Found problems: 85335
Say that a 4-digit positive integer is [i]mixed[/i] if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits. For example, 2013 is mixed. How many 4-digit positive integers are mixed?
Let $ABC$ be a triangle with sides $a,b,c$ and corresponding angles $\alpha,\beta\gamma$ . Prove that if $\alpha = 3\beta$ then $(a^2 -b^2)(a-b) = bc^2$ . Is the converse true?
Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that
$$P(x)^3+Q(x)^3=x^{12}+1.$$
Triangle $ABC$ with $\angle BAC=60^\circ$ is given. The circumcircle of $ABC$ is $\Omega$, and the orthocenter of $ABC$ is $H$. Let $S$ denote the midpoint of the arc $BC$ of $\Omega$ which doesn't contain $A$. Point $P$ was chosen on $\Omega$ so that $\angle HPS=90^\circ$. Prove that there exists a circle that goes through $P$ and $S$ and is tangent to lines $AB$, $AC$.
Let $[n]$ denote the set of integers $0, 1, \ldots, n-1.$ Let $\omega_n=e^{2\pi i/n}.$ Let $$f(n) = \prod_{\overset{i \in [n]}{\gcd(i,n)=1}} \prod_{\overset{j \in [n]}{\gcd(j,n)=1}} (\omega_n^i - \omega_n^j).$$ Then, $f(2024)=2^{e_1} \cdot 11^{e_2} \cdot 23^{e_3}$ for positive integers $e_1, e_2, e_3.$ Find $e_1+e_2+e_3.$
Determine all pairs of integers $(x,y)$ satisfying the equation
$$x^3 +x^2y+xy^2 +y^3 = 8(x^2 +xy+y^2 +1).$$
Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB \parallel AC$. Point $Q$ is such that $AQ = AC$ and $CQ \parallel AB$. Segments $CP$ and $BQ$ meet at point $X$. Prove that the circumcenter of triangle $ABC$ lies on the circumcircle of triangle $PXQ$.
Given a triangle $PQX$ in the plane, with $PQ = 3, PX = 2.6$ and $QX = 3.8$. Construct a right-angled triangle $ABC$ such that the incircle of $\vartriangle ABC$ touches $AB$ at $P$ and $BC$ at $Q$, and point $X$ lies on the line $AC$.
Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$ at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$.
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and let $g : \mathbb{R} \to \mathbb{R}$ be arbitrary. Suppose that the Minkowski sum of the graph of $f$ and the graph of $g$ (i.e., the set $\{( x+y; f(x)+g(y) ) \mid x, y \in \mathbb{R}\}$) has Lebesgue measure zero. Does it follow then that the function $f$ is of the form $f(x) = ax + b$ with suitable constants $a, b \in \mathbb{R}$ ?
The graph of $y=C\sin x$, where $C>0$ is a constant, is drawn on the interval $[0,\pi]$. Suppose that there exists a point $P$ on the graph such that the triangle with vertices $(0,0)$, $(\pi,0)$, and $P$ is equilateral. Find $C^2$.
Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$.
Remark: "Natural numbers" is the set of positive integers.
A convex $n$-gon is divided into triangles by diagonals which do not intersect except at vertices of the n-gon. Each vertex belongs to an odd number of triangles. Show that $n$ must be a multiple of $3$.
(a) The numbers $1, 2,... , 100$ are divided into two groups so that the sum of all numbers in one group is equal to that in the other. Prove that one can remove two numbers from each group so that the sums of all numbers in each group are still the same.
(b) The numbers $1, 2 , ... , n$ are divided into two groups so that the sum of all numbers in one group is equal to that in the other . Is it true that for every such$ n > 4$ one can remove two numbers from each group so that the sums of all numbers in each group are still the same?
(A Shapovalov) [(a) for Juniors, (a)+(b) for Seniors]
The points $A_1,A_2, \ldots , A_{1983}$ are set on the circumference of a circle and each is given one of the values $\pm 1$. Show that if the number of points with the value $+1$ is greater than $1789$, then at least $1207$ of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.
Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.
Prove the inequality
\[\sum_{i<j} \frac{a_ia_j}{a_i \plus{} a_j} \le \frac{n}{2(a_1 \plus{} a_2 \plus{}\cdots \plus{} a_n)}\sum_{i<j} a_ia_j\]
for all positive real numbers $ a_1, a_2,\ldots , a_n$.
Calculate the integral $$\int \frac{dx}{\sin (x - 1) \sin (x - 2)} .$$
Hint: Change $\tan x = t$ .
Let $ABC$ an acute triangle with circumcenter $O$. Points $X$ and $Y$ are chosen such that:
[list]
[*]$\angle XAB = \angle YCB = 90^\circ$[/*]
[*]$\angle ABC = \angle BXA = \angle BYC$[/*]
[*]$X$ and $C$ are in different half-planes with respect to $AB$[/*]
[*]$Y$ and $A$ are in different half-planes with respect to $BC$[/*]
[/list]
Prove that $O$ is the midpoint of $XY$.
The value of
$$\frac{\log_35\log_25}{\log_35+\log_25}$$
can be expressed as $a\log_bc$, where $a$, $b$, and $c$ are positive integers, and $a+b$ is as small as possible. Find $a+2b+3c$.
Find the least positive integers $m,k$ such that
a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube.
b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square.
The author is Vasile Suceveanu
[b](a)[/b] Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other.
[b](b)[/b] Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?
Introduce a standard scalar product in $\mathbb{R}^4.$ Let $V$ be a partial vector space in $\mathbb{R}^4$ produced by $\left(
\begin{array}{c}
1 \\
-1 \\
-1 \\
1
\end{array}
\right),\left(
\begin{array}{c}
1 \\-1 \\
1 \\
-1
\end{array}
\right).$
Find a pair of base of orthogonal complement $W$ for $V$ in $\mathbb{R}^4.$
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?