Found problems: 85335
The positive integers $a,b,c$ are less than $99$ and satisfy $a^2+b^2=c^2+99^2$. . Find the minimum and maximum value of $a+b+c$.
The sequence of numbers $a_0,a_1,a_2,...$ is determined by $a_0 = 0$, and
$$a_n= \begin{cases} 1+a_{n-1} \,\,\, when\,\,\, n \,\,\, is \,\,\, positive \,\,\, and \,\,\, odd \\
3a_{n/2} \,\,\,when \,\,\,n \,\,\,is \,\,\,positive \,\,\,and \,\,\,even\end{cases}$$
How many of these numbers are less than $2007$ ?
(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$.
(b) Show that a corresponding decomposition into $30$ squares is also possible.
[i](Walther Janous)[/i]
What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)
Alexandru Mihalcu
$ a,b,c$ are three different positive real numbers. Prove that:\[ \left|\frac{a\plus{}b}{a\minus{}b}\plus{}\frac{b\plus{}c}{b\minus{}c}\plus{}\frac{c\plus{}a}{c\minus{}a}\right|>1\]
The sides of triangle $BAC$ are in the ratio $2: 3: 4$. $BD$ is the angle-bisector drawn to the shortest side $AC$, dividing it into segments $AD$ and $CD$. If the length of $AC$ is $10$, then the length of the longer segment of $AC$ is:
$\text{(A)} \ 3\frac12 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 5\frac57 \qquad \text{(D)} \ 6 \qquad \text{(E)} \ 7\frac12$
Let $x_1,x_2,x_3,y_1,y_2,y_3$ be nonzero real numbers satisfying $x_1+x_2+x_3=0, y_1+y_2+y_3=0$. Prove that
\[\frac{x_1x_2+y_1y_2}{\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}}+\frac{x_2x_3+y_2y_3}{\sqrt{(x_2^2+y_2^2)(x_3^2+y_3^2)}}+\frac{x_3x_1+y_3y_1}{\sqrt{(x_3^2+y_3^2)(x_1^2+y_1^2)}} \ge -\frac32.\]
[i]Ray Li, Max Schindler.[/i]
5) f twice cont diff, $|f''(x)+2xf'(x)+(x^{2}+1)f(x)|\leq 1$. prove $\lim_{x\rightarrow +\infty} f(x) = 0$
Circles $\omega$ and $\gamma$ are drawn such that $\omega$ is internally tangent to $\gamma$, the distance between their centers are $5$, and the area inside of $\gamma$ but outside of $\omega$ is $100\pi$. What is the sum of the radii of the circles?
[asy]
size(3cm);
real lw=0.4, dr=0.3;
real r1=14, r2=9;
pair A=(0,0), B=(r1-r2,0);
draw(A--B,dashed);
draw(circle(A,r1),linewidth(lw)); draw(circle(B,r2),linewidth(lw));
filldraw(circle(A,dr)); filldraw(circle(B,dr));
label("$5$",(A+B)/2,dir(-90));
label("$\gamma$",A+r1*dir(135),dir(135)); label("$\omega$",B+r2*dir(135),dir(135));
[/asy]
[i]2020 CCA Math Bonanza Individual Round #2[/i]
Find all real $x$ such that $$\sqrt{2+\frac{5}{2}\cos x}\leq\sin x.$$
Prove that if natural numbers $a,b,c,d$ satisfy the equality $ab = cd$, then $\frac{gcd(a,c)gcd(a,d)}{gcd(a,b,c,d)}= a$
Haddaway once asked,``what is love?''. The answer can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are positive integers such that $m^2 + n^2 < 2013$. Find $100m+n$.
[i]Proposed by Evan Chen[/i]
Given are triangle $ABC$ and line $\ell$. The reflections of $\ell$ in $AB$ and $AC$ meet at point $A_1$. Points $B_1, C_1$ are defined similarly. Prove that
a) lines $AA_1, BB_1, CC_1$ concur,
b) their common point lies on the circumcircle of $ABC$
c) two points constructed in this way for two perpendicular lines are opposite.
[url=https://artofproblemsolving.com/community/c677808][b]Cyprus IMO TST 2018[/b][/url]
[url=https://artofproblemsolving.com/community/c6h1666662p10591751][b]Problem 1.[/b][/url] Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.
[url=https://artofproblemsolving.com/community/c6h1666663p10591753][b]Problem 2.[/b][/url] Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.
[url=https://artofproblemsolving.com/community/c6h1666660p10591747][b]Problem 3.[/b][/url] Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression
$$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$obtains its minimum value.
[url=https://artofproblemsolving.com/community/c6h1666661p10591749][b]Problem 4.[/b][/url] Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$.
(a) Prove that
$$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence:
$$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$
Solve the equation $$\log_{10} (x^3+x)=\log_2 x.$$
Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$.
Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.
Let $ABC$ be an acute triangle. Let $D$ and $E$ be the midpoint of segment $AB$ and $AC$ respectively. Suppose $L_1$ and $L_2$ are circumcircle of triangle $ABC$ and $ADE$ respectively. $CD$ intersects $L_1$ and $L_2$ at $M (M \not= C)$ and $N (N \not= D)$. If $DM = DN$, prove that $\triangle ABC$ is isosceles.
Trapezoid $ABCD$ with bases $AD$ and $BC$ is inscribed in a circle, $M$ is the intersection of of its diagonals. A straight line passing through $M$ perpendicular to the bases intersects $BC$ at point$ K$, and the circle at point $L$, where $L$ is the one of the two intersection points for which $M$ lies on the segment $KL$. It is known that $MK = a$, $LM = b$. Find the radius of the circle tangent to the segments $AM$, $BM$ and the circle circumscribed around $ABCD$.
Show that $r = 2$ is the largest real number $r$ which satisfies the following condition:
If a sequence $a_1$, $a_2$, $\ldots$ of positive integers fulfills the inequalities
\[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\]
for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2} = a_n$ for every $n \geq M$.
In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE = 6,$ and $AF = 2$. Which of the following is closest to the area of the rectangle $ABCD$?
[asy]
size(140);
pair A, B, C, D, E, F, X, Y;
real length = 12.5;
real width = 10;
A = origin;
B = (length, 0);
C = (length, width);
D = (0, width);
X = rotate(330, C)*B;
E = extension(C, X, A, B);
Y = rotate(30, C)*D;
F = extension(C, Y, A, D);
draw(E--C--F);
label("$2$", A--F, dir(180));
label("$6$", E--B, dir(270));
draw(A--B--C--D--cycle);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);
label("$A$", A, dir(225));
label("$B$", B, dir(315));
label("$C$", C, dir(45));
label("$D$", D, dir(135));
label("$E$", E, dir(270));
label("$F$", F, dir(180));
[/asy]
$\textbf{(A)} \ 110 \qquad \textbf{(B)} \ 120 \qquad \textbf{(C)} \ 130 \qquad \textbf{(D)} \ 140 \qquad \textbf{(E)} \ 150$
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?
Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.
Let $r,s,t$ positive integers which are relatively prime and $a,b \in G$, $G$ a commutative multiplicative group with unit element $e$, and $a^r=b^s=(ab)^t=e$.
(a) Prove that $a=b=e$.
(b) Does the same hold for a non-commutative group $G$?
Find all positive integers $n$ such that there exists a monic polynomial $P(x)$ of degree $n$ with integers coefficients satisfying
$$P(a)P(b)\neq P(c)$$
for all integers $a,b,c$.
Let $ABCD$ be a square with side length $1$. How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex?
[i]Proposed by Josef Tkadlec - Czech Republic[/i]