Found problems: 85335
A convex polygon is divided into finitely many quadrilaterals. Prove that at least one of these quadrilaterals must also be convex.
$ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$.
Let $n \geq 3$ be an integer. Rowan and Colin play a game on an $n \times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is [i]orderly[/i] if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of $n$, how many orderly colorings are there?
[i]Proposed by Alec Sun[/i]
In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red. Thereafter, $7$ out of every $8$ counted were red. If, in all, $90\%$ or more of the balls counted were red, the maximum value of $n$ is:
$\textbf{(A)}\ 225 \qquad
\textbf{(B)}\ 210 \qquad
\textbf{(C)}\ 200 \qquad
\textbf{(D)}\ 180 \qquad
\textbf{(E)}\ 175$
In an acute-angled triangle, one of the angles is $60^o$. Prove that the line passing through the center of the circumcircle and the intersection point of the medians of the triangle cuts off an equilateral triangle from it.
(A. Zaslavsky)
Let b be a given real number. The sequence of integers $a_1, a_2,a_3, ...$ is such that $a_1 =(b]$ and $a_{n+1}=(a_n+b]$ for all $n\ge 1$ Prove that the sum $a_1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n}$ is an integer number for any natural $n$ .
(In the condition of the problem, $(x]$ denotes the smallest integer that is greater than or equal to $x$)
In $\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$?
[asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE); label("$E$",E,NW); label("$F$",F,1.5*N); [/asy]
$\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2$
Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear.
[i]Proposed by Michael Kural[/i]
Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\
a^2 = 2(a+b+c)\\
\end{array} \right.
\]
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
For any positive reals $x$, $y$, $z$ with $xyz + xy + yz + zx = 4$, prove that
$$\sqrt{\frac{xy+x+y}{z}}+\sqrt{\frac{yz+y+z}{x}}+\sqrt{\frac{zx+z+x}{y}}\geq 3\sqrt{\frac{3(x+2)(y+2)(z+2)}{(2x + 1)(2y + 1)(2z + 1).
}}$$
Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.
Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation
\[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]
Let $P$ be an interior point of the triangle $ABC$ which is not on the median belonging to $BC$ and satisfying $\angle CAP = \angle BCP. \: BP \cap CA = \{B'\} \: , \: CP \cap AB = \{C'\}$ and $Q$ is the second point of intersection of $AP$ and the circumcircle of $ABC. \: B'Q$ intersects $CC'$ at $R$ and $B'Q$ intersects the line through $P$ parallel to $AC$ at $S.$ Let $T$ be the point of intersection of lines $B'C'$ and $QB$ and $T$ be on the other side of $AB$ with respect to $C.$ Prove that
\[\angle BAT = \angle BB'Q \: \Longleftrightarrow \: |SQ|=|RB'| \]
Let $a_n$ be a sequence de fined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$.
Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$.
Find all positive integers $n,k_1,\dots,k_n$ such that $k_1+\cdots+k_n=5n-4$ and
\[ \frac1{k_1}+\cdots+\frac1{k_n}=1. \]
Consider the acute-angled triangle $ABC$. Let $X$ be a point on the side
$BC$, and $Y$ be a point on the side $CA$. The circle $k_1$ with diameter $AX$ cuts
$AC$ again at $E'$ .The circle $k_2$ with diameter $BY$ cuts $BC$ again at $B'$.
(i) Let $M$ be the midpoint of $XY$ . Prove that $A'M = B'M$.
(ii) Suppose that $k_1$ and $k_2$ meet at $P$ and $Q$. Prove that the orthocentre
of $ABC$ lies on the line $PQ$.
We want to place $2012$ pockets, including variously colored balls, into $k$ boxes such that
[b]i)[/b] For any box, all pockets in this box must include a ball with the same color
or
[b]ii)[/b] For any box, all pockets in this box must include a ball having a color which is not included in any other pocket in this box
Find the smallest value of $k$ for which we can always do this placement whatever the number of balls in the pockets and whatever the colors of balls.
