Found problems: 85335
2018 Canadian Mathematical Olympiad Qualification, 1
Determine all real solutions to the following system of equations:
$$
\begin{cases}
y = 4x^3 + 12x^2 + 12x + 3\\
x = 4y^3 + 12y^2 + 12y + 3.
\end{cases}
$$
2023 Bangladesh Mathematical Olympiad, P3
For any positive integer $n$, define $f(n)$ to be the smallest positive integer that does not divide $n$. For example, $f(1)=2$, $f(6)=4$. Prove that for any positive integer $n$, either $f(f(n))$ or $f(f(f(n)))$ must be equal to $2$.
2007 ITest, 40
Let $S$ be the sum of all $x$ such that $1\leq x\leq 99$ and \[\{x^2\}=\{x\}^2.\] Compute $\lfloor S\rfloor$.
2020 LIMIT Category 1, 9
What is the sum of all two-digit positive integer $n<50$ for which the sum of the squares of first $n$ positive integers is not a divisor of $(2n)!$ ?
2019 Brazil Team Selection Test, 6
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
2016 Indonesia TST, 1
Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.
2015 Saudi Arabia BMO TST, 2
Find the number of $6$-tuples $(a_1,a_2, a_3,a_4, a_5,a_6)$ of distinct positive integers satisfying the following two conditions:
(a) $a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 30$
(b) We can write $a_1,a_2, a_3,a_4, a_5,a_6$ on sides of a hexagon such that after a finite number of time choosing a vertex of the hexagon and adding $1$ to the two numbers written on two sides adjacent to the vertex, we obtain
a hexagon with equal numbers on its sides.
Lê Anh Vinh
2005 AMC 10, 23
In trapezoid $ ABCD$ we have $ \overline{AB}$ parallel to $ \overline{DC}$, $ E$ as the midpoint of $ \overline{BC}$, and $ F$ as the midpoint of $ \overline{DA}$. The area of $ ABEF$ is twice the area of $ FECD$. What is $ AB/DC$?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 8$
2011 Switzerland - Final Round, 7
For a given rational number $r$, find all integers $z$ such that \[2^z + 2 = r^2\mbox{.}\]
[i](Swiss Mathematical Olympiad 2011, Final round, problem 7)[/i]
1999 Brazil Team Selection Test, Problem 2
If $a,b,c,d$ are Distinct Real no. such that
$a = \sqrt{4+\sqrt{5+a}}$
$b = \sqrt{4-\sqrt{5+b}}$
$c = \sqrt{4+\sqrt{5-c}}$
$d = \sqrt{4-\sqrt{5-d}}$
Then $abcd = $
2000 Vietnam National Olympiad, 1
For every integer $ n \ge 3$ and any given angle $ \alpha$ with $ 0 < \alpha < \pi$, let $ P_n(x) \equal{} x^n \sin\alpha \minus{} x \sin n\alpha \plus{} \sin(n \minus{} 1)\alpha$.
(a) Prove that there is a unique polynomial of the form $ f(x) \equal{} x^2 \plus{} ax \plus{} b$ which divides $ P_n(x)$ for every $ n \ge 3$.
(b) Prove that there is no polynomial $ g(x) \equal{} x \plus{} c$ which divides $ P_n(x)$ for every $ n \ge 3$.
1996 Turkey Junior National Olympiad, 3
Let $P$ be a point inside of equilateral $\triangle ABC$ such that $m(\widehat{APB})=150^\circ$, $|AP|=2\sqrt 3$, and $|BP|=2$. Find $|PC|$.
2023 MOAA, 9
Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $BC \perp CD$. There exists a point $P$ on $BC$ such that $\triangle{PAD}$ is equilateral. If $PB = 20$ and $PC = 23$, the area of $ABCD$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $b$ is square-free and $a$ and $c$ are relatively prime. Find $a+b+c$.
[i]Proposed by Andy Xu[/i]
2017 Vietnam Team Selection Test, 3
For each integer $n>0$, a permutation $a_1,a_2,\dots ,a_{2n}$ of $1,2,\dots 2n$ is called [i]beautiful[/i] if for every $1\leq i<j \leq 2n$, $a_i+a_{n+i}=2n+1$ and $a_i-a_{i+1}\not \equiv a_j-a_{j+1}$ (mod $2n+1$) (suppose that $a_{2n+1}=a_1$).
a. For $n=6$, point out a [i]beautiful [/i] permutation.
b. Prove that there exists a [i]beautiful[/i] permutation for every $n$.
