Found problems: 85335
2003 Baltic Way, 10
A [i]lattice point[/i] in the plane is a point with integral coordinates. The[i] centroid[/i] of four points $(x_i,y_i )$, $i = 1, 2, 3, 4$, is the point $\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right)$.
Let $n$ be the largest natural number for which there are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that $n = 12$.
2014 Turkey EGMO TST, 5
Let $ABC$ be a triangle with circumcircle $\omega$ and let $\omega_A$ be a circle drawn outside $ABC$ and tangent to side $BC$ at $A_1$ and tangent to $\omega$ at $A_2$. Let the circles $\omega_B$ and $\omega_C$ and the points $B_1, B_2, C_1, C_2$ are defined similarly. Prove that if the lines $AA_1, BB_1, CC_1$ are concurrent, then the lines $AA_2, BB_2, CC_2$ are also concurrent.
2005 Greece Team Selection Test, 3
Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.
2016 USAMO, 4
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$
2016 Ecuador NMO (OMEC), 6
A positive integer $n$ is "[i]olympic[/i]" if there are $n$ non-negative integers $x_1, x_2, ..., x_n$ that satisfy that:
$\bullet$ There is at least one positive integer $j$: $1 \le j \le n$ such that $x_j \ne 0$.
$\bullet$ For any way of choosing $n$ numbers $c_1, c_2, ..., c_n$ from the set $\{-1, 0, 1\}$, where not all $c_i$ are equal to zero, it is true that the sum $c_1x_1 + c_2x_2 +... + c_nx_n$ is not divisible by $n^3$.
Find the largest positive "olympic" integer.
2023 Thailand Online MO, 1
Let $n$ be a positive integer. Chef Kao has $n$ different flavors of ice cream. He wants to serve one small cup and one large cup for each flavor. He arranges the $2n$ ice cream cups into two rows of $n$ cups on a tray. He wants the tray to be colorful, so he arranges the ice cream cups with the following conditions:
[list]
[*]each row contains all ice cream flavors, and
[*]each column has different sizes of ice cream cup.
[/list]Determine the number of ways that Chef Kao can arrange cups of ice cream with the above conditions.
1998 Cono Sur Olympiad, 6
The mayor of a city wishes to establish a transport system with at least one bus line, in which:
- each line passes exactly three stops,
- every two different lines have exactly one stop in common,
- for each two different bus stops there is exactly one line that passes through both.
Determine the number of bus stops in the city.
2021 Polish MO Finals, 3
Let $\omega$ be the circumcircle of a triangle $ABC$. Let $P$ be any point on $\omega$ different than the verticies of the triangle.
Line $AP$ intersects $BC$ at $D$, $BP$ intersects $AC$ at $E$ and $CP$ intersects $AB$ at $F$. Let $X$ be the projection of $D$ onto line passing through midpoints of $AP$ and $BC$, $Y$ be the projection of $E$ onto line passing through $BP$ and $AC$ and let $Z$ be the projection of $F$ onto line passing through midpoints of $CP$ and $AB$. Let $Q$ be the circumcenter of triangle $XYZ$. Prove that all possible points $Q$, corresponding to different positions of $P$ lie on one circle.
1991 Arnold's Trivium, 79
How many solutions has the boundary-value problem
\[u_{xx}+\lambda u=\sin x,\;u(0)=u(\pi)=0\]
2016 ASDAN Math Tournament, 7
What is
$$\sum_{n=1996}^{2016}\lfloor\sqrt{n}\rfloor?$$
2021 Yasinsky Geometry Olympiad, 2
In the triangle $ABC$, it is known that $AB = BC = 20$ cm, and $AC = 24$ cm. The point $M$ lies on the side $BC$ and is equidistant from sides $AB$ and $AC$. Find this distance.
(Alexander Shkolny)
1971 Miklós Schweitzer, 9
Given a positive, monotone function $ F(x)$ on $ (0, \infty)$ such that $ F(x)/x$ is monotone nondecreasing and $ F(x)/x^{1+d}$ is monotone nonincreasing for some positive $ d$, let $ \lambda_n >0$ and $ a_n \geq 0 , \;n \geq 1$. Prove that if \[ \sum_{n=1}^{\infty} \lambda_n F \left( a_n \sum _{k=1}^n \frac{\lambda_k}{\lambda_n} \right) < \infty,\] or \[ \sum_{n=1}^{\infty} \lambda_n F \left( \sum _{k=1}^n a_k \frac{\lambda_k}{\lambda_n} \right) < \infty,\] then $ \sum_{n=1}^ {\infty} a_n$ is convergent.
