Found problems: 85335
2009 AMC 10, 10
Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD\equal{}3$, and $ DC\equal{}4$. What is the area of $ \triangle{ABC}$?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21));
pair D=foot(B,A,C);
pair[] ps={B,C,A,D};
draw(A--B--C--cycle);
draw(B--D);
draw(rightanglemark(B,D,C));
dot(ps);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$3$",midpoint(A--D),NE);
label("$4$",midpoint(D--C),NE);[/asy]$ \textbf{(A)}\ 4\sqrt3 \qquad
\textbf{(B)}\ 7\sqrt3 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 14\sqrt3 \qquad
\textbf{(E)}\ 42$
2013 Korea Junior Math Olympiad, 4
Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.
2010 USAJMO, 3
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.
2009 Mathcenter Contest, 4
Find the values of the real numbers $x,y,z$ that correspond to the system of equations.
$$8(x+\frac{1}{x}) =15(y+\frac{1}{y}) = 17(z+\frac{1}{z})$$
$$xy + yz + zx=1$$
[i](Heir of Ramanujan)[/i]
1975 Miklós Schweitzer, 5
Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$
[i]J. Szucs[/i]
2023 Ecuador NMO (OMEC), 2
Let $ABCD$ a cyclic convex quadrilateral. There is a line $l$ parallel to $DC$ containing $A$. Let $P$ a point on $l$ closer to $A$ than to $B$. Let $B'$ the reflection of $B$ over the midpoint of $AD$. Prove that $\angle B'AP = \angle BAC$
2022 IFYM, Sozopol, 7
A graph $ G$ with $ n$ vertices is given. Some $ x$ of its edges are colored red so that each triangle has at most one red edge. The maximum number of vertices in $ G$ that induce a bipartite graph equals $ y.$ Prove that $ n\ge 4x/y.$
1971 IMO Longlists, 3
Let $a, b, c$ be positive real numbers, $0 < a \leq b \leq c$. Prove that for any positive real numbers $x, y, z$ the following inequality holds:
\[(ax+by+cz) \left( \frac xa + \frac yb+\frac zc \right) \leq (x+y+z)^2 \cdot \frac{(a+c)^2}{4ac}.\]
2014 PUMaC Team, 7
Let us consider a function $f:\mathbb{N}\to\mathbb{N}$ for which $f(1)=1$, $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$. Find the number of values at which the maximum value of $f(n)$ is attained for integer $n$ satisfying $0<n<2014$.
2013 Greece Team Selection Test, 3
Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$
Russian TST 2017, P1
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
2001 Moldova National Olympiad, Problem 5
Show that there are nine distinct nonzero integers such that their sum is a perfect square and the sum of any eight of them is a perfect cube.
Ukrainian TYM Qualifying - geometry, 2015.20
What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?
2005 Thailand Mathematical Olympiad, 6
Let $a, b, c$ be distinct real numbers. Prove that
$$\left(\frac{2a - b}{a -b} \right)^2+\left(\frac{2b - c}{b - c} \right)^2+\left(\frac{2c - a}{c - a} \right)^2 \ge 5$$
2009 Estonia Team Selection Test, 1
For arbitrary pairwise distinct positive real numbers $a, b, c$, prove the inequality
$$\frac{(a^2- b^2)^3 + (b^2-c^2)^3+(c^2-a^2)^3}{(a- b)^3 + (b-c)^3+(c-a)^3}> 8abc$$
2014 Baltic Way, 9
What is the least posssible number of cells that can be marked on an $n \times n$ board such that for each $m >\frac{ n}{2}$ both diagonals of any $m \times m$ sub-board contain a marked cell?
2019 Purple Comet Problems, 3
The mean of $\frac12 , \frac34$ , and $\frac56$ differs from the mean of $\frac78$ and $\frac{9}{10}$ by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2008 Vietnam Team Selection Test, 1
On the plane, given an angle $ xOy$. $ M$ be a mobile point on ray $ Ox$ and $ N$ a mobile point on ray $ Oy$. Let $ d$ be the external angle bisector of angle $ xOy$ and $ I$ be the intersection of $ d$ with the perpendicular bisector of $ MN$. Let $ P$, $ Q$ be two points lie on $ d$ such that $ IP \equal{} IQ \equal{} IM \equal{} IN$, and let $ K$ the intersection of $ MQ$ and $ NP$.
$ 1.$ Prove that $ K$ always lie on a fixed line.
$ 2.$ Let $ d_1$ line perpendicular to $ IM$ at $ M$ and $ d_2$ line perpendicular to $ IN$ at $ N$. Assume that there exist the intersections $ E$, $ F$ of $ d_1$, $ d_2$ from $ d$. Prove that $ EN$, $ FM$ and $ OK$ are concurrent.
2015 Iran MO (3rd round), 5
$p>30$ is a prime number. Prove that one of the following numbers is in form of $x^2+y^2$.
$$ p+1 , 2p+1 , 3p+1 , .... , (p-3)p+1$$
2025 Bulgarian Winter Tournament, 10.4
The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.
2019 ELMO Shortlist, G3
Let $\triangle ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. The incircle touches sides $BC,CA,$ and $AB$ at $D,E,$ and $F$ respectively, and $A'$ is the reflection of $A$ over $O$. The circumcircles of $ABC$ and $A'EF$ meet at $G$, and the circumcircles of $AMG$ and $A'EF$ meet at a point $H\neq G$, where $M$ is the midpoint of $EF$. Prove that if $GH$ and $EF$ meet at $T$, then $DT\perp EF$.
[i]Proposed by Ankit Bisain[/i]
2007 Bulgarian Autumn Math Competition, Problem 8.1
Determine all real $a$, such that the solutions to the system of equations
$\begin{cases}
\frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\
(2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a
\end{cases}$
form an interval with length $\frac{32}{225}$.
2016 Switzerland - Final Round, 9
Let $n \ge 2$ be a natural number. For an $n$-element subset $F$ of $\{1, . . . , 2n\}$ we define $m(F)$ as the minimum of all $lcm \,\, (x, y)$ , where $x$ and $y$ are two distinct elements of $F$. Find the maximum value of $m(F)$.
IV Soros Olympiad 1997 - 98 (Russia), 9.8
There is a king in the lower left corner of a chessboard of dimensions $6$ and $6$. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different paths can the king take to the upper right corner of the board?
2023 Assara - South Russian Girl's MO, 4
In a $50 \times 50$ checkered square, each cell is painted in one of $100$ given colors so that all colors are present and it is impossible to cut a single-color domino from the square (i.e. a $1 \times 2$ rectangle). Galiia wants to recolor all the cells of one of the colors into another color (out of the given $100$ colors) so that this condition is preserved (i.e., it is still impossible to cut out a domino of the same color). Is it true that Galiia will definitely be able to do this?