Found problems: 85335
2022 MIG, 22
Jerry and Aaron both pick two integers from $1$ to $6$, inclusive, and independently and secretly tell their numbers to Dennis.
Dennis then announces, "Aaron's number is at least three times Jerry's number."
Aaron says, "I still don't know Jerry's number."
Jerry then replies, "Oh, now I know Aaron's number."
What is the sum of their numbers?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
1978 Romania Team Selection Test, 1
Show that for every natural number $ a\ge 3, $ there are infinitely many natural numbers $ n $ such that $ a^n\equiv 1\pmod n . $ Does this hold for $ n=2? $
2000 VJIMC, Problem 3
Prove that if m,n are nonnegative integers and 0<=x<=1 then
$(1-x^n)^m + (1-(1-x)^m)^n \ge 1$
1973 AMC 12/AHSME, 23
There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, then the probability that the under-side is also red is
$ \textbf{(A)}\ \frac14 \qquad
\textbf{(B)}\ \frac13 \qquad
\textbf{(C)}\ \frac12 \qquad
\textbf{(D)}\ \frac23 \qquad
\textbf{(E)}\ \frac34$
2010 Romania Team Selection Test, 2
Let $\ell$ be a line, and let $\gamma$ and $\gamma'$ be two circles. The line $\ell$ meets $\gamma$ at points $A$ and $B$, and $\gamma'$ at points $A'$ and $B'$. The tangents to $\gamma$ at $A$ and $B$ meet at point $C$, and the tangents to $\gamma'$ at $A'$ and $B'$ meet at point $C'$. The lines $\ell$ and $CC'$ meet at point $P$. Let $\lambda$ be a variable line through $P$ and let $X$ be one of the points where $\lambda$ meets $\gamma$, and $X'$ be one of the points where $\lambda$ meets $\gamma'$. Prove that the point of intersection of the lines $CX$ and $C'X'$ lies on a fixed circle.
[i]Gazeta Matematica[/i]
1992 Putnam, A4
Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$
2018 Turkey Junior National Olympiad, 2
We are placing rooks on a $n \cdot n$ chess table that providing this condition:
Every two rooks will threaten an empty square at least.
What is the most number of rooks?
2013 IberoAmerican, 2
Let $X$ and $Y$ be the diameter's extremes of a circunference $\Gamma$ and $N$ be the midpoint of one of the arcs $XY$ of $\Gamma$. Let $A$ and $B$ be two points on the segment $XY$. The lines $NA$ and $NB$ cuts $\Gamma$ again in $C$ and $D$, respectively. The tangents to $\Gamma$ at $C$ and at $D$ meets in $P$. Let $M$ the the intersection point between $XY$ and $NP$. Prove that $M$ is the midpoint of the segment $AB$.
2023 SG Originals, Q1
Let $n$ be a positive integer. A sequence $a_1$, $a_2$,$ ...$ , $a_n$ is called [i]good [/i] if the following conditions hold:
$\bullet$ For each $i \in \{1, 2, ..., n\}$, $1 \le a_i \le n$
$\bullet$ For all positive integers $i, j$ with $1 \le i \le j \le n$, the expression $a_i + a_{i+1} + ...+ a_j$ is not divisible by $ n + 1$.
Find the number of good sequences (in terms of $n$).
2004 Moldova Team Selection Test, 5
Let $n\in\mathbb{N}$, the set $A=\{(x_1,x_2...,x_n)|x_i\in\mathbb{R}_{+}, i=1,2,...,n\}$ and the function $$f:A\rightarrow\mathbb{R}, f(x_1,...,x_n)=\frac{1}{x_1}+\frac{1}{2x_2}+\ldots+\frac{1}{(n-1)x_{n-1}}+\frac{1}{nx_n}.$$
Prove that $f(\textstyle\binom{n}{1},\binom{n}{2},...,\binom{n}{n-1},\binom{n}{n})=f(2^{n-1},2^{n-2},...,2,1).$
1998 Tournament Of Towns, 2
On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region.
(P Kozhevnikov)
2014 Junior Balkan Team Selection Tests - Romania, 3
Consider six points in the interior of a square of side length $3$.
Prove that among the six points, there are two whose distance is less than $2$.
1983 All Soviet Union Mathematical Olympiad, 356
The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?
2012 Balkan MO, 4
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
2000 Rioplatense Mathematical Olympiad, Level 3, 3
Let $n>1$ be an integer. For each numbers $(x_1, x_2,\dots, x_n)$ with $x_1^2+x_2^2+x_3^2+\dots +x_n^2=1$, denote
$m=\min\{|x_i-x_j|, 0<i<j<n+1\}$
Find the maximum value of $m$.
2002 AIME Problems, 6
The solutions to the system of equations
\begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*}
are $(x_1,y_1)$ and $(x_2, y_2).$ Find $\log_{30}{(x_1y_1x_2y_2)}.$
2002 IMC, 3
Let $n$ be a positive integer and let $a_k = \dfrac{1}{\binom{n}{k}}, b_k = 2^{k-n},\ (k=1..n)$.
Show that $\sum_{k=1}^n \dfrac{a_k-b_k}{k} = 0$.
2013 JBMO Shortlist, 3
Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.
2012 Princeton University Math Competition, A5
Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts ([u]not ends[/u]) with $2012$. For example, $586$ is a leader since $586^3 =201230056$. How many leaders are there in the set $\{1, 2, 3, ..., 2012\}$?
2019 Benelux, 1
[list=a]
[*]Let $a,b,c,d$ be real numbers with $0\leqslant a,b,c,d\leqslant 1$. Prove that
$$ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.$$[/*]
[*]Find all quadruples $(a,b,c,d)$ of real numbers with $0\leqslant a,b,c,d\leqslant 1$ for which equality holds in the above inequality.
[/list]
2012 China National Olympiad, 1
Let $f(x)=(x + a)(x + b)$ where $a,b>0$. For any reals $x_1,x_2,\ldots ,x_n\geqslant 0$ satisfying $x_1+x_2+\ldots +x_n =1$, find the maximum of $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}} $.
2000 Belarus Team Selection Test, 4.1
Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$
2024 Rioplatense Mathematical Olympiad, 2
Let $ABC$ be a triangle with $AB < AC$, incentre $I$, and circumcircle $\omega$. Let $D$ be the intersection of the external bisector of angle $\widehat{ BAC}$ with line $BC$. Let $E$ be the midpoint of the arc $BC$ of $\omega$ that does not contain $A$. Let $M$ be the midpoint of $DI$, and $X$ the intersection of $EM$ with $\omega$. Prove that $IX$ and $EM$ are perpendicular.
2004 China Western Mathematical Olympiad, 3
Let $\ell$ be the perimeter of an acute-angled triangle $ABC$ which is not an equilateral triangle. Let $P$ be a variable points inside the triangle $ABC$, and let $D,E,F$ be the projections of $P$ on the sides $BC,CA,AB$ respectively. Prove
that \[ 2(AF+BD+CE ) = \ell \] if and only if $P$ is collinear with the incenter and the circumcenter of the triangle $ABC$.
2015 Kazakhstan National Olympiad, 4
$P_k(n) $ is the product of all positive divisors of $n$ that are divisible by $k$ (the empty product is equal to $1$). Show that $P_1(n)P_2(n)\cdots P_n(n)$ is a perfect square, for any positive integer $n$.