Found problems: 85335
2001 Croatia National Olympiad, Problem 3
Let there be given triples of integers $(r_j,s_j,t_j),~j=1,2,\ldots,N$, such that for each $j$, $r_j,t_j,s_j$ are not all even. Show that one can find integers $a,b,c$ such that $ar_j+bs_j+ct_j$ is odd for at least $\frac{4N}7$ of the indices $j$.
2015 India National Olympiad, 2
For any natural number $n > 1$ write the finite decimal expansion of $\frac{1}{n}$ (for example we write $\frac{1}{2}=0.4\overline{9}$ as its infinite decimal expansion not $0.5)$. Determine the length of non-periodic part of the (infinite) decimal expansion of $\frac{1}{n}$.
2013 Harvard-MIT Mathematics Tournament, 5
Thaddeus is given a $2013 \times 2013$ array of integers each between $1$ and $2013$, inclusive. He is allowed two operations:
1. Choose a row, and subtract $1$ from each entry.
2. Chooses a column, and add $1$ to each entry.
He would like to get an array where all integers are divisible by $2013$. On how many arrays is this possible?
2015 AIME Problems, 9
Let $S$ be the set of all ordered triples of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n\ge 4$. Find the number of such sequences for which $a_n=0$ for some $n$.
2017 South East Mathematical Olympiad, 2
Let $x_i \in \{0,1\}(i=1,2,\cdots ,n)$,if the value of function $f=f(x_1,x_2, \cdots ,x_n)$ can only be $0$ or $1$,then we call $f$ a $n$-var Boole function,and we denote $D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}.$
$(1)$ Find the number of $n$-var Boole function;
$(2)$ Let $g$ be a $n$-var Boole function such that $g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2$,
Find the number of elements of the set $D_n(g)$,and find the maximum of $n \in \mathbb{N}_+$ such that
$\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.$
1983 Brazil National Olympiad, 5
Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$.
Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.
2008 USA Team Selection Test, 7
Let $ ABC$ be a triangle with $ G$ as its centroid. Let $ P$ be a variable point on segment $ BC$. Points $ Q$ and $ R$ lie on sides $ AC$ and $ AB$ respectively, such that $ PQ \parallel AB$ and $ PR \parallel AC$. Prove that, as $ P$ varies along segment $ BC$, the circumcircle of triangle $ AQR$ passes through a fixed point $ X$ such that $ \angle BAG = \angle CAX$.
2007 France Team Selection Test, 1
For a positive integer $a$, $a'$ is the integer obtained by the following method: the decimal writing of $a'$ is the inverse of the decimal writing of $a$ (the decimal writing of $a'$ can begin by zeros, but not the one of $a$); for instance if $a=2370$, $a'=0732$, that is $732$.
Let $a_{1}$ be a positive integer, and $(a_{n})_{n \geq 1}$ the sequence defined by $a_{1}$ and the following formula for $n \geq 1$:
\[a_{n+1}=a_{n}+a'_{n}. \]
Can $a_{7}$ be prime?
2012 Hanoi Open Mathematics Competitions, 8
Determine the greatest number m such that the
system $x^2$ + $y^2$ = 1; |$x^3$-$y^3$|+|x-y|=$m^3$ has a solution.
2012 IMO Shortlist, N2
Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and
\[x^3(y^3+z^3)=2012(xyz+2).\]
2014 Saudi Arabia Pre-TST, 1.2
Let $D$ be the midpoint of side $BC$ of triangle $ABC$ and $E$ the midpoint of median $AD$. Line $BE$ intersects side $CA$ at $F$. Prove that the area of quadrilateral $CDEF$ is $\frac{5}{12}$ the area of triangle $ABC$.
2014 Purple Comet Problems, 5
The diagram below shows a large triangle with area $72$. Each side of the triangle has been trisected, and line segments have been drawn between these trisection points parallel to the sides of the triangle. Find the area of the shaded region.
