Found problems: 85335
Let $ABC$ be a right angled triangle with hypotenuse $AC$, and let $H$ be the foot of the altitude from $B$ to $AC$. Knowing that there is a right-angled triangle with side-lengths $AB, BC, BH$, determine all the possible values of $\frac{AH}{CH}$
Let $ ABCD$ be a (non-degenerate) quadrangle and $ N$ the intersection of $ AC$ and $ BD$. Denote by $ a,b,c,d$ the length of the altitudes from $ N$ to $ AB,BC,CD,DA$, respectively.
Prove that $ \frac{1}{a}\plus{}\frac{1}{c} \equal{} \frac{1}{b}\plus{}\frac{1}{d}$ if $ ABCD$ has an incircle.
Extension: Prove that the converse is true, too.
[If this has already been posted, I humbly apologize. A quick search turned up nothing.]
Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
Consider the set $I=\{ 1,2, \cdots, 2020 \}$. Let $W= \{w(a,b)=(a+b)+ab | a,b \in I \} \cap I$, $Y=\{y(a,b)=(a+b) \cdot ab | a,b \in I \} \cap I$ be its two subsets. Further, let $X= W \cap Y$.
[b](1)[/b] Find the sum of maximal and minimal elements in $X$.
[b](2)[/b] An element $n=y(a,b) (a \le b)$ in $Y$ is called [i]excellent[/i], if its representation is not unique (for instance, $20=y(1,5)=y(2,3)$). Find the number of [i]excellent[/i] elements in $Y$.
[hide=Note][b](2)[/b] is only for Grade 11.[/hide]
The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$).
Prove that there are two consecutive positive integers such that
the largest prime divisor of one of them is $p$ and that of the other is $q$.
Find all non-negative integers $n$, $a$, and $b$ satisfying
\[2^a + 5^b + 1 = n!.\]
Let $ABC$ be an acute angled isosceles triangle with $AB=AC$ and circumcentre $O$. Lines $BO$ and $CO$ intersect $AC, AB$ respectively at $B', C'$. A straight line $l$ is drawn through $C'$ parallel to $AC$. Prove that the line $l$ is tangent to the circumcircle of $\triangle B'OC$.
Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.
Evaluate $50(\cos 39^{\circ}\cos21^{\circ}+\cos129^{\circ}\cos69^{\circ})$
Show that if for non-negative integers $m$, $n$, $N$, $k$ the equation \[(n^2+1)^{2^k}\cdot(44n^3+11n^2+10n+2)=N^m\] holds, then $m = 1$.
Given an interger $n\geq 2$, determine the maximum value the sum $\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_{n-1}}{a_n}$ may achieve, and the points at which the maximum is achieved, as $a_1,a_2,...a_n$ run over all positive real numers subject to $a_k\geq a_1+a_2...+a_{k-1}$, for $k=2,...n$
11.4 Let $AA_1$ and $BB_1$ are altitudes of an acute non-isosceles triangle $ABC$, $A'$ is a midpoint of $BC$ and $B'$ is a midpoint of $AC$. A segement $A_1B_1$ intersects $A'B'$ at point $C'$. Prove that $CC'\perp HO$, where $H$ is a orthocenter and $O$ is a circumcenter of $ABC$.
([i]L. Emel'yanov[/i])
Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel.
Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.
In a $3 \times 25$ board, $1 \times 3$ pieces are placed (vertically or horizontally) so that they occupy entirely $3$ boxes on the board and do not have a common point.
What is the maximum number of pieces that can be placed, and for that number, how many configurations are there?
[hide=original formulation]
Num tabuleiro 3 × 25 s˜ao colocadas pe¸cas 1 × 3 (na vertical ou na horizontal) de modo que ocupem inteiramente 3 casas do tabuleiro e n˜ao se toquem em nenhum ponto.
Qual ´e o n´umero m´aximo de pe¸cas que podem ser colocadas, e para esse n´umero,
quantas configura¸c˜oes existem?
