Found problems: 85335
The midpoint $M$ of $[AC]$ of a triangle $\triangle ABC$ is between $C$ and the feet $H$ of the altitude from $B$. If $m(\widehat{ABH}) = m(\widehat{MBC})$, $m(\widehat{ACB}) = 15^{\circ}$, and $|HM|=2\sqrt{3}$, then $|AC|=?$
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 5 \sqrt 2 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ \frac{16}{\sqrt3} \qquad \textbf{(E)}\ 10$
Let $n_1,\ldots,n_k$ be positive integers, and define $d_1=1$ and $d_i=\frac{(n_1,\ldots,n_{i-1})}{(n_1,\ldots,n_{i})}$, for $i\in \{2,\ldots,k\}$, where $(m_1,\ldots,m_{\ell})$ denotes the greatest common divisor of the integers $m_1,\ldots,m_{\ell}$. Prove that the sums \[\sum_{i=1}^k a_in_i\] with $a_i\in\{1,\ldots,d_i\}$ for $i\in\{1,\ldots,k\}$ are mutually distinct $\mod n_1$.
Aaron the Ant is somewhere on the exterior of a hollow cube of side length $2$ inches, and Fred the Flea is on the inside, at one of the vertices. At some instant, Fred flies in a straight line towards the opposite vertex, and simultaneously Aaron begins crawling on the exterior of the cube towards that same vertex. Fred moves at $\sqrt{3}$ inches per second and Aaron moves at $\sqrt{2}$ inches per second. If Aaron arrives before Fred, the area of the surface on the cube from which Aaron could have started can be written as $a\pi+\sqrt{b}+c$ where $a$, $b$, and $c$ are integers. Find $a+b+c.$
A $7 \times 7$ square is divided into unit squares by lines parallel to its sides. Some Swiss crosses (obtained by removing corner unit squares from a square of side $3$) are to be put on the large square, with the edges along division lines. Find the smallest number of unit squares that need to be marked in such a way that every cross covers at least one marked square.
In triangle $ABC$, $D$ is the middle point of side $BC$ and $M$ is a point on segment $AD$ such that $AM=3MD$.
The barycenter of $ABC$ and $M$ are on the inscribed circumference of $ABC$.
Prove that $AB+AC>3BC$.
Numbers $-1011, -1010, \ldots, -1, 1, \ldots, 1011$ in some order form the sequence $a_1,a_2,\ldots, a_{2022}$.
Find the maximum possible value of the sum $$|a_1|+|a_1+a_2|+\ldots+|a_1+\ldots+a_{2022}|$$
Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.
Sequence $x_1 , x_2 , ..., $ with $x_1=20$ ; $x_2=12$ for all $n\geq 1$ such that $x_{n+2}=x_n+x_{n+1}+2\sqrt{x_{n}*x_{n+1}+121} $then prove that $x_{2013}$ is an integer number.
Given a triangle $\vartriangle ABC$,construct squares $BAQP$ and $ACRS$ outside the triangle $ABC$ (with vertices in that listed in counterclockwise order).Show that the line from $A$ perpendicular to $BC$ passes through the midpoint of the segment $QS$.
A ladder $10\text{ m}$ long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of $1\text{ m/s}$, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is $6\text{ m}$ from the wall?
Let $ABC$ be an acute triangle. Let $D$ be a point on the line parallel to $AC$ that passes through $B$, such that $\angle BDC = 2\angle BAC$ as well as such that $ABDC$ is a convex quadrilateral. Show that $BD + DC = AC$.
The prime number $p > 2$ and the integer $n$ are given. Prove that the number $pn^2$ has no more than one divisor $d$ for which $n^2+d$ is the square of the natural number.
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One bisector of a given triangle is parallel to one sideline of its Nagel triangle. Prove that one of two remaining bisectors is parallel to another sideline of the Nagel triangle.
Proposed by:L.Emelyanov
A natural number $n$ is called [i]perfect [/i] if it is equal to the sum of all its natural divisors other than $n$. For example, the number $6$ is perfect because $6 = 1 + 2 + 3$. Find all even perfect numbers that can be given as the sum of two cubes positive integers.
Given some red and blue points. Some of them are connected by the segments. Let us call "exclusive" the point, if its colour differs from the colour of more than half of the connected points. Every move one arbitrary "exclusive" point is repainted to the other colour. Prove that after the finite number of moves there will remain no "exclusive" points.
A real function has been assigned to every cell of an $n \times n$ table. Prove that a function can be assigned to each row and each column of this table such that the function assigned to each cell is equivalent to the combination of functions assigned to the row and the column containing it.
Consider a latin square of size $n$. We are allowed to choose a $1 \times 1$ square in the table, and add $1$ to any number on the same row and column as the chosen square (the original square will be counted aswell) , or we can add $-1$ to all of them instead. Can we with doing finitly many operation , reach any latin square of size $n?$
Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
While Steve and LeRoy are fishing $ 1$ mile from shore, their boat springs a leak, and water comes in at a constant rate of $ 10$ gallons per minute. The boat will sink if it takes in more than $ 30$ gallons of water. Steve starts rowing toward the shore at a constant rate of $ 4$ miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 10$
In two circles intersecting at points $A$ and $B$, parallel chords $A_1B_1$ and $A_2B_2$ are drawn. The lines $AA_1$ and $BB_2$ intersect at the point $X, AA_2$ and $BB_1$ intersect at the point $Y$. Prove that $XY // A_1B_1$.
Let $\alpha\geq 1$ be a real number. Define the set
$$A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\}$$
Suppose that all the positive integers that [b]does not belong[/b] to the $A(\alpha)$ are exactly the positive integers that have the same remainder $r$ in the division by $2021$ with $0\leq r<2021$. Determine all the possible values of $\alpha$.
Let $ p\geq 7$ be a prime number, $ \zeta$ a primitive $ p$th root of unity, $ c$ a rational number. Prove that in the additive group generated by the numbers $ 1,\zeta,\zeta^2,\zeta^3\plus{}\zeta^{\minus{}3}$ there are only finitely many elements whose norm is equal to $ c$. (The norm is in the $ p$th cyclotomic field.)
[i]K. Gyory[/i]
A circle is divided by $2N$ points into $2N$ arcs of length $1$. These points are joined in pairs to form $N$ chords. Each chord divides the circle into two arcs, the length of each being an even integer. Prove that $N$ is even.
Let $p\ge 2$ be a prime number and $a\ge 1$ be an integer different from $p$. Find all pairs $(a, p)$ such that $a + p | a^2 + p^2$.
Prove that there is a positive integer $n$ such that the decimal representation of $7^n$ contains a block of at least $m$ consecutive zeros, where $m$ is any given positive integer.