[i]L. Leindler[/i]
2020 Tournament Of Towns, 2
$ What~ is~ the~ maximum~ number~ of~ distinct~ integers~ in~ a~ row~ such~ that~ the~sum~ of~ any~ 11~ consequent~ integers~ is~ either~ 100~ or~ 101~?$
I'm posting this problem for people to discuss
2011 ELMO Shortlist, 2
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
2018 Serbia National Math Olympiad, 2
Let $n>1$ be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by $n$. Prove that the number of consecutive beautiful numbers is less or equal to $1+\lfloor \sqrt{3n} \rfloor$.
1989 IMO Longlists, 89
155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.
1992 AMC 12/AHSME, 7
The ratio of $w$ to $x$ is $4:3$, of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y$?
$ \textbf{(A)}\ 1:3\qquad\textbf{(B)}\ 16:3\qquad\textbf{(C)}\ 20:3\qquad\textbf{(D)}\ 27:4\qquad\textbf{(E)}\ 12:1 $
2004 Unirea, 2
Let be two matrices $ A,N\in\mathcal{M}_2(\mathbb{R}) $ that commute and such that $ N $ is nilpotent. Show that:
[b]a)[/b] $ \det (A+N)=\det (A) $
[b]b)[/b] if $ A $ is general linear, then the matrix $ A+N $ is invertible and $ (A+N)^{-1}=(A-N)A^{-2} . $
1986 IMO Longlists, 50
Let $D$ be the point on the side $BC$ of the triangle $ABC$ such that $AD$ is the bisector of $\angle CAB$. Let $I$ be the incenter of$ ABC.$
[i](a)[/i] Construct the points $P$ and $Q$ on the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$ and the perimeter of the triangle $APQ$ is equal to $k \cdot BC$, where $k$ is a given rational number.
[i](b) [/i]Let $R$ be the intersection point of $PQ$ and $AD$. For what value of $k$ does the equality $AR = RI$ hold?
[i](c)[/i] In which case do the equalities $AR = RI = ID$ hold?
2023 239 Open Mathematical Olympiad, 3
Let $n>1$ be a natural number and $x_k{}$ be the residue of $n^2$ modulo $\lfloor n^2/k\rfloor+1$ for all natural $k{}$. Compute the sum \[\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.\]
2015 Mathematical Talent Reward Programme, MCQ: P 14
$z=x+i y$ where $x$ and $y$ are two real numbers. Find the locus of the point $(x, y)$ in the plane, for which $\frac{z+i}{z-i}$ is purely imaginary (that is, it is of the form $i b$ where $b$ is a real number). [Here, $i=\sqrt{-1}$
[list=1]
[*] A straight line
[*] A circle
[*] A parabole
[*] None of these
[/list]
2020 China Girls Math Olympiad, 8
Let $n$ be a given positive integer. Let $\mathbb{N}_+$ denote the set of all positive integers.
Determine the number of all finite lists $(a_1,a_2,\cdots,a_m)$ such that:
[b](1)[/b] $m\in \mathbb{N}_+$ and $a_1,a_2,\cdots,a_m\in \mathbb{N}_+$ and $a_1+a_2+\cdots+a_m=n$.
[b](2)[/b] The number of all pairs of integers $(i,j)$ satisfying $1\leq i<j\leq m$ and $a_i>a_j$ is even.
For example, when $n=4$, the number of all such lists $(a_1,a_2,\cdots,a_m)$ is $6$, and these lists are $(4),$ $(1,3),$ $(2,2),$ $(1,1,2),$ $(2,1,1),$ $(1,1,1,1)$.
2018 Brazil Undergrad MO, 23
How many prime numbers $ p $ the number $ p ^ 3-4 p + 9 $ is a perfect square
2007 Princeton University Math Competition, 1
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?
1979 IMO Longlists, 49
Let there be given two sequences of integers $f_i(1), f_i(2), \cdots (i = 1, 2)$ satisfying:
$(i) f_i(nm) = f_i(n)f_i(m)$ if $\gcd(n,m) = 1$;
$(ii)$ for every prime $P$ and all $k = 2, 3, 4, \cdots$, $f_i(P^k) = f_i(P)f_i(P^{k-1}) - P^2f(P^{k-2}).$
Moreover, for every prime $P$:
$(iii) f_1(P) = 2P,$
$(iv) f_2(P) < 2P.$
Prove that $|f_2(n)| < f_1(n)$ for all $n$.