Found problems: 85335
2014-2015 SDML (Middle School), 1
Given that each unit square in the grid below is a $1\times1$ square, find the area of the shaded region in square units.
[asy]
fill((3,0)--(4,0)--(6,3)--(4,4)--(4,3)--(0,2)--(2,2)--cycle, grey);
draw((0,0)--(6,0));
draw((0,1)--(6,1));
draw((0,2)--(6,2));
draw((0,3)--(6,3));
draw((0,4)--(6,4));
draw((0,0)--(0,4));
draw((1,0)--(1,4));
draw((2,0)--(2,4));
draw((3,0)--(3,4));
draw((4,0)--(4,4));
draw((5,0)--(5,4));
draw((6,0)--(6,4));
[/asy]
$\text{(A) }8\qquad\text{(B) }9\qquad\text{(C) }10\qquad\text{(D) }11\qquad\text{(E) }12$
Ukraine Correspondence MO - geometry, 2003.11
Let $ABCDEF$ be a convex hexagon, $P, Q, R$ be the intersection points of $AB$ and $EF$, $EF$ and $CD$, $CD$ and $AB$. $S, T,UV$ are the intersection points of $BC$ and $DE$, $DE$ and $FA$, $FA$ and $BC$, respectively. Prove that if $$\frac{AB}{PR}=\frac{CD}{RQ}=\frac{EF}{QP},$$ then $$\frac{BC}{US}=\frac{DE}{ST}=\frac{FA}{TU}.$$
2004 Federal Math Competition of S&M, 1
Suppose that $a,b,c$ are positive numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is an integer. Show that $abc$ is a perfect cube.
2019 China Team Selection Test, 6
Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time.
Determine all $k$ such that $A$ can always win the game.
2005 Bundeswettbewerb Mathematik, 2
Let be $x$ a rational number.
Prove: There are only finitely many triples $(a,b,c)$ of integers with $a<0$ and $b^2-4ac=5$ such that $ax^2+bx+c$ is positive.
2008 National Olympiad First Round, 20
Each of the integers $a_1,a_2,a_3,\dots,a_{2008}$ is at least $1$ and at most $5$. If $a_n < a_{n+1}$, the pair $(a_n, a_{n+1})$ will be called as an increasing pair. If $a_n > a_{n+1}$, the pair $(a_n, a_{n+1})$ will be called as an decreasing pair. If the sequence contains $103$ increasing pairs, at least how many decreasing pairs are there?
$
\textbf{(A)}\ 21
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 36
\qquad\textbf{(D)}\ 102
\qquad\textbf{(E)}\ \text{None of the above}
$
2010 Today's Calculation Of Integral, 574
Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,
2022 Junior Balkan Team Selection Tests - Romania, P3
Determine all pairs of positive integers $(a,b)$ such that the following fraction is an integer: \[\frac{(a+b)^2}{4+4a(a-b)^2}.\]
2012 Sharygin Geometry Olympiad, 4
Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices.
(B.Frenkin)
2023 Austrian MO National Competition, 4
The number $2023$ is written $2023$ times on a blackboard. On one move, you can choose two numbers $x, y$ on the blackboard, delete them and write $\frac{x+y} {4}$ instead. Prove that when one number remains, it is greater than $1$.
TNO 2008 Junior, 8
A traffic accident involved three cars: one blue, one green, and one red. Three witnesses spoke to the police and gave the following statements:
**Person 1:** The red car was guilty, and either the green or the blue one was involved.
**Person 2:** Either the green car or the red car was guilty, but not both.
**Person 3:** Only one of the cars was guilty, but it was not the blue one.
The police know that at least one car was guilty and that at least one car was not. However, the police do not know if any of the three witnesses lied.
Which car(s) were responsible for the accident?
2016 Cono Sur Olympiad, 1
Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers.
[b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.
2006 Hanoi Open Mathematics Competitions, 6
On the circle of radius $30$ cm are given $2$ points A,B with $AB = 16$ cm and $C$ is a midpoint of $AB$. What is the perpendicular distance from $C$ to the circle?
2022 Czech and Slovak Olympiad III A, 6
Consider any graph with $50$ vertices and $225$ edges. We say that a triplet of its (mutually distinct) vertices is [i]connected[/i] if the three vertices determine at least two edges. Determine the smallest and the largest possible number of connected triples.
