Found problems: 85335
Circles $O_1, O_2$ intersects at $A, B$. The circumcircle of $O_1BO_2$ intersects $O_1, O_2$ and line $AB$ at $R, S, T$ respectively. Prove that $TR = TS$
Let $F_1=1, F_2=1,$ and $F_{n+2}=F_{n+1}+F_n.$ Then, $$S = \sum_{n=1}^{\infty} \arctan\left(\frac{1}{F_n}\right)\arctan\left(\frac{1}{F_{n+1}}\right)$$ Find $\lfloor 80S \rfloor.$
(Hint: it may be useful to note that $\arctan(\tfrac{1}{1}) = \arctan(\tfrac{1}{2})+\arctan(\tfrac{1}{3}).$)
Let $a{}$ and $b{}$ be positive integers. Prove that for any real number $x{}$ \[\sum_{j=0}^a\binom{a}{j}\big(2\cos((2j-a)x)\big)^b=\sum_{j=0}^b\binom{b}{j}\big(2\cos((2j-b)x)\big)^a.\]
Georg is putting his $250$ stamps in a new album. On the first page he places one stamp and then on every page just as many or twice as many stamps as on the preceding page. In this way he ends up precisely having put all $250$ stamps in the album. How few pages are sufficient for him?
A football is made by sewing together some black and white leather
patches. The black patches are regular pentagons of the same size. The white
patches are regular hexagons of the same size. Each pentagon is bordered by 5
hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football.
How many hexagons are needed to make one football?
Prove that the integer $1^1 + 3^3 + 5^5 + .. + (2^n - 1)^{2^n-1}$ is a multiple of $2^n$ but not a multiple of $2^{n+1}$.
Let $x$ and $y$ be real numbers such that $x^2+y^2-22x-16y+113=0.$ Determine the smallest possible value of $x.$
Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\]
[i]Proposed by Otgonbayar Uuye.[/i]
Show that a convex polygon with more than four sides cannot be decomposed into two others, both similar to the first (directly or inversely), by means of a single rectilinear cut. Reasonably specify which are the quadrilaterals and triangles that admit a decomposition of this type.
Find all pairs of positive integers $(a, b)$, such that $S(a^{b+1})=a^b$, where $S(m)$ denotes the digit sum of $m$.
Sofiya and Marquis are playing a game. Sofiya announces to Marquis that she's thinking of a polynomial of the form $f(x)=x^3+px+q$ with three integer roots that are not necessarily distinct. She also explains that all of the integer roots have absolute value less than (and not equal to) $N$, where $N$ is some fixed number which she tells Marquis. As a "move" in this game, Marquis can ask Sofiya about any number $x$ and Sofiya will tell him whether $f(x)$ is positive negative, or zero. Marquis's goal is to figure out Sofiya's polynomial.
If $N=3\cdot 2^k$ for some positive integer $k$, prove that there is a strategy which allows Marquis to identify the polynomial after making at most $2k+1$ "moves".
Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$.
Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$
about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.
Say that a complex number $z$ is [i]three-presentable[/i] if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$?
For positive integer $n \ge 3$, find the number of ordered pairs $(a_1, a_2, ... , a_n)$ of integers that satisfy the following two conditions
[list=disc]
[*]For positive integer $i$ such that $1\le i \le n$, $1 \le a_i \le i$
[*]For positive integers $i,j,k$ such that $1\le i < j < k \le n$, if $a_i = a_j$ then $a_j \ge a_k$
[/list]
Prove that \[\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\geq \frac{a+b+c}{3}\]
for positive reals $a,b,c$
(S Tokarev)
Let $k, m, n$ be integers such that $1<n\le m-1 \le k$. Determine the maximum size of a subset $S$ of the set $\{ 1,2, \cdots, k \}$ such that no $n$ distinct elements of $S$ add up to $m$.
In the sequence $ (a_n)$ with general term $ a_n \equal{} n^3 \minus{} (2n \plus{} 1)^2$, does there exist a term that is divisible by 2006?
Show that every integer $k>1$ has a multiple less than $k^4$ whose decimal expansion has at most four distinct digits.
Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$.
Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.
Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$, $x_2y_1-x_1y_2=5$, and $x_1y_1+5x_2y_2=\sqrt{105}$. Find the value of $y_1^2+5y_2^2$
Points $E$, $F$, $G$, and $H$ lie on sides $AB$, $BC$, $CD$, and $DA$ of a convex quadrilateral $ABCD$ such that
$\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA}=1$.
Points $A$, $B$, $C$, and $D$ lie on sides $H_1E_1$, $E_1F_1$, $F_1G_1$, and $G_1H_1$ of a convex quadrilateral $E_1F_1G_1H_1$ such that $E_1F_1 \parallel EF$, $F_1G_1 \parallel FG$, $G_1H_1 \parallel GH$, and $H_1E_1 \parallel HE$. Given that $\frac{E_1A}{AH_1}=a$, express $\frac{F_1C}{CG_1}$ in terms of $a$.

In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$. The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$. Prove that for each positive integer $n \geq 3$, the equation: \[ (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) \] is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$, and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$.
\[\int_0^{\frac \pi 2} \left(\frac{1}{1-\cos(x)}-\frac{2}{x^2}\right)\,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
Let $ABCD$ be a parallelogram with sides $AB=10$ and $BC=6$. The circles $\omega_1$ and $\omega_2$ pass through $B$ and have centers $A$ and $C$ respectively. An arbitrary circle with center $D$ intersects $\omega_1$ at points $P_1\neq Q_1$ and $\omega_2$ at points $P_2 \neq Q_2$.
Calculate the ratio $\dfrac{P_1Q_1}{P_2Q_2}$.
A game of Jai Alai has eight players and starts with players $P_1$ and $P_2$ on court and the other players $P_3, P_4, P_5, P_6, P_7, P_8$ waiting in a queue. After each point is played, the loser goes to the end of the queue; the winner adds $1$ point to his score and stays on the court; and the player at the head of the queue comes on to contest the next point. Play continues until someone has scored $7$ points. At that moment, we observe that a total of $37$ points have been scored by all eight players. Determine who has won and justify your answer.