Found problems: 85335
2019 Estonia Team Selection Test, 6
It is allowed to perform the following transformations in the plane with any integers $a$:
(1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$,
(2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$.
Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to:
a) Vertices of a square,
b) Vertices of a rectangle with unequal side lengths?
2023 Germany Team Selection Test, 1
Does there exist a positive odd integer $n$ so that there are primes $p_1$, $p_2$ dividing $2^n-1$ with $p_1-p_2=2$?
2022 Canadian Mathematical Olympiad Qualification, 7
Let $ABC$ be a triangle with $|AB| < |AC|$, where $| · |$ denotes length. Suppose $D, E, F$ are points on side $BC$ such that $D$ is the foot of the perpendicular on $BC$ from $A$, $AE$ is the angle bisector of $\angle BAC$, and $F$ is the midpoint of $BC$. Further suppose that $\angle BAD = \angle DAE = \angle EAF = \angle FAC$. Determine all possible values of $\angle ABC$.
2015 BMT Spring, 17
There exist real numbers $x$ and $y$ such that $x(a^3 + b^3 + c^3) + 3yabc \ge (x + y)(a^2b + b^2c + c^2a)$ holds for all positive real numbers $a, b$, and $c$. Determine the smallest possible value of $x/y$.
.
1983 Miklós Schweitzer, 4
For which cardinalities $ \kappa$ do antimetric spaces of cardinality $ \kappa$ exist?
$ (X,\varrho)$ is called an $ \textit{antimetric space}$ if $ X$ is a nonempty set, $ \varrho : X^2 \rightarrow [0,\infty)$ is a symmetric map, $ \varrho(x,y)\equal{}0$ holds iff $ x\equal{}y$, and for any three-element subset $ \{a_1,a_2,a_3 \}$ of $ X$ \[ \varrho(a_{1f},a_{2f})\plus{}\varrho(a_{2f},a_{3f}) < \varrho(a_{1f},a_{3f})\] holds for some permutation $ f$ of $ \{1,2,3 \}$.
[i]V. Totik[/i]
2020 AMC 8 -, 18
Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$
[asy]
// diagram by SirCalcsALot
draw(arc((0,0),17,180,0));
draw((-17,0)--(17,0));
fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey);
draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle);
dot("$A$",(8,0), 1.25*S);
dot("$B$",(8,15), 1.25*N);
dot("$C$",(-8,15), 1.25*N);
dot("$D$",(-8,0), 1.25*S);
dot("$E$",(17,0), 1.25*S);
dot("$F$",(-17,0), 1.25*S);
label("$16$",(0,0),N);
label("$9$",(12.5,0),N);
label("$9$",(-12.5,0),N);
[/asy]
$\textbf{(A) }240 \qquad \textbf{(B) }248 \qquad \textbf{(C) }256 \qquad \textbf{(D) }264 \qquad \textbf{(E) }272$
2012 AMC 10, 1
Cagney can frost a cupcake every $20$ seconds and Lacey can frost a cupcake every $30$ seconds. Working together, how many cupcakes can they frost in $5$ minutes?
$ \textbf{(A)}\ 10
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 25
\qquad\textbf{(E)}\ 30
$
2010 Portugal MO, 2
On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.
1974 Miklós Schweitzer, 7
Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$.
[i]G. Halasz[/i]
2012 Hanoi Open Mathematics Competitions, 4
What is the largest integer less than or equal to $4x^3 - 3x$, where $x=\frac{\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}}{2}$ ?
(A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) None of the above.
2019 Harvard-MIT Mathematics Tournament, 8
There is a unique function $f: \mathbb{N} \to \mathbb{R}$ such that $f(1) > 0$ and such that
\[\sum_{d \mid n} f(d) f\left(\frac{n}{d}\right) = 1\]
for all $n \ge 1$. What is $f(2018^{2019})$?
2003 Romania National Olympiad, 3
Let be a circumcircle of radius $ 1 $ of a triangle whose centered representation in the complex plane is given by the affixes of $ a,b,c, $ and for which the equation $ a+b\cos x +c\sin x=0 $ has a real root in $ \left( 0,\frac{\pi }{2} \right) . $ prove that the area of the triangle is a real number from the interval $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $
[i]Gheorghe Iurea[/i]
2013 Online Math Open Problems, 9
David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
[i]Ray Li[/i]
2009 Argentina National Olympiad, 4
You have $100$ equal rods. It is allowed to split each rod into two or three shorter rods, not necessarily the same. The objective is that by rearranging the pieces (and using them all) $q>200$ can be assembled new rods, all of equal length. Find the values of $q$ for whom this can be done.
