Found problems: 85335
2014 Turkey Team Selection Test, 3
Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds.
Prove that
\[ED=AC-AB \iff R=2r+r_a.\]
2021 CCA Math Bonanza, L1.2
A square is inscribed in a circle of radius $6$. A quarter circle is inscribed in the square, as shown in the diagram below. Given the area of the region inside the circle but outside the quarter circle is $n\pi$ for some positive integer $n$, what is $n$?
[asy]
size(5 cm);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw(circle((1,1),1.41));
draw(arc((0,0),2,0,90));[/asy]
[i]2021 CCA Math Bonanza Lightning Round #1.2[/i]
2022 Harvard-MIT Mathematics Tournament, 9
Let $\Gamma_1$ and $\Gamma_2$ be two circles externally tangent to each other at $N$ that are both internally tangent to $\Gamma$ at points $U$ and $V$ , respectively. A common external tangent of $\Gamma_1$ and $\Gamma_2$ is tangent to $\Gamma_1$ and $\Gamma_2$ at $P$ and $Q$, respectively, and intersects $\Gamma$ at points $X$ and $Y$ . Let $M$ be the midpoint of the arc $XY$ that does not contain $U$ and $V$ . Let $Z$ be on $\Gamma$ such $MZ \perp NZ$, and suppose the circumcircles of $QVZ$ and $PUZ$ intersect at $T\ne Z$. Find, with proof, the value of $T U + T V$ , in terms of $R$, $r_1$, and $r_2$, the radii of $\Gamma$, $\Gamma_1$, and $\Gamma_2$, respectively.
2007 Brazil National Olympiad, 3
Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements.
2021 Saudi Arabia IMO TST, 6
Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying
\[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\]
for all integers $a$ and $b$
2016 EGMO, 6
Let $S$ be the set of all positive integers $n$ such that $n^4$ has a divisor in the range $n^2 +1, n^2 + 2,...,n^2 + 2n$. Prove that there are infinitely many elements of $S$ of each of the forms $7m, 7m+1, 7m+2, 7m+5, 7m+6$ and no elements of $S$ of the form $7m+3$ and $7m+4$, where $m$ is an integer.
2015 CCA Math Bonanza, L2.3
Find the last digit of the number $$\frac{400!}{(200!)(2^{200})}$$
[i]2015 CCA Math Bonanza Lightning Round #2.3[/i]
1997 Moldova Team Selection Test, 11
Let $P(X)$ be a polynomial with real coefficients such that $\{P(n)\}\leq\frac{1}{n}, \forall n\in\mathbb{N}$, where $\{a\}$ is the fractional part of the number $a$. Show that $P(n)\in\mathbb{Z}, \forall n\in\mathbb{N}$.
2014 ELMO Shortlist, 1
Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.
[i]Proposed by Sammy Luo[/i]
2017 Harvard-MIT Mathematics Tournament, 32
Let $a$, $b$, $c$ be non-negative real numbers such that $ab + bc + ca = 3$. Suppose that
\[a^3 b + b^3 c + c^3 a + 2abc(a + b + c) = \frac{9}{2}.\]
What is the maximum possible value of $ab^3 + bc^3 + ca^3$?
2008 Hanoi Open Mathematics Competitions, 1
How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?
2019 Online Math Open Problems, 5
Consider the set $S$ of lattice points $(x,y)$ with $0\le x,y\le 8$. Call a function $f:S\to \{1,2,\dots, 9\}$ a [i]Sudoku function[/i] if:
[list]
[*] $\{ f(x,0), f(x,1), \dots, f(x,8)\} = \{1,2,\dots, 9\}$ for each $0\le x\le 8$ and $\{ f(0,y), f(1,y), \dots, f(8,y) \} = \{1,2,\dots, 9\}$ for each $0\le y\le 8$.
[*] For any integers $0\le m,n\le 2$ and any $0\le i_1,j_1,i_2,j_2\le 2$, $f(3m+i_1, 3n+j_1)\neq f(3m+i_2, 3n+j_2)$ unless $i_1=i_2$ and $j_1=j_2$.
[/list]
Over all Sudoku functions $f$, compute the maximum possible value of $\sum_{0\le i\le 8} f(i,i) + \sum_{0\le i\le 7} f(i, i+1)$.
[i]Proposed by Brandon Wang[/i]
2015 JBMO Shortlist, A3
If $a,b,c$ are positive real numbers prove that: $\frac{a}{b}+\sqrt{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}>2.$
2015 Junior Balkan Team Selection Tests - Romania, 2
Two players, $A$ and $B,$ alternatively take stones from a pile of $n \geq 2$ stones. $A$ plays first and in his first move he must take at least one stone and at most $n-1$ stones. Then each player must take at least one stone and at most as many stones as his opponent took in the previous move. The player who takes the last stone wins. Which player has a winning strategy?
2022 CCA Math Bonanza, T1
Let $a$, $b$, $c$, and $d$ be positive integers such that $77^a \cdot 637^b = 143^c \cdot 49^d$. Compute the minimal value of $a+b+c+d$.
[i]2022 CCA Math Bonanza Team Round #1[/i]
Today's calculation of integrals, 862
Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$
2021 USA TSTST, 2
Let $a_1<a_2<a_3<a_4<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence
\[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\]
[i]Merlijn Staps[/i]
1991 ITAMO, 2
Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.
2006 India Regional Mathematical Olympiad, 4
A $ 6\times 6$ square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always [b]two[/b] congruent rectangles.
2009 Ukraine National Mathematical Olympiad, 1
Pairwise distinct real numbers $a, b, c$ satisfies the equality
\[a +\frac 1b =b + \frac 1c =c+\frac 1a.\]
Find all possible values of $abc .$
2018 Junior Regional Olympiad - FBH, 2
On blackboard is written $3$ digit number so all three digits are distinct than zero. Out of it, we made three $2$ digit numbers by crossing out first digit of original number, crossing out second digit of original number and crossing out third digit of original number. Sum of those three numbers is $293$. Which number is written on blackboard?
1995 Bulgaria National Olympiad, 3
Two players $A$ and $B$ take stones one after the other from a heap with $n \ge 2$ stones. $A$ begins the game and takes at least one stone, but no more than $n -1$ stones. Thereafter, a player on turn takes at least one, but no more than the other player has taken before him. The player who takes the last stone wins. Who of the players has a winning strategy?
2011 Ukraine Team Selection Test, 3
Given a positive integer $ n> 2 $. Prove that there exists a natural $ K $ such that for all integers $ k \ge K $ on the open interval $ ({{k} ^{n}}, \ {{(k + 1)} ^{n}}) $ there are $n$ different integers, the product of which is the $n$-th power of an integer.
2023 Chile TST Ibero., 4
Let \(ABC\) be a triangle with \(AB < AC\) and let \(\omega\) be its circumcircle. Let \(M\) denote the midpoint of side \(BC\) and \(N\) the midpoint of arc \(BC\) of \(\omega\) that contains \(A\). The circumcircle of triangle \(AMN\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. Prove that \(BP = CQ\).
2003 Romania National Olympiad, 3
Prove that the midpoints of the altitudes of a triangle are collinear if and only if the triangle is right.
[i]Dorin Popovici[/i]