Found problems: 85335
Find the derivative of the solution of the equation $\ddot{x} =x + A\dot{x}^2$, with initial conditions $x(0) = 1$, $\dot{x}(0) = 0$, with respect to the parameter $A$ for $A = 0$.
The point $O$ is the center of the circle circumscribed of the acute triangle $ABC$, and $H$ is the point of intersection of the heights of this triangle. Let $A_1, B_1, C_1$ be the points diametrically opposed to the vertices $A, B , C$ respectively of the triangle, and $A_2, B_2, C_2$ be the midpoints of the segments $[AH], [BH] ¸[CH]$ respectively . Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ are concurrent .
Prove that for each $n \ge 4$ a parallelogram can be dissected in $n$ cyclic quadrilaterals.
Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?
In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.
$ABC$ is a non-isosceles triangle. $O, I, H$ are respectively the center of its circumscribed circle, the inscribed circle and its orthocenter.
prove that $\widehat{OIH}$ is obtuse.
Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.
Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.
Find all prime numbers $p$ such that $\frac{3^{p-1} - 1}{p}$ is a perfect square.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(x)^2+f(2y+1)=x^2+f(y)+y+1\] for all reals $x$, $y$.
[i](Proposed by Lim Yun Zhe)[/i]
Let $n,k$ be given natural numbers. Determine all ordered n-tuples of non-negative real numbers $(x_1,x_2,...,x_n)$ that satisfy the system of equations
$$x_1^k+x_2^k+...+x_n^k=1$$
$$(1+x_1)(1+x_2)...(1+x_n)=2$$
Jonathan has a magical coin machine which takes coins in amounts of $7, 8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ coins, what is the minimum number of entries he can make to end up with $4$ coins?
Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.
Two players $ A$, $ B$ and another 2001 people form a circle, such that $ A$ and $ B$ are not in consecutive positions. $ A$ and $ B$ play in alternating turns, starting with $ A$. A play consists of touching one of the people neighboring you, which such person once touched leaves the circle. The winner is the last man standing.
Show that one of the two players has a winning strategy, and give such strategy.
Note: A player has a winning strategy if he/she is able to win no matter what the opponent does.
Given an integer $n>1$, let $S_n$ be the group of permutations of the numbers $1,\;2,\;3,\;\ldots,\;n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_n$. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group $S_n$. The player who made the last move loses the game. The first move is made by A. Which player has a winning strategy?
[i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]
The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime.
Show that at least one of these numbers is prime.
If $xy = 15$ and $x + y = 11$, calculate the value of $x^3 + y^3$.
Let $a,b,c,d \in [0,1] .$ Prove that$$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+d}+\frac{1}{1+d+a}\leq \frac{4}{1+2\sqrt[4]{abcd}}$$
Given a sequence $\{a_n\}$ whose terms are non-zero real numbers. For any positive integer $n$, the equality
\[(\sum_{i=1}^{n}a_i)^2=\sum_{i=1}^{n}a_i^3\]
holds.
[b](1)[/b] If $n=3$, find all possible sequence $a_1,a_2,a_3$;
[b](2)[/b] Does there exist such a sequence $\{a_n\}$ such that $a_{2011}=-2012$?
In the triangle $ABC$ points $M$ and $N$ are the midpoints of sides $AC$ and $AB$ respectively. $I$ is the incenter of the triangle. It is known that the angle $MIC$ is a right angle.
Find the angle $NIB$.
Let $ABC$ be an acute triangle. Let $M$ be the midpoint of the segment $BC$. Let $I, J, K$ be the incenters of triangles $ABC$, $ABM$, $ACM$, respectively. Let $P, Q$ be points on the lines $MK$, $MJ$, respectively, such that $\angle AJP=\angle ABC$ and $\angle AKQ=\angle BCA$. Let $R$ be the intersection of the lines $CP$ and $BQ$. Prove that the lines $IR$ and $BC$ are perpendicular.
[color=darkred]Let $a$ , $b$ and $c$ be three complex numbers such that $a+b+c=0$ and $|a|=|b|=|c|=1$ . Prove that:
\[3\le |z-a|+|z-b|+|z-c|\le 4,\]
for any $z\in\mathbb{C}$ , $|z|\le 1\, .$[/color]
What are the last $3$ digits of $1!+2!+\cdots +100!$
Points $P$, $Q$, and $M$ lie on a circle $\omega$ such that $M$ is the midpoint of minor arc $PQ$ and $MP=MQ=3$. Point $X$ varies on major arc $PQ$, $MX$ meets segment $PQ$ at $R$, the line through $R$ perpendicular to $MX$ meets minor arc $PQ$ at $S$, $MS$ meets line $PQ$ at $T$. If $TX=5$ when $MS$ is minimized, what is the minimum value of $MS$?
[i]2019 CCA Math Bonanza Team Round #9[/i]
In a class, there are $n \geqslant 3$ students and a teacher with $M$ marbles. The teacher then play a [i]Marble distribution[/i] according to the following rules. At the start, the teacher distributed all her marbles to students, so that each student receives at least $1$ marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of $2(n-1)$. That chosen student then equally distribute half of his marbles to $n-1$ other students. The same goes on until the teacher is not able to choose anymore student.
Find all integer $M$, such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.