Found problems: 85335
Let $ABC$ be an acute triangle. Construct a point $X$ on the different side of $C$ with respect to the line $AB$ and construct a point $Y$ on the different side of $B$ with respect to the line $AC$ such that $BX=AC$, $CY=AB$, and $AX=AY$. Let $A'$ be the reflection of $A$ across the perpendicular bisector of $BC$. Suppose that $X$ and $Y$ lie on different sides of the line $AA'$, prove that points $A$, $A'$, $X$, and $Y$ lie on a circle.
Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.
Without using tables, find the exact value of the product:
\[P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).\]
Let $n\in\mathbb{N}, x_0=0$ and $x_1,x_2,\ldots,x_n$ be postive real numbers such that $x_1+x_2+\ldots+x_n=1$. Show that $$1\leq\sum_{i=1}^{n}\frac{x_i}{\sqrt{1+x_0+x_1+\ldots+x_{i-1}}\cdot\sqrt{x_i+x_{i+1}+\ldots+x_n}}<\frac{\pi}{2}.$$
Let $p>2$ be a prime number. For $k=1,2,\ldots,p-1$ we denote by $a_k$ the remainder when $k^p$ is divided by $p^2$. Prove that
$$a_1+a_2+\ldots+a_{p-1}=\frac{p^3-p^2}2.$$
The Fibonacci sequence is defined by \[ a_{n+1} = a_n + a_{n-1}, n \geq 1, a_0 = 0, a_1 = a_2 = 1. \] Find the greatest common divisor of the 1960-th and 1988-th terms of the Fibonacci sequence.
For any positive integer $a$, denote $f(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{1,9\}\}|$ and $g(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{3,7\}\}|$. Prove that $f(a)\geq g(a)\forall a\in\mathbb{N}$.
In a class of $30$ kids are distributed $430 $ apples . Prove that at least two kids will take the same number of apples.
It is possible to compose a triangle from the altitudes of a given triangle. Can we conclude that it is possible to compose a triangle from its bisectors?
The degree of the polynomial $P(x)$ is $2017.$ Prove that the number of distinct real roots of the equation $P(P(x)) = 0$ is not less than the number of distinct real roots of the equation $P(x) = 0.$
Let $s < t$ be positive integers. Define a sequence by: $a_1 = s, a_2 = t$; $a_3$ is the smallest integer that's greater than $a_2$ and divisible by $a_1$; in general, $a_{n + 1}$ is the smallest integer greater than $a_n$ that's divisible by $a_1, a_2, ..., a_{n - 2}, a_{n - 1}$.
[b]a)[/b] What is the maximum number of odd integers that can appear in such a sequence? (Justify your answer)
[b]b)[/b] Prove that $a_{2025}$ is divisible by $2^{808}$, regardless of the choice of $s$ and $t$.
Proposed by [i]Ilija Jovcevski[/i]
Let $z_1$, $z_2$, and $z_3$ be the complex roots of the equation $(2z -3\overline{z})^3 = 54i+54$. Compute the area of the triangle formed by $z_1$, $z_2$, and $z_3$ when plotted in the complex plane.
We number both the rows and the columns of an $8 \times 8$ chessboard with the numbers $1$ to $8$. Some grains of rice are placed on each square, in such a way that the number of grains on each square is equal to the product of the row and column numbers of the square. How many grains of rice are there on the entire chessboard?
Let $x_1,x_2,\dots, x_n$ be positive real numbers such that $x_1+x_2+\cdots + x_n=1$. Prove that $$\sum_{i=1}^n \frac{\min\{x_{i-1},x_i\}\cdot \max\{x_i,x_{i+1}\}}{x_i}\leq 1,$$ where we denote $x_0=x_n$ and $x_{n+1}=x_1$.
One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$
is an integer.
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]
If $ g(x)\equal{}1\minus{}x^2$ and $ f(g(x)) \equal{} \frac{1\minus{}x^2}{x^2}$ when $ x\neq0$, then $ f(1/2)$ equals
$ \textbf{(A)}\ 3/4 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ \sqrt2/2 \qquad
\textbf{(E)}\ \sqrt2$
At the two ends $A, B$ of a diameter (of length $2r$) of a pavement horizontal circular rise two vertical columns, of equal height h, whose ends support a beam $A' B' $ of length equal to the before mentioned diameter. It forms a covered by placing numerous taut cables (which are admitted to be rectilinear), joining points of the beam $A'B'$ with points of the circumference edge of the pavement, so that the cables are perpendicular to the beam $A'B'$ . You want to find out the volume enclosed between the roof and the pavement.
[hide=original wording]En los dos extremos A, B de un di´ametro (de longitud 2r) de un pavimento circular horizontal se levantan sendas columnas verticales, de igual altura h, cuyos extremos soportan una viga A' B' de longitud igual al diametro citado. Se forma una cubierta colocando numerosos cables tensos (que se admite que quedan rectilıneos), uniendo puntos de la viga A'B' con puntos de la circunferencia borde del pavimento, de manera que los cables queden perpendiculares a la viga A'B' . Se desea averiguar el volumen encerrado entre la cubierta y el pavimento.[/hide]
In a certain game, Auntie Hall has four boxes $B_1$, $B_2$, $B_3$, $B_4$, exactly one of which contains a valuable gemstone; the other three contain cups of yogurt. You are told the probability the gemstone lies in box $B_n$ is $\frac{n}{10}$ for $n=1,2,3,4$.
Initially you may select any of the four boxes; Auntie Hall then opens one of the other three boxes at random (which may contain the gemstone) and reveals its contents. Afterwards, you may change your selection to any of the four boxes, and you win if and only if your final selection contains the gemstone. Let the probability of winning assuming optimal play be $\tfrac mn$, where $m$ and $n$ are relatively prime integers. Compute $100m+n$.
[i]Proposed by Evan Chen[/i]
Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$
Evaluate $ \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.$
In a school with double schooling, in the morning the language teacher divided the students into $200$ groups for an activity. In the afternoon, the math teacher divided the same students into $300$ groups for another activity. A student is considered [i]special[/i] if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.
[b]Note:[/b] Each group has at least one student.
What is the smallest perimeter of the convex $32$-gon, having all the vertices in the nodes of cross-lined paper with the sides of its squares equal to $1$?
Inside the triangle $ABC$, the point $P$ is chosen. Let $ a, b, c $ be the lengths of the sides $ BC $, $ CA $, $ AB $, respectively, and $ x, y, z $ the distances of the point $ P $ from the vertices $ B, C, A $. Prove that if
$$ x^2 + xy + y^2 = a^2 $$
$$y^2 + yz + z^2 = b^2 $$
$$z^2 + zx + x^2 = c^2$$
this
$$ a^2 + ab + b^2 > c^2.$$