Found problems: 85335
2009 Macedonia National Olympiad, 5
Solve the following equation in the set of integer numbers:
\[ x^{2010}-2006=4y^{2009}+4y^{2008}+2007y. \]
2004 Germany Team Selection Test, 2
Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles.
[i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]
I Soros Olympiad 1994-95 (Rus + Ukr), 9.4
Use a compass and a ruler to construct a triangle, given the intersection point of its median, the orthocenter, and one from the vertices.
2005 Morocco National Olympiad, 4
$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that
\[bc<2a^2<4bc\]
2007 Baltic Way, 20
Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.
2025 AIME, 3
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such way that each square has $2$ red sides and blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings.
[asy]
size(4cm);
defaultpen(linewidth(1.2));
draw((0, 0) -- (2, 0) -- (2, 1));
draw((0, 1) -- (1, 1) -- (1, 2) -- (2,2));
draw((0, 0) -- (0, 1), dotted);
draw((1, 0) -- (1, 1) -- (2, 1) -- (2, 2), dotted);
draw((0, 1) -- (0, 2) -- (1, 2), dotted);
[/asy]
2014 JHMMC 7 Contest, 26
Alex is training to make $\text{MOP}$. Currently he will score a $0$ on $\text{the AMC,}\text{ the AIME,}\text{and the USAMO}$. He can expend $3$ units of effort to gain $6$ points on the $\text{AMC}$, $7$ units of effort to gain $10$ points on the $\text{AIME}$, and $10$ units of effort to gain $1$ point on the $\text{USAMO}$. He will need to get at least $200$ points on $\text{the AMC}$ and $\text{AIME}$ combined and get at least $21$ points on $\text{the USAMO}$ to make $\text{MOP}$. What is the minimum amount of effort he can expend to make $\text{MOP}$?
2018 Estonia Team Selection Test, 6
We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)
2020 Canadian Junior Mathematical Olympiad, 2
Ziquan makes a drawing in the plane for art class. He starts by placing his pen at the origin, and draws a series of line segments, such that the $n^{th}$ line segment has length $n$. He is not allowed to lift his pen, so that the end of the $n^{th}$ segment is the start of the $(n + 1)^{th}$ segment. Line segments drawn are allowed to intersect and even overlap previously drawn segments.
After drawing a finite number of line segments, Ziquan stops and hands in his drawing to his art teacher. He passes the course if the drawing he hands in is an $N$ by $N$ square, for some positive integer $N$, and he fails the course otherwise. Is it possible for Ziquan to pass the course?
Kyiv City MO Juniors 2003+ geometry, 2021.8.41
On the sides $AB$ and $BC$ of the triangle $ABC$, the points $K$ and $M$ are chosen so that $KM \parallel AC$. The segments $AM$ and $KC$ intersect at the point $O$. It is known that $AK =AO$ and $KM =MC$. Prove that $AM=KB$.
1967 IMO, 2
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
1997 Romania National Olympiad, 3
The triangle $ABC$ has $\angle ACB = 30^o$, $BC = 4$ cm and $AB = 3$ cm . Compute the altitudes of the triangle.
2010 Stanford Mathematics Tournament, 4
Compute $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}...}$
2004 Switzerland - Final Round, 10
Let $n > 1$ be an odd natural number. The squares of an $n \times n$ chessboard are alternately colored white and black so that the four corner squares are black. An $L$-triomino is an $L$-shaped piece that covers exactly three squares of the board. For which values of $n$ is it possible to cover all black squares with $L$-triominoes, so that no two $L$-triominos overlap? For these values of $n$ determine the smallest possible number of $L$-triominoes that are necessary for this.
2010 AMC 12/AHSME, 11
The solution of the equation $ 7^{x\plus{}7}\equal{}8^x$ can be expressed in the form $ x\equal{}\log_b 7^7$. What is $ b$?
$ \textbf{(A)}\ \frac{7}{15} \qquad
\textbf{(B)}\ \frac{7}{8} \qquad
\textbf{(C)}\ \frac{8}{7} \qquad
\textbf{(D)}\ \frac{15}{8} \qquad
\textbf{(E)}\ \frac{15}{7}$
2023 Stanford Mathematics Tournament, 1
Compute the area of the polygon formed by connecting the roots of
\[x^{10} + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + x + 1\]
graphed in the complex plane with line segments in counterclockwise order.
2023 AMC 12/AHSME, 17
Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. What is the probability that Flora will eventually land at $10$?
$\textbf{(A) } \frac{5}{512} \qquad \textbf{(B) } \frac{45}{1024} \qquad \textbf{(C) } \frac{127}{1024} \qquad \textbf{(D) } \frac{511}{1024} \qquad \textbf{(E) } \frac{1}{2}$
2006 AIME Problems, 15
Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|$.
2001 Greece JBMO TST, 3
$4$ men stand at the entrance of a dark tunnel. Man $A$ needs $10$ minutes to pass through the tunnel, man $B$ needs $5$ minutes, man $C$ needs $2$ minutes and man $D$ needs $1$ minute. There is only one torch, that may be used from anyone that passes through the tunnel. Additionaly, at most $2$ men can pass through at the same time using the existing torch.
Determine the smallest possible time the four men need to reach the exit of the tunnel.
1986 IMO Longlists, 64
Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n.$
2023 Math Prize for Girls Olympiad, 1
Let $n \ge 2023$ be an integer. Prove that there exists a permutation $(p_1, p_2, \dots, p_n)$ of $(1, 2, \dots, n)$ such that
\[
p_1 + 2p_2 + 3p_3 + \dots + np_n
\]
is divisible by $n$.
1994 AIME Problems, 6
The graphs of the equations \[ y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k, \] are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.$ These 63 lines cut part of the plane into equilateral triangles of side $2/\sqrt{3}.$ How many such triangles are formed?
1995 Tournament Of Towns, (466) 4
From the vertex $A$ of a triangle $ABC$, three segments are drawn: the bisectors $AM$ and $AN$ of its interior and exterior angles and the tangent $AK$ to the circumscribed circle of the triangle (the points $M$, $K$ and $N$ lie on the line $BC$). Prove that $MK = KN$.
(I Sharygin)
2011 Balkan MO, 3
Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.
2025 EGMO, 4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.