Found problems: 85335
2024-IMOC, N7
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that
$$|xf(y)-yf(x)|$$
is a perfect square for every $x,y \in \mathbb{N}$
1997 Belarusian National Olympiad, 3
$$Problem3;$$If distinct real numbers x,y satisfy $\{x\} = \{y\}$ and $\{x^3\}=\{y^3\}$
prove that $x$ is a root of a quadratic equation with integer coefficients.
1988 AMC 12/AHSME, 25
$X$, $Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets $X$, $Y$, $Z$, $X \cup Y$, $X \cup Z$ and $Y \cup Z$ are given in the table below.
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
\rule{0pt}{1.1em} Set & $X$ & $Y$ & $Z$ & $X\cup Y$ & $X\cup Z$ & $Y\cup Z$\\[0.5ex] \hline \rule{0pt}{2.2em} \shortstack{Average age of \\ people in the set} & 37 & 23 & 41 & 29 & 39.5 & 33\\[1ex]\hline\end{tabular}
Find the average age of the people in set $X \cup Y \cup Z$.
$ \textbf{(A)}\ 33\qquad\textbf{(B)}\ 33.5\qquad\textbf{(C)}\ 33.6\overline{6}\qquad\textbf{(D)}\ 33.83\overline{3}\qquad\textbf{(E)}\ 34 $
2019 China Team Selection Test, 2
A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ .
2016 Kosovo National Mathematical Olympiad, 2
Sum of all coefficients of polynomial $P(x)$ is equal with $2$ . Also the sum of coefficients which are at odd exponential in $x^k$ are equal to sum of coefficients which are at even exponential in $x^k$ . Find the residue of polynomial $P(x)$ when it is divide by $x^2-1$ .
2019-2020 Winter SDPC, 7
Let $a,b$ be positive integers. Find, with proof, the maximum possible value of $a\lceil b\lambda \rceil - b \lfloor a \lambda \rfloor$ for irrational $\lambda$.
2022 Sharygin Geometry Olympiad, 8.8
An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$ QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.
2009 AMC 12/AHSME, 17
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?
$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {3}{16}\qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {3}{8}\qquad \textbf{(E)}\ \frac {1}{2}$
2010 APMO, 3
Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?
1991 AMC 12/AHSME, 1
If for any three distinct numbers $a$, $b$ and $c$ we define \[\boxed{a,b,c} = \frac{c + a}{c - b},\] then $\boxed{1,-2,-3}=$
$ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -\frac{2}{5}\qquad\textbf{(C)}\ -\frac{1}{4}\qquad\textbf{(D)}\ \frac{2}{5}\qquad\textbf{(E)}\ 2 $
2003 National Olympiad First Round, 13
Let $ABC$ be a triangle such that $|AB|=8$ and $|AC|=2|BC|$. What is the largest value of altitude from side $[AB]$?
$
\textbf{(A)}\ 3\sqrt 2
\qquad\textbf{(B)}\ 3\sqrt 3
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ \dfrac {16}3
\qquad\textbf{(E)}\ 6
$
2013 Vietnam National Olympiad, 3
Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$.
[b]a)[/b] Prove that $D,I,J$ collinear.
[b]b)[/b] $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.
1986 India National Olympiad, 2
Solve
\[ \left\{ \begin{array}{l}
\log_2 x\plus{}\log_4 y\plus{}\log_4 z\equal{}2 \\
\log_3 y\plus{}\log_9 z\plus{}\log_9 x\equal{}2 \\
\log_4 z\plus{}\log_{16} x\plus{}\log_{16} y\equal{}2 \\
\end{array} \right.\]
1989 AMC 8, 3
Which of the following numbers is the largest?
$\text{(A)}\ .99 \qquad \text{(B)}\ .9099 \qquad \text{(C)}\ .9 \qquad \text{(D)}\ .909 \qquad \text{(E)}\ .9009$
2003 Romania National Olympiad, 1
Find the maximum number of elements which can be chosen from the set $ \{1,2,3,\ldots,2003\}$ such that the sum of any two chosen elements is not divisible by 3.
2022 Czech-Austrian-Polish-Slovak Match, 4
Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.
2025 District Olympiad, P4
Find all triplets of matrices $A,B,C\in\mathcal{M}_2(\mathbb{R})$ which satisfy \begin{align*}
A=BC-CB \\
B=CA-AC \\
C=AB-BA
\end{align*}
[i]Proposed by David Anghel[/i]
III Soros Olympiad 1996 - 97 (Russia), 11.3
A chord $AB$ is drawn in a certain circle. The smaller of the two arcs $AB$ corresponds to a central angle of $120^o$. A tangent $p$ to this arc is drawn. Two circles with radii $R$ and $r$ are constructed, touching this smaller arc $AB$ and straight lines $AB$ and $p$. Find the radius of the original circle.
2020 Brazil Team Selection Test, 1
Consider an $n\times n$ unit-square board. The main diagonal of the board is the $n$ unit squares along the diagonal from the top left to the bottom right. We have an unlimited supply of tiles of this form:
[asy]
size(1.5cm);
draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0));
[/asy]
The tiles may be rotated. We wish to place tiles on the board such that each tile covers exactly three unit squares, the tiles do not overlap, no unit square on the main diagonal is covered, and all other unit squares are covered exactly once. For which $n\geq 2$ is this possible?
[i]Proposed by Daniel Kohen[/i]
2016 All-Russian Olympiad, 8
Medians $AM_A,BM_B,CM_C$ of triangle $ABC$ intersect at $M$.Let $\Omega_A$ be circumcircle of triangle passes through midpoint of $AM$ and tangent to $BC$ at $M_A$.Define $\Omega_B$ and $\Omega_C$ analogusly.Prove that $\Omega_A,\Omega_B$ and $\Omega_C$ intersect at one point.(A.Yakubov)
[hide=P.S]sorry for my mistake in translation :blush: :whistling: .thank you jred for your help :coolspeak: [/hide]
2008 Peru IMO TST, 3
Given a positive integer $n$, consider the sequence $(a_i)$, $1 \leq i \leq 2n$, defined as follows:
$a_{2k-1} = -k, 1 \leq k \leq n$
$a_{2k} = n-k+1, 1 \leq k \leq n.$
We call a pair of numbers $(b,c)$ good if the following conditions are met:
$i) 1 \leq b < c \leq 2n,$
$ii) \sum_{j=b}^{c}a_j = 0$
If $B(n)$ is the number of good pairs corresponding to $n$, prove that there are infinitely many $n$ for which $B(n) = n$.
2011 China Second Round Olympiad, 6
In a tetrahedral $ABCD$, given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$, $AD=BD=3$, and $CD=2$. Find the radius of the circumsphere of $ABCD$.
2022 Bulgarian Autumn Math Competition, Problem 8.1
Solve the equation:
\[4x^2+|9-6x|=|10x-15|+6(2x+1)\]
2007 Hanoi Open Mathematics Competitions, 10
What is the smallest possible value of $x^2+2y^2-x-2y-xy$?
2009 Princeton University Math Competition, 6
In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees.
[asy]
pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15;
pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd);
[/asy]