\circ}, XY=1,$ and $XZ=2,$ mark a point $Q$ on $YZ$ such that $\frac{ZQ}{ZY}=\frac{1}{3}.$ A laser beam is shot from $Q$ perpendicular to $YZ,$ and it reflects off the sides of $XYZ$ indefinitely. How many bounces does it take for the laser beam to get back to $Q$ for the first time (not including the release from $Q$ and the return to $Q$)?
2006 Kyiv Mathematical Festival, 2
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $x,y>0$ and $xy\ge1.$ Prove that $x^3+y^3+4xy\ge x^2+y^2+x+y+2.$
Let $x,y>0$ and $xy\ge1.$ Prove that $2(x^3+y^3+xy+x+y)\ge5(x^2+y^2).$
2020 Iranian Our MO, 6
Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$
[i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]
2010 Kosovo National Mathematical Olympiad, 2
The equation is given
$x^2-(m+3)x+m+2=0$.
If $x_1$ and $x_2$ are its solutions find all $m$ such that
$\frac{x_1}{x_1+1}+\frac{x_2}{x_2+1}=\frac{13}{10}$.
2002 Switzerland Team Selection Test, 1
In space are given $24$ points, no three of which are collinear. Suppose that there are exactly $2002$ planes determined by three of these points. Prove that there is a plane containing at least six points.
2004 Mediterranean Mathematics Olympiad, 2
In a triangle $ABC$, the altitude from $A$ meets the circumcircle again at $T$ . Let $O$ be the circumcenter. The lines $OA$ and $OT$ intersect the side $BC$ at $Q$ and $M$, respectively. Prove that
\[\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .\]