1998 Iran MO (2nd round), 2
Let $ABC$ be a triangle. $I$ is the incenter of $\Delta ABC$ and $D$ is the meet point of $AI$ and the circumcircle of $\Delta ABC$. Let $E,F$ be on $BD,CD$, respectively such that $IE,IF$ are perpendicular to $BD,CD$, respectively. If $IE+IF=\frac{AD}{2}$, find the value of $\angle BAC$.
2018 District Olympiad, 4
Let $n$ and $q$ be two natural numbers, $n\ge 2$, $q\ge 2$ and $q\not\equiv 1 (\text{mod}\ 4)$ and let $K$ be a finite field which has exactly $q$ elements. Show that for any element $a$ from $K$, there exist $x$ and $y$ in $K$ such that $a = x^{2^n} + y^{2^n}$. (Every finite field is commutative).
2015 Belarus Team Selection Test, 3
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
[i]Proposed by Netherlands[/i]
2018 BMT Spring, 7
A line in the $xy$-plane has positive slope, passes through the point $(x, y) = (0, 29)$, and lies tangent to the ellipse defined by $\frac{x^2}{100} +\frac{y^2}{400} = 1$. What is the slope of the line?
2010 USA Team Selection Test, 7
In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on segment $BC$. Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$.
2017 Mediterranean Mathematics Olympiad, Problem 4
Let $x,y,z$ and $a,b,c$ be positive real numbers with $a+b+c=1$. Prove that
$$\left(x^2+y^2+z^2\right) \left( \frac{a^3}{x^2+2y^2} + \frac{b^3}{y^2+2z^2} + \frac{c^3}{z^2+2x^2} \right) \ge\frac19.$$
2024 Macedonian Mathematical Olympiad, Problem 3
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equation
$$f(f(x+y))=f(x+y)+f(x)f(y)-xy,$$
for any two real numbers $x$ and $y$.
2020 Final Mathematical Cup, 1
Find all such functions $f:\mathbb{R} \to \mathbb{R}$ that for any real $x,y$ the following equation is true.
$$f(f(x)+y)+1=f(x^2+y)+2f(x)+2y$$
2011 Tokyo Instutute Of Technology Entrance Examination, 2
For a positive real number $t$, in the coordiante space, consider 4 points $O(0,\ 0,\ 0),\ A(t,\ 0,\ 0),\ B(0,\ 1,\ 0),\ C(0,\ 0,\ 1)$.
Let $r$ be the radius of the sphere $P$ which is inscribed to all faces of the tetrahedron $OABC$.
When $t$ moves, find the maximum value of $\frac{\text{vol[P]}}{\text{vol[OABC]}}.$
1998 AMC 8, 23
If the pattern in the diagram continues, what fraction of the interior would be shaded in the eighth triangle?
[asy]
unitsize(5);
draw((0,0)--(12,0)--(6,6sqrt(3))--cycle);
draw((15,0)--(27,0)--(21,6sqrt(3))--cycle);
fill((21,0)--(18,3sqrt(3))--(24,3sqrt(3))--cycle,black);
draw((30,0)--(42,0)--(36,6sqrt(3))--cycle);
fill((34,0)--(32,2sqrt(3))--(36,2sqrt(3))--cycle,black);
fill((38,0)--(36,2sqrt(3))--(40,2sqrt(3))--cycle,black);
fill((36,2sqrt(3))--(34,4sqrt(3))--(38,4sqrt(3))--cycle,black);
draw((45,0)--(57,0)--(51,6sqrt(3))--cycle);
fill((48,0)--(46.5,1.5sqrt(3))--(49.5,1.5sqrt(3))--cycle,black);
fill((51,0)--(49.5,1.5sqrt(3))--(52.5,1.5sqrt(3))--cycle,black);
fill((54,0)--(52.5,1.5sqrt(3))--(55.5,1.5sqrt(3))--cycle,black);
fill((49.5,1.5sqrt(3))--(48,3sqrt(3))--(51,3sqrt(3))--cycle,black);
fill((52.5,1.5sqrt(3))--(51,3sqrt(3))--(54,3sqrt(3))--cycle,black);
fill((51,3sqrt(3))--(49.5,4.5sqrt(3))--(52.5,4.5sqrt(3))--cycle,black);
[/asy]
$\text{(A)}\ \dfrac{3}{8} \qquad \text{(B)}\ \dfrac{5}{27} \qquad \text{(C)}\ \dfrac{7}{16} \qquad \text{(D)}\ \dfrac{9}{16} \qquad \text{(E)}\ \dfrac{11}{45}$
2013 Brazil Team Selection Test, 4
Let $a, b, c$ be non-negative reals with $a + b + c \le 2$. prove that $$\sqrt{b^2+ac} + \sqrt{a^2+bc} + \sqrt{c^2+ab} \le 3$$