[i](Jan Mazak, Josef Tkadlec)[/i]
2023 India Regional Mathematical Olympiad, 6
Consider a set of $16$ points arranged in $4 \times 4$ square grid formation. Prove that if any $7$ of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.
1994 French Mathematical Olympiad, Problem 1
For each positive integer $n$, let $I_n$ denote the number of integers $p$ for which $50^n<7^p<50^{n+1}$.
(a) Prove that, for each $n$, $I_n$ is either $2$ or $3$.
(b) Prove that $I_n=3$ for infinitely many $n\in\mathbb N$, and find at least one such $n$.
2011 Purple Comet Problems, 7
Find the prime number $p$ such that $71p + 1$ is a perfect square.
1961 Miklós Schweitzer, 3
[b]3.[/b] Let $f(x)= x^n +a_1 x^(n-1)+ \dots + a_n$ ($n\geq 1$) be an irreducible polynomial over the field $K$. Show that every non-zero matrix commuting with the matrix
$
\begin{bmatrix}
0 & 1 & 0 & \dots & 0 & 0 \\
0 & 0 & 1 & \dots & 0 & 0 \\
\dots & \dots & \dots & \dots & \dots & \dots \\
0 & 0 & 0 & \dots & 0 & 1 \\
-a_n & -a_{n-1} & -a_{n-2} & \dots & -a_2 & -a_1
\end{bmatrix} $
is invertible. [b](A. 4)[/b]
2021 Francophone Mathematical Olympiad, 2
Albert and Beatrice play a game. $2021$ stones lie on a table. Starting with Albert, they alternatively remove stones from the table, while obeying the following rule. At the $n$-th turn, the active player (Albert if $n$ is odd, Beatrice if $n$ is even) can remove from $1$ to $n$ stones. Thus, Albert first removes $1$ stone; then, Beatrice can remove $1$ or $2$ stones, as she wishes; then, Albert can remove from $1$ to $3$ stones, and so on.
The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player's moves?
1965 All Russian Mathematical Olympiad, 068
Given two relatively prime numbers $p>0$ and $q>0$. An integer $n$ is called "good" if we can represent it as $n = px + qy$ with nonnegative integers $x$ and $y$, and "bad" in the opposite case.
a) Prove that there exist integer $c$ such that in a pair $\{n, c-n\}$ always one is "good" and one is "bad".
b) How many there exist "bad" numbers?
1994 Cono Sur Olympiad, 2
Consider a circle $C$ with diameter $AB=1$. A point $P_0$ is chosen on $C$, $P_0 \ne A$, and starting in $P_0$ a sequence of points $P_1, P_2, \dots, P_n, \dots$ is constructed on $C$, in the following way: $Q_n$ is the symmetrical point of $A$ with respect of $P_n$ and the straight line that joins $B$ and $Q_n$ cuts $C$ at $B$ and $P_{n+1}$ (not necessary different). Prove that it is possible to choose $P_0$ such that:
[b]i[/b] $\angle {P_0AB} < 1$.
[b]ii[/b] In the sequence that starts with $P_0$ there are $2$ points, $P_k$ and $P_j$, such that $\triangle {AP_kP_j}$ is equilateral.
2007 Iran Team Selection Test, 1
Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]
1980 Canada National Olympiad, 3
Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.
2008 Middle European Mathematical Olympiad, 3
Let $ ABC$ be an acute-angled triangle. Let $ E$ be a point such $ E$ and $ B$ are on distinct sides of the line $ AC,$ and $ D$ is an interior point of segment $ AE.$ We have $ \angle ADB \equal{} \angle CDE,$ $ \angle BAD \equal{} \angle ECD,$ and $ \angle ACB \equal{} \angle EBA.$ Prove that $ B, C$ and $ E$ lie on the same line.
2016 All-Russian Olympiad, 8
Medians $AM_A,BM_B,CM_C$ of triangle $ABC$ intersect at $M$.Let $\Omega_A$ be circumcircle of triangle passes through midpoint of $AM$ and tangent to $BC$ at $M_A$.Define $\Omega_B$ and $\Omega_C$ analogusly.Prove that $\Omega_A,\Omega_B$ and $\Omega_C$ intersect at one point.(A.Yakubov)
[hide=P.S]sorry for my mistake in translation :blush: :whistling: .thank you jred for your help :coolspeak: [/hide]