Found problems: 85335
2020 CMIMC Team, 15
Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose $M$ is the midpoint of segment $\overline{BC}$, $N$ is the midpoint of $\overline{AM}$, and $E$ and $F$ are the feet of the altitudes of $M$ onto $\overline{AB}$ and $\overline{AC}$, respectively. Further suppose $BC$ intersects $NE$ at $S$ and $NF$ at $T$, and let $X$ and $Y$ be the circumcenters of $\triangle MES$ and $\triangle MFT$, respectively. If $XY$ is tangent to the circumcircle of $\triangle ABC$, what is the area of $\triangle ABC$?
2022 Germany Team Selection Test, 1
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
2010 China Second Round Olympiad, 4
the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide.
find the number of all possible codes(in terms of $n$).
2010 Turkey Team Selection Test, 2
For an interior point $D$ of a triangle $ABC,$ let $\Gamma_D$ denote the circle passing through the points $A, \: E, \: D, \: F$ if these points are concyclic where $BD \cap AC=\{E\}$ and $CD \cap AB=\{F\}.$ Show that all circles $\Gamma_D$ pass through a second common point different from $A$ as $D$ varies.
2023 Korea - Final Round, 5
Given a positive integer $n$, there are $n$ boxes $B_1,...,B_n$. The following procedure can be used to add balls.
$$\text{(Procedure) Chosen two positive integers }n\geq i\geq j\geq 1\text{, we add one ball each to the boxes }B_k\text{ that }i\geq k\geq j.$$
For positive integers $x_1,...,x_n$ let $f(x_1,...,x_n)$ be the minimum amount of procedures to get all boxes have its amount of balls to be a multiple of 3, starting with $x_i$ balls for $B_i(i=1,...,n)$. Find the largest possible value of $f(x_1,...,x_n)$. (If $x_1,...,x_n$ are all multiples of 3, $f(x_1,...,x_n)=0$.)
2018 Canadian Senior Mathematics Contest, A4
Suppose that $n$ is a positive integer and that $a$ is the integer equal to $\frac{10^{2n}-1}{3\left(10^n+1\right)}.$
If the sum of the digits of $a$ is 567, what is the value of $n$?
2014 USAMTS Problems, 3:
Let $P$ be a square pyramid whose base consists of the four vertices $(0, 0, 0), (3, 0, 0), (3, 3, 0)$, and $(0, 3, 0)$, and whose apex is the point $(1, 1, 3)$. Let $Q$ be a square pyramid whose base is the same as the base of $P$, and whose apex is the point $(2, 2, 3)$. Find the volume of the intersection of the interiors of $P$ and $Q$.
2017 AMC 12/AHSME, 3
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically?
$\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\
$\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\
$\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\
$\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\
$\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.
2011 Korea - Final Round, 1
Prove that there is no positive integers $x,y,z$ satisfying
\[ x^2 y^4 - x^4 y^2 + 4x^2 y^2 z^2 +x^2 z^4 -y^2 z^4 =0 \]
2018 Thailand TSTST, 4
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way:
\[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \]
Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]
2011 NIMO Problems, 7
The number $ \left (2+2^{96} \right )!$ has $2^{93}$ trailing zeroes when expressed in base $B$.
[b]
a)[/b] Find the minimum possible $B$.
[b]b)[/b] Find the maximum possible $B$.
[b]c)[/b] Find the total number of possible $B$.
[i]Proposed by Lewis Chen[/i]
1997 German National Olympiad, 5
We are given $n$ discs in a plane, possibly overlapping, whose union has the area $1$. Prove that we can choose some of them which are mutually disjoint and have the total area greater than $1/9$.
2003 Tuymaada Olympiad, 4
Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite.
Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$
[i]Proposed by F. Petrov[/i]
[hide="For those of you who liked this problem."]
Check [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62259]this thread[/url] out.[/hide]
2018 AMC 8, 2
What is the value of the product$$\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?$$
$\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{4}{3}\qquad\textbf{(C) }\frac{7}{2}\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
1997 Iran MO (2nd round), 1
Let $x_1,x_2,x_3,x_4$ be positive reals such that $x_1x_2x_3x_4=1$. Prove that:
\[ \sum_{i=1}^{4}{x_i^3}\geq\max\{ \sum_{i=1}^{4}{x_i},\sum_{i=1}^{4}{\frac{1}{x_i}} \}. \]
2015 ASDAN Math Tournament, 16
Find the maximum value of $c$ such that
\begin{align*}
1&=-cx+y\\
-7&=x^2+y^2+8y
\end{align*}
has a unique real solution $(x,y)$.
1997 National High School Mathematics League, 12
Let $a=\lg z+\lg\left[x(yz)^{-1}+1\right],b=\lg x^{-1}+\lg(xyz+1),c=\lg y+\lg\left[(xyz)^{-1}+1\right]$, if $M=\max\{a,b,c\}$, then the minumum value of $M$ is________.
1997 National High School Mathematics League, 8
Line $l$ that passes right focal point of hyperbola $x^2-\frac{y^2}{2}=1$ intersects the hyperbola at $A,B$. The number of line $l$ that $|AB|=\lambda$ is 3, then $\lambda=$________.
Russian TST 2016, P1
For which even natural numbers $d{}$ does there exists a constant $\lambda>0$ such that any reduced polynomial $f(x)$ of degree $d{}$ with integer coefficients that does not have real roots satisfies the inequality $f(x) > \lambda$ for all real numbers?
1987 ITAMO, 1
Show that $3x^5 +5x^3 -8x$ is divisible by $120$ for any integer $x$
2013 Kazakhstan National Olympiad, 3
Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$. Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$. Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.
2001 IMO Shortlist, 4
Let $p \geq 5$ be a prime number. Prove that there exists an integer $a$ with $1 \leq a \leq p-2$ such that neither $a^{p-1}-1$ nor $(a+1)^{p-1}-1$ is divisible by $p^2$.
2018 AIME Problems, 7
Triangle $ABC$ has sides $AB=9,BC = 5\sqrt{3},$ and $AC=12$. Points $A=P_0, P_1, P_2, \dots, P_{2450} = B$ are on segment $\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\dots,2449$, and points $A=Q_0, Q_1, Q_2, \dots ,Q_{2450} = C$ for $k=1,2,\dots,2449$. Furthermore, each segment $\overline{P_kQ_k}, k=1,2,\dots,2449$, is parallel to $\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions have the same area. Find the number of segments $\overline{P_kQ_k}, k=1,2 ,\dots,2450$, that have rational length.
2024 pOMA, 3
Let $ABC$ be a triangle with circumcircle $\Omega$, and let $P$ be a point on the arc $BC$ of $\Omega$ not containing $A$. Let $\omega_B$ and $\omega_C$ be circles respectively passing through $B$ and $C$ and such that both of them are tangent to line $AP$ at point $P$. Let $R$, $R_B$, $R_C$ be the radii of $\Omega$, $\omega_B$, and $\omega_C$, respectively.
Prove that if $h$ is the distance from $A$ to line $BC$, then
\[
\frac{R_B+R_C}{R} \le \frac{BC}{h}.
\]
2023 Brazil National Olympiad, 5
Let $m$ be a positive integer with $m \leq 2024$. Ana and Banana play a game alternately on a $1\times2024$ board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any $k \leq m$ white squares on the board and painting them all green. Each Banana play consists of choosing any sequence of consecutive green squares and painting them all white. What is the smallest value of $m$ for which Ana can guarantee that, after one of her moves, the entire board will be painted green?