Found problems: 85335
Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$.
Alice and Bob are playing "factoring game." On the paper, $270000(=2^43^35^4)$ is written and each person picks one number from the paper(call it $N$) and erase $N$ and writes integer $X,Y$ such that $N=XY$ and $\text{gcd}(X,Y)\ne1.$ Alice goes first and the person who can no longer make this factoring loses. If two people use optimal strategy, prove that Alice always win.
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $f(0) = 0$ and for any real numbers $x, y$ the following equality holds \[f(x^2+yf(x))+f(y^2+xf(y))=f(x+y)^2.\]
We have sheet of paper, divided on $100\times 100$ unit squares. In some squares we put rightangled isosceles triangles with leg =$1$ ( Every triangle lies in one unit square and is half of this square). Every unit grid segment( boundary too) is under one leg of triangle. Find maximal number of unit squares, that don`t contains triangles.
In acute $\triangle ABC,$ let $I$ denote the incenter and suppose that line $AI$ intersects segment $BC$ at a point $D.$ Given that $AI=3, ID=2,$ and $BI^2+CI^2=64,$ compute $BC^2.$
[i]Proposed by Kyle Lee[/i]
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
(Nigeria)
Outside an arbitrary triangle $ABC$, triangles $ADB$ and $BCE$ are constructed such that $\angle ADB=\angle BEC=90^{\circ}$ and $\angle DAB=\angle EBC=30^{\circ}$. On the segment $AC$ the point $F$ with $AF=3FC$ is chosen. Prove that $\angle DFE=90^{\circ}$ and $\angle FDE=30^{\circ}$.
Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that
\[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \]
where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.
Determine all positive integers $k$ such that \[\frac{d(n^{2})}{d(n)}= k\] for some $n \in \mathbb{N}$.
Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$
What is the area of the smallest possible square that can be drawn around a regular hexagon of side length $2$ such that the hexagon is contained entirely within the square?
[i]2018 CCA Math Bonanza Individual Round #9[/i]
Let $m$ be the smallest odd positive integer for which $1+ 2 +...+ m$ is a square of an integer and let $n$ be the smallest even positive integer for which $1 + 2 + ... + n$ is a square of an integer. What is the value of $m + n$?
Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$
Find all possible values of $a$.
Into triangle $ABC$ gives point $K$ lies on bisector of $ \angle BAC$. Line $CK$ intersect circumcircle $ \omega$ of triangle $ABC$ at $M \neq C$. Circle $ \Omega$ passes through $A$, touch $CM$ at $K$ and intersect segment $AB$ at $P \neq A$ and $\omega $ at $Q \neq A$.
Prove, that $P$, $Q$, $M$ lies at one line.
Consider the sequence $x_n>0$ defined with the following recurrence relation:
\[x_1 = 0\]
and for $n>1$ \[(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.\]
Show that if $n$ is a prime number larger or equal to $5$, then $x_n$ is an integer.
Compute the number of ways to color $3$ cells in a $3\times 3$ grid so that no two colored cells share an edge.
Let $n \ge 2$ be a positive integer, with divisors \[1=d_{1}< d_{2}< \cdots < d_{k}=n \;.\] Prove that \[d_{1}d_{2}+d_{2}d_{3}+\cdots+d_{k-1}d_{k}\] is always less than $n^{2}$, and determine when it divides $n^{2}$.
Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1$.
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }9\qquad\textbf{(E) }\text{infinitely many}$
Let $ABCD$ be a convex cyclic quadilateral. Suppose $P$ is a point in the plane of the quadilateral such that the sum of its distances from the vertices of $ABCD$ is the least. If $$\{PC, PB, PC, PD\} = \{3, 4, 6, 8\}$$, what is the maxumum possible area of $ABCD$?
Source: 2018 Canadian Open Math Challenge Part C Problem 2
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Alice has two boxes $A$ and $B$. Initially box $a$ contains $n$ coins and box $B$ is empty. On each turn, she may either move a coin from box $a$ to box $B$, or remove $k$ coins from box $A$, where $k$ is the current number of coins in box $B$. She wins when box $A$ is empty.
$\text{(a)}$ If initially box $A$ contains 6 coins, show that Alice can win in 4 turns.
$\text{(b)}$ If initially box $A$ contains 31 coins, show that Alice cannot win in 10 turns.
$\text{(c)}$ What is the minimum number of turns needed for Alice to win if box $A$ initially contains 2018 coins?
Circle $k$ passes through $A$ and intersects the sides of $\Delta ABC$ in $P,Q$, and $L$. Prove that:
$\frac{S_{PQL}}{S_{ABC}}\leq \frac{1}{4} (\frac{PL}{AQ})^2$.
Prove that the sum of the squares of five consecutive integers cannot be a perfect square.
Jeffrey jogs $3$ miles at $4$ miles per hour. Then, he walks $4$ miles at $3$ miles an hour. What was Jeffrey's average speed in miles per hour for the entire trip? Express your answer as a decimal to the nearest hundredth.
Are there positive integers $m>4$ and $n$, such that
a) ${m \choose 3}=n^2;$
b) ${m \choose 4}=n^2+9?$
Find all the integers written as $\overline{abcd}$ in decimal representation and $\overline{dcba}$ in base $7$.