2005 MOP Homework, 6
Solve the system of equations:
$x^2=\frac{1}{y}+\frac{1}{z}$,
$y^2=\frac{1}{z}+\frac{1}{x}$,
$z^2=\frac{1}{x}+\frac{1}{y}$.
in the real numbers.
1995 Belarus Team Selection Test, 3
Show that there is no infinite sequence an of natural numbers such that \[a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2\] for all $n\geq 2$
2008 AMC 12/AHSME, 17
Let $ a_1,a_2,\dots$ be a sequence of integers determined by the rule $ a_n\equal{}a_{n\minus{}1}/2$ if $ a_{n\minus{}1}$ is even and $ a_n\equal{}3a_{n\minus{}1}\plus{}1$ if $ a_{n\minus{}1}$ is odd. For how many positive integers $ a_1 \le 2008$ is it true that $ a_1$ is less than each of $ a_2$, $ a_3$, and $ a_4$?
$ \textbf{(A)}\ 250 \qquad
\textbf{(B)}\ 251 \qquad
\textbf{(C)}\ 501 \qquad
\textbf{(D)}\ 502 \qquad
\textbf{(E)}\ 1004$
1996 IMC, 7
Prove that if $f:[0,1]\rightarrow[0,1]$ is a continuous function, then the sequence of iterates $x_{n+1}=f(x_{n})$ converges if and only if
$$\lim_{n\to \infty}(x_{n+1}-x_{n})=0$$
2017 IMO Shortlist, C6
Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation.
It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.
2024 Cono Sur Olympiad, 4
Let $N$ be a positive integer with $2k$ digits. Its chunks are defined by the two numbers formed by the digits from $1$ to $k$ and $k+1$ to $2k$ (e.g. the chunks of 142856 are 142 and 856). We define the $N$-[i]reverse[/i] as the number formed by switching its chunks (e.g. the reverse of 142856 is 856142 and for 1401 it is 114). We call a number [i]cearense[/i] is it satisfies the following conditions:
[list=i]
[*] Has an even number of digits
[*] Its chunks are relatively prime
[*]Divides its reverse
[/list]
Find the two smallest cearense integer.
2019 CCA Math Bonanza, I9
Isosceles triangle $\triangle{ABC}$ has $\angle{BAC}=\angle{ABC}=30^\circ$ and $AC=BC=2$. If the midpoints of $BC$ and $AC$ are $M$ and $N$, respectively, and the circumcircle of $\triangle{CMN}$ meets $AB$ at $D$ and $E$ with $D$ closer to $A$ than $E$ is, what is the area of $MNDE$?
[i]2019 CCA Math Bonanza Individual Round #9[/i]
1996 Moldova Team Selection Test, 7
Let $ABCDA_1B_1C_1D_1$ be a cube. On the sides $AB{}$ and $AD{}$ there are the points $M{}$ and $N{}$, respectively, such that $AM+AN=AB$. Show that the measure of the dihedral angle between the planes $(MA_1C)$ and $(NA_1C)$ doe not depend on the positions of $M{}$ and $N{}$. Find this measure.
2010 Belarus Team Selection Test, 2.1
Point $D$ is marked inside a triangle $ABC$ so that $\angle ADC = \angle ABC + 60^o$, $\angle CDB =\angle CAB + 60^o$, $\angle BDA = \angle BCA + 60^o$. Prove that $AB \cdot CD = BC \cdot AD = CA \cdot BD$.
(A. Levin)
2020 CIIM, 3
Let $(m,r,s,t)$ be positive integers such that $m\geq s+1$ and $r\geq t$. Consider $m$ sets $A_1, A_2, \dots, A_m$ with $r$ elements each one. Suppose that, for each $1\leq i\leq m$, there exist at least $t$ elements of $A_i$, such that each one(element) belongs to (at least) $s$ sets $A_j$ where $j\neq i$. Determine the greatest quantity of elements in the following set $A_1 \cup A_2 \cup A_3 \dots \cup A_m$.
1985 Brazil National Olympiad, 5
$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$.