[asy]
size(4cm);
pair A,B1,B2,B3,C1,C2,C3,M,I,J;
A=origin;
dotfactor=4;
B1=dir(49);
B2=2*B1;
B3=3*B1;
C1=1.35*dir(127);
C2=2*C1;
C3=3*C1;
M=(B2+C2)/2;
I=B1+C2;
J=C1+B2;
pair d[] = {A,B1,B2,B3,C1,C2,C3,M,I,J};
filldraw(C1--B1--B2--J--I--C2--cycle,rgb(.76,.76,.76));
draw(A--C3--B3--cycle);
draw(C1--J^^C2--B2^^B1--I);
for(int i=0;i<10;++i){
dot(d[i]);
}
[/asy]
2015 India Regional MathematicaI Olympiad, 5
Two circles X and Y in the plane intersect at two distinct points A and B such that the centre of Y lies on X. Let points C and D be on X and Y respectively, so that C, B and D are collinear. Let point E on Y be such that DE is parallel to AC. Show that AE = AB.
2021 Brazil Team Selection Test, 6
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
2023 Estonia Team Selection Test, 3
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
2024 ISI Entrance UGB, P1
Find, with proof, all possible values of $t$ such that
\[\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c\]
for some real $c>0$. Also find the corresponding values of $c$.
1981 All Soviet Union Mathematical Olympiad, 318
The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the triangle $ABC$ .
$$\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$$
Prove that the perimeter $P$ of the triangle $ABC$ and the perimeter $p$ of the triangle $A_1B_1C_1$ , satisfy inequality $$\frac{P}{2} < p < \frac{3P}{4}$$
2008 Sharygin Geometry Olympiad, 6
(A. Myakishev, 8--9) In the plane, given two concentric circles with the center $ A$. Let $ B$ be an arbitrary point on some of these circles, and $ C$ on the other one. For every triangle $ ABC$, consider two equal circles mutually tangent at the point $ K$, such that one of these circles is tangent to the line $ AB$ at point $ B$ and the other one is tangent to the line $ AC$ at point $ C$. Determine the locus of points $ K$.
2012 Sharygin Geometry Olympiad, 16
Given right-angled triangle $ABC$ with hypothenuse $AB$. Let $M$ be the midpoint of $AB$ and $O$ be the center of circumcircle $\omega$ of triangle $CMB$. Line $AC$ meets $\omega$ for the second time in point $K$. Segment $KO$ meets the circumcircle of triangle $ABC$ in point $L$. Prove that segments $AL$ and $KM$ meet on the circumcircle of triangle $ACM$.
1961 AMC 12/AHSME, 8
Let the two base angles of a triangle be $A$ and $B$, with $B$ larger than $A$. The altitude to the base divides the vertex angle $C$ into two parts, $C_1$ and $C_2$, with $C_2$ adjacent to side $a$. Then:
${{{ \textbf{(A)}\ C_1+C_2=A+B \qquad\textbf{(B)}\ C_1-C_2=B-A \qquad\textbf{(C)}\ C_1-C_2=A-B} \qquad\textbf{(D)}\ C_1+C_2=B-A}\qquad\textbf{(E)}\ C_1-C_2=A+B} $
2004 Harvard-MIT Mathematics Tournament, 1
Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.
2022 Durer Math Competition Finals, 12
Csongi taught Benedek how to fold a duck in 8 steps from a $24$ cm $\times 24$ cm piece of paper. The paper is meant to be folded along the dashed line in the direction of the arrow. Once Benedek folded the duck, he undid all the steps, finding crease lines on the square sheet of paper. On one side of the paper, he drew in blue the folds which opened towards Benedek, and in red the folds which opened toward the table. How many cm is the difference between the total length of the blue lines and the red lines?
[img]https://cdn.artofproblemsolving.com/attachments/0/1/358a3b2c3b959a85406b94e34c182fd1c2e28d.png[/img]
2017 Junior Balkan Team Selection Tests - Romania, 2
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$
2004 Alexandru Myller, 1
Show that the equation $ (x+y)^{-1}=x^{-1}+y^{-1} $ has a solution in the field of integers modulo $ p $ if and only if $ p $ is a prime congruent to $ 1 $ modulo $ 3. $
[i]Mihai Piticari[/i]
2004 China National Olympiad, 1
Let $EFGH,ABCD$ and $E_1F_1G_1H_1$ be three convex quadrilaterals satisfying:
i) The points $E,F,G$ and $H$ lie on the sides $AB,BC,CD$ and $DA$ respectively, and $\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1$;
ii) The points $A,B,C$ and $D$ lie on sides $H_1E_1,E_1F_1,F_1,G_1$ and $G_1H_1$ respectively, and $E_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE$.
Suppose that $\frac{E_1A}{AH_1}=\lambda$. Find an expression for $\frac{F_1C}{CG_1}$ in terms of $\lambda$.
[i]Xiong Bin[/i]