Found problems: 85335
2007 Moldova Team Selection Test, 1
Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.
2024 AMC 10, 6
A rectangle has integer side lengths and an area of $2024$. What is the least possible perimeter of the rectangle?
$
\textbf{(A) }160 \qquad
\textbf{(B) }180 \qquad
\textbf{(C) }222 \qquad
\textbf{(D) }228 \qquad
\textbf{(E) }390 \qquad
$
1956 AMC 12/AHSME, 31
In our number system the base is ten. If the base were changed to four you would count as follows: $ 1,2,3,10,11,12,13,20,21,22,23,30,\ldots$ The twentieth number would be:
$ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 38 \qquad\textbf{(C)}\ 44 \qquad\textbf{(D)}\ 104 \qquad\textbf{(E)}\ 110$
2017 AMC 10, 21
In $\triangle ABC,$ $AB=6, AC=8, BC=10,$ and $D$ is the midpoint of $\overline{BC}.$ What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC?$
$\textbf{(A)} \sqrt{5} \qquad \textbf{(B)} \frac{11}{4}\qquad \textbf{(C)} 2\sqrt{2} \qquad \textbf{(D)} \frac{17}{6} \qquad \textbf{(E)} 3$
2013 Swedish Mathematical Competition, 4
A robotic lawnmower is located in the middle of a large lawn. Due a manufacturing defect, the robot can only move straight ahead and turn in directions that are multiples of $60^o$. A fence must be set up so that it delimits the entire part of the lawn that the robot can get to, by traveling along a curve with length no more than $10$ meters from its starting position, given that it is facing north when it starts. How long must the fence be?
1992 IMO Longlists, 11
Let $\phi(n,m), m \neq 1$, be the number of positive integers less than or equal to $n$ that are coprime with $m.$ Clearly, $\phi(m,m) = \phi(m)$, where $\phi(m)$ is Euler’s phi function. Find all integers $m$ that satisfy the following inequality:
\[\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}\]
for every positive integer $n.$
2020 IMO Shortlist, A2
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
\begin{align*}
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\end{align*}
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
2014-2015 SDML (Middle School), 12
Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$
1997 Miklós Schweitzer, 1
Define a class of graphs $G_k$ for each positive integer k as follows. A graph G = ( V , E ) is an element of $G_k$ if and only if there exists an edge coloring $\psi: E\to [ k ] = \{1,2, ..., k\}$ such that for all vertex coloring $\phi: V\to [ k ]$ there exist an edge e = { x , y } such that $\phi ( x ) = \phi( y ) = \psi( e )$. Prove that there exist $c_1< c_2$ positive constants with the following two properties:
(i) each graph in $G_k$ has at least $c_1 k^2$ vertices;
(ii) there is a graph in $G_k$ which has at most $c_2 k^2$ vertices.
1960 AMC 12/AHSME, 3
Applied to a bill for $\$10,000$ the difference between a discount of $40\%$ and two successive discounts of $36\%$ and $4\%$, expressed in dollars, is:
$ \textbf{(A) }0\qquad\textbf{(B) }144\qquad\textbf{(C) }256\qquad\textbf{(D) }400\qquad\textbf{(E) }416 $
1954 Czech and Slovak Olympiad III A, 1
Solve the equation $$ax^2+2(a-1)x+a-5=0$$ in real numbers with respect to (real) parametr $a$.
1998 Poland - First Round, 3
In the isosceles triangle $ ABC$ the angle $ BAC$ is a right angle. Point $ D$ lies on the side $ BC$ and satisfies $ BD \equal{} 2 \cdot CD$. Point $ E$ is the foot of the perpendicular of the point $ B$ on the line $ AD$. Find the angle $ CED$.
1967 IMO Shortlist, 5
Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality
\[af^2 + bfg +cg^2 \geq 0\]
holds if and only if the following conditions are fulfilled:
\[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]
1979 VTRMC, 5
Show, for all positive integers $n = 1,2 , \dots ,$ that $14$ divides $ 3 ^ { 4 n + 2 } + 5 ^ { 2 n + 1 }$.
2006 JHMT, 3
Rectangle $ABCD$ is folded in half so that the vertices $D$ and $B$ coincide, creating the crease $\overline{EF}$, with $E$ on $\overline{AD}$ and $F$ on $\overline{BC}$. Let $O$ be the midpoint of $\overline{EF}$. If triangles $DOC$ and $DCF$ are congruent, what is the ratio $BC : CD$?
1990 Greece National Olympiad, 2
If $a+b=1$, $ \in \mathbb{R}$ and $ab \ne 0$, prove that $$\frac{a}{b^3-1}+\frac{b}{a^3-1}=\frac{2(ab-2)}{a^2b^2+3}$$
2017 CMIMC Individual Finals, 3
Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.
2010 IMAC Arhimede, 6
Consider real numbers $a, b ,c \ge0$ with $a+b+c=2$. Prove that:
$\frac{bc}{\sqrt[4]{3a^2+4}}+\frac{ca}{\sqrt[4]{3b^2+4}}+\frac{ab}{\sqrt[4]{3c^2+4}} \le \frac{2*\sqrt[4] {3}}{3}$
2025 Japan MO Finals, 4
Find all integer-coefficient polynomials $f(x)$ satisfying the following conditions for every integer $n \geqslant 2$:
[list]
[*] $f(n) > 0$.
[*] $f(n)$ divides $n^{f(n)} - 1$.
[/list]
2022 MIG, 10
The diagram below shows a square of area $36$ separated into two rectangles and a smaller square. One of the rectangles has an area of $12$. What is the smallest rectangle's area?
[asy]
size(70);
draw((0,0)--(2,0)--(2,6)--(0,6)--cycle);
draw((2,2)--(6,2)--(6,6)--(2,6)--cycle);
draw((2,2)--(6,2)--(6,0)--(2,0)--cycle);
label("$12$",(1,3));
label("$?$",(4,4));
label("$?$",(4,1));
[/asy]
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }\text{Not Enough Information}$
2014 Harvard-MIT Mathematics Tournament, 3
$ABC$ is a triangle such that $BC = 10$, $CA = 12$. Let $M$ be the midpoint of side $AC$. Given that $BM$ is parallel to the external bisector of $\angle A$, find area of triangle $ABC$. (Lines $AB$ and $AC$ form two angles, one of which is $\angle BAC$. The external angle bisector of $\angle A$ is the line that bisects the other angle.
2012 France Team Selection Test, 1
Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$:
\[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\]
For which $k$ does there exist a $k$-tastrophic function?
2017 CCA Math Bonanza, I12
Let $a_1,a_2,\ldots,a_{2017}$ be the $2017$ distinct complex numbers which satisfy $a_i^{2017}=a_i+1$ for $i=1,2,\ldots,2017$. Compute $$\displaystyle\sum_{i=1}^{2017}\frac{a_i}{a_i^2+1}.$$
[i]2017 CCA Math Bonanza Individual Round #12[/i]
2006 Thailand Mathematical Olympiad, 2
Triangle $\vartriangle ABC$ has side lengths $AB = 2$, $CA = 3$ and $BC = 4$. Compute the radius of the circle centered on $BC$ that is tangent to both $AB$ and $AC$.
1993 Baltic Way, 2
Do there exist positive integers $a>b>1$ such that for each positive integer $k$ there exists a positive integer $n$ for which $an+b$ is a $k$-th power of a positive integer?