Found problems: 85335
2019 Jozsef Wildt International Math Competition, W. 11
Let $(s_n)_{n\geq 1}$ be a sequence given by $s_n=-2\sqrt{n}+\sum \limits_{k=1}^n\frac{1}{\sqrt{k}}$ with $\lim \limits_{n \to \infty}s_n=s=$Ioachimescu constant and $(a_n)_{n\geq 1}$ , $(b_n)_{n\geq 1}$ be a positive real sequences such that $$\lim \limits_{n\to \infty}\frac{a_{n+1}}{na_n}=a\in \mathbb{R}^*_+, \lim \limits_{n\to \infty}\frac{b_{n+1}}{b_n\sqrt{n}}=b\in \mathbb{R}^*_+$$Compute$$\lim \limits_{n\to \infty}\left(1+e^{s_n}-e^{s_{n+1}}\right)^{\sqrt[n]{a_nb_n}}$$
2002 Moldova National Olympiad, 4
Prove that there are infinitely many triplets $ (a,b,c)$ that satisfy the following equalities:
$ \dfrac{2a\minus{}b\plus{}6}{4a\plus{}c\plus{}2}\equal{}\dfrac{b\minus{}2c}{a\minus{}c}\equal{}\dfrac{2a\plus{}b\plus{}2c\minus{}2}{6a\plus{}2c\minus{}2}$
2019 CMIMC, 11
Let $S$ be a subset of the natural numbers such that $0\in S$, and for all $n\in\mathbb N$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?
2016 Brazil Team Selection Test, 1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
2020 Sharygin Geometry Olympiad, 21
The diagonals of bicentric quadrilateral $ABCD$ meet at point $L$. Given are three segments equal to $AL$, $BL$, $CL$. Restore the quadrilateral using a compass and a ruler.
2006 India National Olympiad, 4
Some 46 squares are randomly chosen from a $9 \times 9$ chess board and colored in [color=red]red[/color]. Show that there exists a $2\times 2$ block of 4 squares of which at least three are colored in [color=red]red[/color].
1996 Tournament Of Towns, (516) 3
The parabola $y = x^2$ is drawn in the coordinate plane and then the axes are erased so that the whole parabola stays on the picture but the origin is not shown on it. Reconstruct the axes with compass and ruler alone.
(A Egorov)
2005 Federal Math Competition of S&M, Problem 2
Suppose that in a convex hexagon, each of the three lines connecting the midpoints of two opposite sides divides the hexagon into two parts of equal area. Prove that these three lines intersect in a point.
2020 Harvard-MIT Mathematics Tournament, 1
Let $n$ be a positive integer. Define a sequence by $a_0 = 1$, $a_{2i+1} = a_i$, and $a_{2i+2} = a_i + a_{i+1}$ for each $i \ge 0$. Determine, with proof, the value of $a_0 + a_1 + a_2 + \dots + a_{2^n-1}$.
[i]Proposed by Kevin Ren.[/i]
1997 Israel Grosman Mathematical Olympiad, 3
Find all real solutions of $\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3$.
2013 AMC 10, 24
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other's school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
$\textbf{(A)} \ 540 \qquad \textbf{(B)} \ 600 \qquad \textbf{(C)} \ 720 \qquad \textbf{(D)} \ 810 \qquad \textbf{(E)} \ 900$
2021 IMC, 6
For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.
2018 Junior Balkan MO, 2
Find max number $n$ of numbers of three digits such that :
1. Each has digit sum $9$
2. No one contains digit $0$
3. Each $2$ have different unit digits
4. Each $2$ have different decimal digits
5. Each $2$ have different hundreds digits
2013 Korea National Olympiad, 5
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying
\[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \]
for all positive integer $m,n$.
2012 AMC 10, 6
The product of two positive numbers is $9$. The reciprocal of one of these numbers is $4$ times the reciprocal of the other number. What is the sum of the two numbers?
$ \textbf{(A)}\ \dfrac{10}{3}
\qquad\textbf{(B)}\ \dfrac{20}{3}
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ \dfrac{15}{2}
\qquad\textbf{(E)}\ 8
$
2006 Princeton University Math Competition, 5
Find the largest integer $k$ such that $12^k | 66!$.
LMT Speed Rounds, 15
Find the least positive integer $n$ greater than $1$ such that $n^3 -n^2$ is divisible by $7^2 \times 11$.
[i]Proposed by Jacob Xu[/i]
2009 Romanian Master of Mathematics, 1
For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$.
Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer.
[i]Dan Schwarz, Romania[/i]
1982 IMO Longlists, 14
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
PEN P Problems, 14
Let $n$ be a non-negative integer. Find all non-negative integers $a$, $b$, $c$, $d$ such that \[a^{2}+b^{2}+c^{2}+d^{2}= 7 \cdot 4^{n}.\]
2020 Taiwan TST Round 1, 2
Let point $H$ be the orthocenter of a scalene triangle $ABC$. Line $AH$ intersects with the circumcircle $\Omega$ of triangle $ABC$ again at point $P$. Line $BH, CH$ meets with $AC,AB$ at point $E$ and $F$, respectively. Let $PE, PF$ meet $\Omega$ again at point $Q,R$, respectively. Point $Y$ lies on $\Omega$ so that lines $AY,QR$ and $EF$ are concurrent. Prove that $PY$ bisects $EF$.
1983 Putnam, B5
Let $\lVert u\rVert$ denote the distance from the real number $u$ to the nearest integer. For positive integers $n$, let
$$a_n=\frac1n\int^n_1\left\lVert\frac nx\right\rVert dx.$$Determine $\lim_{n\to\infty}a_n$.
2010 Peru Iberoamerican Team Selection Test, P2
For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system.
Find all positive integers N for which there exist positive integers $a$,$b$,$c$, coprime two by two, such that:
$S(ab) = S(bc) = S(ca) = N$.
2000 JBMO ShortLists, 9
Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2$.
2012 Turkey Team Selection Test, 2
In an acute triangle $ABC,$ let $D$ be a point on the side $BC.$ Let $M_1, M_2, M_3, M_4, M_5$ be the midpoints of the line segments $AD, AB, AC, BD, CD,$ respectively and $O_1, O_2, O_3, O_4$ be the circumcenters of triangles $ABD, ACD, M_1M_2M_4, M_1M_3M_5,$ respectively. If $S$ and $T$ are midpoints of the line segments $AO_1$ and $AO_2,$ respectively, prove that $SO_3O_4T$ is an isosceles trapezoid.