A baseball league has $64$ people, each with a different $6$-digit binary number whose base-$10$ value ranges from $0$ to $63$. When any player bats, they do the following: for each pitch, they swing if their corresponding bit number is a $1$, otherwise, they decide to wait and let the ball pass. For example, the player with the number $11$ has binary number $001011$. For the first and second pitch, they wait; for the third, they swing, and so on. Pitchers follow a similar rule to decide whether to throw a splitter or a fastball, if the bit is $0$, they will throw a splitter, and if the bit is $1$, they will throw a fastball.
If a batter swings at a fastball, then they will score a hit; if they swing on a splitter, they will miss and get a “strike.” If a batter waits on a fastball, then they will also get a strike. If a batter waits on a splitter, then they get a “ball.” If a batter gets $3$ strikes, then they are out; if a batter gets $4$ balls, then they automatically get a hit. For example, if player $11$ pitched against player $6$ (binary is $000110$), the batter would get a ball for the first pitch, a ball for the second pitch, a strike for the third pitch, a strike for the fourth pitch, and a hit for the fifth pitch; as a result, they will count that as a “hit.” If player $11$ pitched against player $5$ (binary is $000101$), however, then the fifth pitch would be the batter’s third strike, so the batter would be “out.”
Each player in the league plays against every other player exactly twice; once as batter, and once as pitcher. They are then given a score equal to the number of outs they throw as a pitcher plus the number of hits they get as a batter. What is the highest score received?
Let a sequence $(a_i)_{i=10}^{\infty}$ be defined as follows:
[list=a]
[*] $a_{10}$ is some positive integer, which can of course be written in base 10.
[*] For $i \geq 10$ if $a_i > 0$, let $b_i$ be the positive integer whose base-$(i + 1)$ representation is the same as $a_i$'s base-$i$ representation. Then let $a_{i + 1} = b_i - 1$. If $a_i = 0$, $a_{i + 1} = 0$.
[/list]
For example, if $a_{10} = 11$, then $b_{10} = 11_{11} (= 12_{10})$; $a_{11} = 11_{11} - 1 = 10_{11} (= 11_{10})$; $b_{11} = 10_{12} (= 12_{10})$; $a_{12} = 11$.
Does there exist $a_{10}$ such that $a_i$ is strictly positive for all $i \geq 10$?
Let $C_1$ be a circle with center $O$ and let $B$ and $C$ be points in $C_1$ such that $BOC$ is an equilateral triangle. Let $D$ be the midpoint of the minor arc $BC$ of $C_1$. Let $C_2$ be the circle with center $C$ that passes through $B$ and $O$. Let $E$ be the second intersection of $C_1$ and $C_2$. The parallel to $DE$ through $B$ intersects $C_1$ for second time in $A$. Let $C_3$ be the circumcircle of triangle $AOC$. The second intersection of $C_2$ and $C_3$ is $F$. Show that $BE$ and $BF$ trisect the angle $\angle ABC$.
For any $n\in\mathbb N$, denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$, where the last summand consists of $n$ digits of $2$. Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$.
Let $A$ be a set of $n$ non-negative integers. We say it has property $\mathcal P$ if the set $\{x + y \mid x, y \in A\}$ has $\binom{n}{2}$ elements. We call the largest element of $A$ minus the smallest element, the diameter of $A$. Let $f(n)$ be the smallest diameter of any set $A$ with property $\mathcal P$. Show that $n^2 \leq 4 f(n) < 4 n^3$.
[hide="Comment"](If you have some amount of time, try a best estimative for $f(n)$, such that $f(p)<2p^2$ for prime $p$).[/hide]
There are $2022$ students in winter school. Two arbitrary students are friend or enemy each other. Each turn, we choose a student $S$, make friends of $S$ enemies, and make enemies of $S$ friends. This continues until it satisfies the final condition.
[b]Final Condition[/b] : For any partition of students into two non-empty groups $A$, $B$, there exist two students $a$, $b$ such that $a\in A$, $b\in B$, and $a$, $b$ are friend each other.
Determine the minimum value of $n$ such that regardless of the initial condition, we can satisfy the final condition with no more than $n$ turns.
We say a non-negative integer $n$ "[i]contains[/i]" another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ [i]contains[/i] $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not [i]contain[/i] a multiple of $7$.