Found problems: 85335
2017 ASDAN Math Tournament, 13
Let $S_1$ be a square of side length $3$. For $i=2,3,4,\dots$, inscribe a square $S_i$ inside $S_{i-1}$ such that the sides of the inner square form four $30^\circ-60^\circ-90^\circ$ triangles with the outer square. Compute the total sum
$$\sum_{i=1}^\infty\text{area}(S_i).$$
2019 AMC 8, 4
Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$?
[asy]
unitsize(1cm);
draw((0,1)--(2,2)--(4,1)--(2,0)--cycle);
dot("$A$",(0,1),W);
dot("$D$",(2,2),N);
dot("$C$",(4,1),E);
dot("$B$",(2,0),S);
[/asy]
$\textbf{(A) } 60
\qquad\textbf{(B) } 90
\qquad\textbf{(C) } 105
\qquad\textbf{(D) } 120
\qquad\textbf{(E) } 144$
2016 IFYM, Sozopol, 8
Find all triples of natural numbers $(x,y,z)$ for which:
$xyz=x!+y^x+y^z+z!$.
2006 Czech and Slovak Olympiad III A, 3
In a scalene triangle $ABC$,the bisectors of angle $A,B$ intersect their corresponding sides at $K,L$ respectively.$I,O,H$ denote respectively the incenter,circumcenter and orthocenter of triangle $ABC$. Prove that $A,B,K,L,O$ are concyclic iff $KL$ is the common tangent line of the circumcircles of the three triangles $ALI,BHI$ and $BKI$.
2020 Princeton University Math Competition, A2/B4
How many ordered triples of nonzero integers $(a, b, c)$ satisfy $2abc = a + b + c + 4$?
2019 USA EGMO Team Selection Test, 5
Let the excircle of a triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively. Denote by $\gamma$ the circumcircle of triangle $A_1B_1C_1$ and assume that $\gamma$ passes through vertex $A$.
[list = a]
[*] Show that $\overline{AA_1}$ is a diameter of $\gamma$.
[*] Show that the incenter of $\triangle ABC$ lies on line $B_1C_1$.
[/list]
2003 Moldova National Olympiad, 12.8
Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by
$F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]
2021 Federal Competition For Advanced Students, P1, 2
Let $ABC$ denote a triangle. The point $X$ lies on the extension of $AC$ beyond $A$, such that $AX = AB$. Similarly, the point $Y$ lies on the extension of $BC$ beyond $B$ such that $BY = AB$. Prove that the circumcircles of $ACY$ and $BCX$ intersect a second time in a point different from $C$ that lies on the bisector of the angle $\angle BCA$.
(Theresia Eisenkölbl)
2010 Thailand Mathematical Olympiad, 6
Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$
2000 All-Russian Olympiad, 6
A perfect number, greater than $6$, is divisible by $3$. Prove that it is also divisible by $9$.
1997 Chile National Olympiad, 3
Let $ ABCD $ be a quadrilateral, whose diagonals intersect at $ O $. The triangles $ \triangle AOB $, $ \triangle BOC $, $ \triangle COD $ have areas $1, 2, 4$, respectively. Find the area of $ \triangle AOD $ and prove that $ ABCD $ is a trapezoid.
2002 France Team Selection Test, 2
Let $ ABC$ be a non-equilateral triangle. Denote by $ I$ the incenter and by $ O$ the circumcenter of the triangle $ ABC$. Prove that $ \angle AIO\leq\frac{\pi}{2}$ holds if and only if $ 2\cdot BC\leq AB\plus{}AC$.
2012 Junior Balkan Team Selection Tests - Romania, 3
Consider the triangle $ABC$ and the points $D \in (BC)$ and $M \in (AD)$. Lines $BM$ and $AC$ meet at $E$, lines $CM$ and $AB$ meet at $F$, and lines $EF$ and $AD$ meet at $N$. Prove that $$\frac{AN}{DN}=\frac{1}{2}\cdot \frac{AM}{DM}$$
2012 Today's Calculation Of Integral, 772
Given are three points $A(2,\ 0,\ 2),\ B(1,\ 1,\ 0),\ C(0,\ 0,\ 3)$ in the coordinate space. Find the volume of the solid of a triangle $ABC$ generated by a rotation about $z$-axis.
2003 Tournament Of Towns, 5
A paper tetrahedron is cut along some of so that it can be developed onto the plane. Could it happen that this development cannot be placed on the plane in one layer?
1967 IMO Longlists, 36
Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.
2019 Stars of Mathematics, 3
Let $ABC$ be a triangle. Let $M$ be a variable point interior to the segment $AB$, and let $\gamma_B$ be the circle through $M$ and tangent at $B$ to $BC$. Let $P$ and $Q$ be the touch points of $\gamma_B$ and its tangents from $A$, and let $X$ be the midpoint of the segment $PQ$. Similarly, let $N$ be a variable point interior to the segment $AC$, and let $\gamma_C$ be the circle through $M$ and tangent at $C$ to $BC$. Let $R$ and $S$ be the touch points of $\gamma_C$ and its tangents from $A$, and let $Y$ be the midpoint of the segment $RS$. Prove that the line through the centers of the circles $AMN$ and $AXY$ passes through a fixed point.
2016 Mathematical Talent Reward Programme, MCQ: P 12
Let $f(x)=(x-1)(x-2)(x-3)$. Consider $g(x)=min\{f(x),f'(x)\}$. Then the number of points of discontinuity are
[list=1]
[*] 0
[*] 1
[*] 2
[*] More than 2
[/list]
2014 District Olympiad, 1
Prove that:
[list=a][*]$\displaystyle\left( \frac{1}{2}\right) ^{3}+\left( \frac{2}{3}\right)^{3}+\left( \frac{5}{6}\right) ^{3}=1$
[*]$3^{33}+4^{33}+5^{33}<6^{33}$[/list]
2015 Princeton University Math Competition, A4/B6
A number is [i]interesting [/i]if it is a $6$-digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers?
2008 Moldova National Olympiad, 12.8
Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.
2014 Middle European Mathematical Olympiad, 1
Determine the lowest possible value of the expression
\[ \frac{1}{a+x} + \frac{1}{a+y} + \frac{1}{b+x} + \frac{1}{b+y} \]
where $a,b,x,$ and $y$ are positive real numbers satisfying the inequalities
\[ \frac{1}{a+x} \ge \frac{1}{2} \] \[\frac{1}{a+y} \ge \frac{1}{2} \] \[ \frac{1}{b+x} \ge \frac{1}{2} \] \[ \frac{1}{b+y} \ge 1. \]
2010 Balkan MO Shortlist, N2
Solve the following equation in positive integers:
$x^{3} = 2y^{2} + 1 $
2002 Austria Beginners' Competition, 2
Prove that there are no $x\in\mathbb{R}^+$ such that $$x^{\lfloor x \rfloor }=\frac92.$$
2022 IOQM India, 5
In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$