[url=https://www.obm.org.br/content/uploads/2018/09/Provas_OMCPLP_2018.pdf]source[/url][/hide]
Steve has a tricycle which has a front wheel with a radius of $30$ cm and back wheels with radii of $10$ cm and $9$ cm. The axle passing through the centers of the back wheels has a length of $40$ cm and is perpendicular to both planes containing the wheels. Since the tricycle is tilted, it goes in a circle as Steve pedals. Steve rides the tricycle until it reaches its original position, so that all of the wheels do not slip or leave the ground. The tires trace out concentric circles on the ground, and the radius of the circle the front wheel traces is the average of the radii of the other two traced circles. Compute the total number of degrees the front wheel rotates. (Express your answer in simplest radical form.)
Find all pairs of positive integers $(a, b)$ such that $$11ab \le a^3 - b^3 \le 12ab.$$
Seven equally-spaced points are drawn on a circle of radius $1$. Three distinct points are chosen uniformly at random. What is the probability that the center of the circle lies in the triangle formed by the three points?
Calculate the given expression $$\sum_{k=0}^{n} \frac{2^k}{3^{2^k}+1}$$
Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$
Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that \[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\] Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.
Given sets $A = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}$, $B = \{2, 3, 8, 10, 12, 13, 14, 15, 18\}$, if a positive integer leaves a remainder (the smallest non-negative remainder) that belongs to $A$ when divided by 19, then that positive integer is called an $\alpha$ number. If a positive integer leaves a remainder that belongs to $B$ when divided by 19, then that positive integer is called a $\beta$ number.
(1) For what positive integer $n$, among all its positive divisors, are the numbers of $\alpha$ divisors and $\beta$ divisors equal?
(2) For which positive integers $k$, are the numbers of $\alpha$ divisors less than the numbers of $\beta$ divisors? For which positive integers $l$, are the numbers of $\alpha$ divisors greater than the numbers of $\beta$ divisors?
In the Cartesian coordinate system, we define a [i]Bongo-line[/i] as a sequence of integer points $\alpha = (\ldots, A_{-1}, A_0, A_1, \ldots)$ such that:
- $A_iA_{i+1} = \sqrt{2}$ for every $i \in \mathbb{Z}$;
- the polyline $\ldots A_{-1}A_0A_1 \ldots$ has no self-intersections.
Let $\alpha = (\ldots, A_{-1}, A_0, A_1, \ldots)$ and $\beta = (\ldots, B_{-1}, B_0, B_1, \ldots)$ be two Bongo-lines such that there exists a bijection $f : \mathbb{Z} \to \mathbb{Z}$ such that $A_iA_{i+1}$ and $B_{f(i)}B_{f(i)+1}$ halve each other. Prove that all vertices of $\alpha$ and $\beta$ lie on two lines.
[i]Proposed by Pavle Martinović[/i]
Given the true statements: (1) If $ a$ is greater than $ b$, then $ c$ is greater than $ d$ (2) If $ c$ is less than $ d$, then $ e$ is greater than $ f$. A valid conclusion is:
$ \textbf{(A)}\ \text{If }{a}\text{ is less than }{b}\text{, then }{e}\text{ is greater than }{f}\qquad \\
\textbf{(B)}\ \text{If }{e}\text{ is greater than }{f}\text{, then }{a}\text{ is less than }{b}\qquad \\
\textbf{(C)}\ \text{If }{e}\text{ is less than }{f}\text{, then }{a}\text{ is greater than }{b}\qquad \\
\textbf{(D)}\ \text{If }{a}\text{ is greater than }{b}\text{, then }{e}\text{ is less than }{f}\qquad \\
\textbf{(E)}\ \text{none of these}$
In a long line of people, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
$
\textbf{(A) }2021 \qquad
\textbf{(B) }2022 \qquad
\textbf{(C) }2023 \qquad
\textbf{(D) }2024 \qquad
\textbf{(E) }2025 \qquad
$