Found problems: 85335
Find out if there is a convex pentagon $A_1A_2A_3A_4A_5$ such that, for each $i = 1, \dots , 5$, the lines $A_iA_{i+3}$ and $A_{i+1}A_{i+2}$ intersect at a point $B_i$ and the points $B_1,B_2,B_3,B_4,B_5$ are collinear. (Here $A_{i+5} = A_i$.)
Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.
For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.
(a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$.
(b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.
Let $n$ be a positive integer. Suppose there are exactly $M$ squarefree integers $k$ such that $\left\lfloor\frac nk\right\rfloor$ is odd in the set $\{ 1, 2,\ldots, n\}$. Prove $M$ is odd.
An integer is [i]squarefree[/i] if it is not divisible by any square other than $1$.
The circle $\omega_1$ intersects the circle $\omega_2$ in the points $A$ and $B$, a tangent line to this circles intersects $\omega_1$ and $\omega_2$ in the points $E$ and $F$ respectively. Suppose that $A$ is inside of the triangle $BEF$, let $H$ be the orthocenter of $BEF$ and $M$ is the midpoint of $BH$. Prove that the centers of the circles $\omega_1$ and $\omega_2$ and the point $M$ are collinears.
Given a simple graph with $ 4038 $ vertices. Assume we arbitrarily choose $ 2019 $ vertices as a group (the other $ 2019 $ is another group, of course), there are always $ k $ edges that connect two groups. Find all possible value of $ k $.
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?
$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$
$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$
$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$
An ordered pair $(n,p)$ is [i]juicy[/i] if $n^{2} \equiv 1 \pmod{p^{2}}$ and $n \equiv -1 \pmod{p}$ for positive integer $n$ and odd prime $p$. How many juicy pairs exist such that $n,p \leq 200$?
Proposed by Harry Chen (Extile)
Prove that for any positive integer $ k$, there exists an arithmetic sequence $ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3}, ... ,\frac{a_k}{b_k}$ of rational numbers, where $ a_i, b_i$ are relatively prime positive integers for each $ i \equal{} 1,2,...,k$ such that the positive integers $ a_1, b_1, a_2, b_2, ..., a_k, b_k$ are all distinct.
Let be given two positive integers $ m$, $ n$ with $ m < 2001$, $ n < 2002$. Let distinct real numbers be written in the cells of a $ 2001 \times 2002$ board (with $ 2001$ rows and $ 2002$ columns). A cell of the board is called [i]bad[/i] if the corresponding number is smaller than at least $ m$ numbers in the same column and at least $ n$ numbers in the same row. Let $ s$ denote the total number of [i]bad[/i] cells. Find the least possible value of $ s$.
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $T$. Let $X$ be a point on circle $\omega_1$. Line $l_1$ is tangent to circle $\omega_1$ and $X$, and line $l$ intersects circle $\omega_2$ at $A$ and $B$. Line $XT$ meets circle $\omega$ at $S$. Point $C$ lies on arc $TS$ (of circle $\omega_2$, not containing points $A$ and $B$). Point $Y$ lies on circle $\omega_1$ and line $YC$ is tangent to circle $\omega_1$. Let $I$ be the intersection of lines $XY$ ad $SC$. Prove that...
a) points $C$, $T$, $Y$, $I$ lie on a circle
(B) $I$ is an excenter of triangle $ABC$.
Given the $7$-element set $A = \{ a ,b,c,d,e,f,g \}$, find a collection $T$ of $3$-element subsets of $A$ such that each pair of elements from $A$ occurs exactly once on one of the subsets of $T$.
A convex hexagon $A_1B_1A_2B_2A_3B_3$ it is inscribed in a circumference $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concurrent in $X$. For each $i=1,2,3$ let $\omega_i$ tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain the other vertices of the hexagon; let $r_i$ the radius of $\omega_i$.
$(a)$ Prove that $R\geq r_1+r_2+r_3$
$(b)$ If $R= r_1+r_2+r_3$, prove that the six points of tangency of the circumferences $\omega_i$ with the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic
For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer $n$ satisfying $s(n)=s(n+864)=20$.
$2025$ light bulbs are operated by some switches. Each switch works on a subset of the light bulbs. When we use a switch, all the light bulbs in the subset change their state: bulbs that were on turn off, and bulbs that were off turn on. We know that every light bulb is operated by at least one of the switches. Initially, all lamps were off. Find the biggest number $k$ for which we can surely turn on at least $k$ light bulbs.
[i]Based on an OKTV problem[/i]
Find all possible values of $$\frac{1}{a}\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{b+c}\right)+\frac{1}{b}\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{c+a}\right)+\frac{1}{c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{a+b}\right)-\frac{1}{a+b+c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$$ where $a$, $b$ and $c$ are positive real numbers such that $ab+bc+ca=abc$
Pedro and Juan are playing the following game:
$-$ There are $2$ piles of rocks, with $X$ rocks in one pile and $Y$ rocks in the other pile ($X < 12, Y < 11$).
$-$ Each player can draw:
-- $1$ rock from one of the piles, or
-- $2$ rocks from one of the piles, or
-- $1$ rock from each pile, or
-- $2$ rock from one pile and $1$ from the other pile.
Each player must perform one of these four operations in their turns.
The looser is the one who takes the last rock.
Pedro plays first and has a winning strategy.
What are the three maximum possible values of ($X+Y$)?
Byan rolls a $12$-sided die, a $14$-sided die, a $20$-sided die, and a $24$-sided die. The probability the sum of the numbers the die landed on is divisible by $7$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Lightning 3.1[/i]
When $p = \sum_{k=1}^{6} k \ln{k}$, the number $e^p$ is an integer. What is the largest power of $2$ that is a factor of $e^p$?
${\textbf{(A)}\ 2^{12}\qquad\textbf{(B)}\ 2^{14}\qquad\textbf{(C)}\ 2^{16}\qquad\textbf{(D)}}\ 2^{18}\qquad\textbf{(E)}\ 2^{20} $
Let $[AB]$ be a diameter of a circle with center $O$ and radius $1$. Consider $P$ a point on the circumference, different from $A$ and $B$ and let $Q$ be the midpoint of the arc $AP$. The line parallel to $PQ$ that passes through $O$ intersects the line $PB$ at point $S$. Determine $\overline{PS}$.
Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with
every possible height-weight combination.
The USAMTS wants to field a competitive team, so there are some strict requirements.
[list]
[*] If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team.
[*] If person $P$ is on the team, then no one whose weight is the same as $P$’s height can also be on the team.
[/list]
Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed?
Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square their numbers to get a new number and see that the sum of these new numbers is half of $169$. Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's by $5070$. What was the sum of their original numbers?
$\text{(A) }-4\qquad\text{(B) }-3\qquad\text{(C) }1\qquad\text{(D) }2\qquad\text{(E) }5$
Each cell of the board with $2021$ rows and $2022$ columns contains exactly one of the three letters $T$, $M$, and $O$ in a way that satisfies each of the following conditions:
[list]
[*] In total, each letter appears exactly $2021\times 674$ of times on the board.
[*] There are no two squares that share a common side and contain the same letter.
[*] Any $2\times 2$ square contains all three letters $T$, $M$, and $O$.
[/list]
Prove that each letter $T$, $M$, and $O$ appears exactly $674$ times on every row.
Let $ABCD$ be a rectangle with sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $ A$ intersects $BD$ at the point $H$. We denote by $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the length of segment $MN$.
Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.