Found problems: 85335
2002 Indonesia MO, 1
Prove that $n^4 - n^2$ is divisible by $12$ for all integers $n > 1$.
1963 AMC 12/AHSME, 31
The number of solutions in positive integers of $2x+3y=763$ is:
$\textbf{(A)}\ 255 \qquad
\textbf{(B)}\ 254\qquad
\textbf{(C)}\ 128 \qquad
\textbf{(D)}\ 127 \qquad
\textbf{(E)}\ 0$
2017 NIMO Problems, 8
Let $N$ be the number of integer sequences $a_1, a_2, \dots, a_{2^{16}-1}$ satisfying \[0 \le a_{2k + 1} \le a_k \le a_{2k + 2} \le 1\] for all $1 \le k \le 2^{15}-1$. Find the number of positive integer divisors of $N$.
[i]Proposed by Ankan Bhattacharya[/i]
2014 Abels Math Contest (Norwegian MO) Final, 3b
Nine points are placed on a circle. Show that it is possible to colour the $36$ chords connecting them using four colours so that for any set of four points, each of the four colours is used for at least one of the six chords connecting the given points
2016 VJIMC, 1
Let $f: \mathbb{R} \to (0, \infty)$ be a continuously differentiable function. Prove that there exists $\xi \in (0,1)$ such that $$e^{f'(\xi)} \cdot f(0)^{f(\xi)} = f(1)^{f(\xi)}$$
2019 LIMIT Category C, Problem 10
A right circular cylinder is inscribed in a sphere of radius $\sqrt3$. What is the height of the cylinder when its volume is maximal?
2020 CCA Math Bonanza, I7
Define the binary operation $a\Delta b=ab+a-1$. Compute
\[
10 \Delta(10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta 10))))))))
\]
where $10$ is written $10$ times.
[i]2020 CCA Math Bonanza Individual Round #7[/i]
1970 AMC 12/AHSME, 10
Let $F=.48181\cdots$ be an infinite repeating decimal with the digits $8$ and $1$ repeating. When $F$ is written as a fraction in lowest terms, the denominator exceeds the numerator by
$\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }29\qquad\textbf{(D) }57\qquad \textbf{(E) }126$
2016 Azerbaijan BMO TST, 4
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\] where $n\in \mathbb{N}.$
2019 Hong Kong TST, 1
Let $a$ be a real number. Suppose the function $f(x) = \frac{a}{x-1} + \frac{1}{x-2} + \frac{1}{x-6}$ defined in the interval $3 < x < 5$ attains its maximum at $x=4$. Find the value of $a.$
2009 China Team Selection Test, 2
In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.
2023 AMC 10, 2
The weight of $\frac 13$ of a large pizza together with $3 \frac 12$ cups of orange slices is the same as the weight of $\frac 34$ of a large pizza together with $\frac 12$ cup of orange slices. A cup of orange slices weighs $\frac 14$ of a pound. What is the weight, in pounds, of a large pizza?
$\textbf{(A)}~1\frac45\qquad\textbf{(B)}~2\qquad\textbf{(C)}~2\frac25\qquad\textbf{(D)}~3\qquad\textbf{(E)}~3\frac35$
2012 China National Olympiad, 3
Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers.
[i]Proposed by Huawei Zhu[/i]
2005 Georgia Team Selection Test, 2
In triangle $ ABC$ we have $ \angle{ACB} \equal{} 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD \equal{} AC$ and $ DB \equal{} DC$. Prove that $ \angle{BAC} \equal{} 3\angle{BAD}$.
2024 IMC, 9
A matrix $A=(a_{ij})$ is called [i]nice[/i], if it has the following properties:
(i) the set of all entries of $A$ is $\{1,2,\dots,2t\}$ for some integer $t$;
(ii) the entries are non-decreasing in every row and in every column: $a_{i,j} \le a_{i,j+1}$ and $a_{i,j} \le a_{i+1,j}$;
(iii) equal entries can appear only in the same row or the same column: if $a_{i,j}=a_{k,\ell}$, then either $i=k$ or $j=\ell$;
(iv) for each $s=1,2,\dots,2t-1$, there exist $i \ne k$ and $j \ne \ell$ such that $a_{i,j}=s$ and $a_{k,\ell}=s+1$.
Prove that for any positive integers $m$ and $n$, the number of nice $m \times n$ matrixes is even.
For example, the only two nice $2 \times 3$ matrices are $\begin{pmatrix} 1 & 1 & 1\\2 & 2 & 2 \end{pmatrix}$ and $\begin{pmatrix} 1 & 1 & 3\\2 & 4 & 4 \end{pmatrix}$.
1996 IMO Shortlist, 3
Let $ a > 2$ be given, and starting $ a_0 \equal{} 1, a_1 \equal{} a$ define recursively:
\[ a_{n\plus{}1} \equal{} \left(\frac{a^2_n}{a^2_{n\minus{}1}} \minus{} 2 \right) \cdot a_n.\]
Show that for all integers $ k > 0,$ we have: $ \sum^k_{i \equal{} 0} \frac{1}{a_i} < \frac12 \cdot (2 \plus{} a \minus{} \sqrt{a^2\minus{}4}).$
PEN S Problems, 26
Prove that there does not exist a natural number which, upon transfer of its initial digit to the end, is increased five, six or eight times.
2013 Moldova Team Selection Test, 1
Let $A=20132013...2013$ be formed by joining $2013$, $165$ times. Prove that $2013^2 \mid A$.
2015 Canadian Mathematical Olympiad Qualification, 1
Find all integer solutions to the equation $7x^2y^2 + 4x^2 = 77y^2 + 1260$.
2022 India National Olympiad, 3
For a positive integer $N$, let $T(N)$ denote the number of arrangements of the integers $1, 2, \cdots N$ into a sequence $a_1, a_2, \cdots a_N$ such that $a_i > a_{2i}$ for all $i$, $1 \le i < 2i \le N$ and $a_i > a_{2i+1}$ for all $i$, $1 \le i < 2i+1 \le N$. For example, $T(3)$ is $2$, since the possible arrangements are $321$ and $312$
(a) Find $T(7)$
(b) If $K$ is the largest non-negative integer so that $2^K$ divides $T(2^n - 1)$, show that $K = 2^n - n - 1$.
(c) Find the largest non-negative integer $K$ so that $2^K$ divides $T(2^n + 1)$
2017 India PRMO, 7
Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} < 11$.
2022 Purple Comet Problems, 5
Below is a diagram showing a $6 \times 8$ rectangle divided into four $6 \times 2$ rectangles and one diagonal line. Find the total perimeter of the four shaded trapezoids.
1988 ITAMO, 5
Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?
2023 Sharygin Geometry Olympiad, 10
Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$. The perpendicular from $H$ to $EF$ meets the line $\ell$ passing through $A$ and parallel to $BC$ at point $P$. The bisectors of two angles between $\ell$ and $HP$ meet $BC$ at points $S$ and $T$. Prove that the circumcircles of triangles $ABC$ and $PST$ are tangent.
2012 Dutch BxMO/EGMO TST, 5
Let $A$ be a set of positive integers having the following property:
for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$.
Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.