Found problems: 85335
The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of:
[b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes;
[b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes;
[b](c)[/b] squares in total, with vertices on the lattice.
Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {{2020}\choose{k}} \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $n=12=2\times 2\times 3$, then the answer would be $2+2+3=7$.)
[i]Proposed by Ankit Bisain[/i]
A regular pentagon $ABCDE$ is given. The point $X$ is on his circumcircle, on the arc $\overarc{AE}$. Prove that $|AX|+|CX|+|EX|=|BX|+|DX|$.
[u][b]Remark:[/b][/u] Here's a more general version of the problem: Prove that for any point $X$ in the plane, $|AX|+|CX|+|EX|\ge|BX|+|DX|$, with equality only on the arc $\overarc{AE}$.
Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that
\[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\]
for all $x,y \in \mathbb{R} $.
Given triangle $ ABC$ with base $ AB$ fixed in length and position. As the vertex $ C$ moves on a straight line, the intersection point of the three medians moves on:
$ \textbf{(A)}\ \text{a circle} \qquad
\textbf{(B)}\ \text{a parabola} \qquad
\textbf{(C)}\ \text{an ellipse} \qquad
\textbf{(D)}\ \text{a straight line} \qquad
\textbf{(E)}\ \text{a curve here not listed}$
In an archery competition of 30 contestants, the target is divided into two zones, zone 1 and zone 2. Each arrow hitting the zone 1 gets 10 points, when hitting zone 2 will get 5 points and no score for miss. Each contestant throws 16 arrows. At the end of the competition, the statistics show that more than 50% of the arrows hit zone 2. The number of arrows that hit zone 1 is equal to the arrows which are missed. Prove than there are two contestants having equal score.
Consider a positive integer $n$ and $A = \{ 1,2,...,n \}$. Call a subset $X \subseteq A$ [i][b]perfect[/b][/i] if $|X| \in X$. Call a perfect subset $X$ [i][b]minimal[/b][/i] if it doesn't contain another perfect subset. Find the number of minimal subsets of $A$.
The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$
The incircle of a right-angled triangle $ABC$ touches hypotenus $AB$ at $P$, $BC$ and $AC$ at $R$ and $Q$ respectively. $C_1$ and $C_2$ are reflections of $C$ in $PQ$ and $PR$.
Find the angle $C_1IC_2$, where $I$ is the incenter of $ABC$.
Solve the equation: $\log_{x} 2 \cdot \log_{2x} 2 = \log_{4x} 2$.
In the tetrahedron $SABC $ at the height $SH$ the following point $O$ is chosen, such that: $$\angle AOS + \alpha = \angle BOS + \beta = \angle COS + \gamma = 180^o, $$ where $\alpha, \beta, \gamma$ are dihedral angles at the edges $BC, AC, AB $, respectively, at this point $H$ lies inside the base $ABC$. Let ${{A} _ {1}}, \, {{B} _ {1}}, \, {{C} _ {1}} $be the points of intersection of lines and planes: ${{A} _ {1}} = AO \cap SBC $, ${{B} _ {1}} = BO \cap SAC $, ${{C} _ {1}} = CO \cap SBA$ . Prove that if the planes $ABC $ and ${{A} _ {1}} {{B} _ {1}} {{C} _ {1}} $ are parallel, then $SA = SB = SC $.
(Alexey Klurman)
Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.
[i]Bulgaria[/i]
Find all pairs of positive integers $(a,b)$ such that $8b+1$ is a multiple of $a$ and $8a+1$ is a multiple of $b$.
Let $AD$ and $BE$ be the altitudes of acute triangle $ABC.$ The circles with diameters $AD$ and $BE$ intersect at points $S$ and $T$. Prove that $\angle ACS=\angle BCT.$
Given a circle and two points $P$ and $Q$, construct a right triangle inscribed in the circle such that its two legs pass through the points $P$ and $Q$ respectively. For what positions of $P$ and $Q$ is this construction impossible?
A positive integer $n{}$ is called [i]discontinuous[/i] if for all its natural divisors $1 = d_0 < d_1 <\cdots<d_k$, written out in ascending order, there exists $1 \leqslant i \leqslant k$ such that $d_i > d_{i-1}+\cdots+d_1+d_0+1$. Prove that there are infinitely many positive integers $n{}$ such that $n,n+1,\ldots,n+2019$ are all discontinuous.
Circle $\Gamma$ touches the circumcircle of triangle $ABC$ at point $R$, and it touches the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. Rays $PQ$ and $BC$ intersect at point $X$. The tangent line at point $R$ to the circle $\Gamma$ meets the segment $QX$ at point $Y$. The line segment $AX$ intersects the circumcircle of triangle $APQ$ at point $Z$. Prove that the circumscribed circles of triangles $ABC$ and $XY Z$ are tangent.
Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
Tanya wrote numbers in forms $n^7-1$ for $n=2,3,...$ and noticed that for $n=8$ she got number divisible by $337$.
For what minimal $n$ did she get number divisible by $2022$?
Let $(a_n)_{n\ge 1}$ be a sequence given by $a_1 = 45$ and $$a_n = a^2_{n-1} + 15a_{n-1}$$ for $n > 1$. Prove that the sequence contains no perfect squares.
Let $AC$ be a diameter of a circle $ \omega $ of radius $1$, and let $D$ be a point on $AC$ such that $CD=\frac{1}{5}$. Let $B$ be the point on $\omega$ such that $DB$ is perpendicular to $AC$, and $E$ is the midpoint of $DB$. The line tangent to $\omega$ at $B$ intersects line $CE$ at the point $X$. Compute $AX$.
Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$.
Given three non-collinear points $A,B,C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ are parallel.
[b]p1.[/b] Find the value of the parameter $a$ for which the following system of equations does not have solutions:
$$ax + 2y = 1$$
$$2x + ay = 1$$
[b]p2.[/b] Find all solutions of the equation $\cos(2x) - 3 \sin(x) + 1 = 0$.
[b]p3.[/b] A circle of a radius $r$ is inscribed into a triangle. Tangent lines to this circle parallel to the sides of the triangle cut out three smaller triangles. The radiuses of the circles inscribed in these smaller triangles are equal to $1,2$ and $3$. Find $r$.
[b]p4.[/b] Does there exist an integer $k$ such that $\log_{10}(1 + 49367 \cdot k)$ is also an integer?
[b]p5.[/b] A plane is divided by $3015$ straight lines such that neither two of them are parallel and neither three of them intersect at one point. Prove that among the pieces of the plane obtained as a result of such division there are at least $2010$ triangular pieces.
PS. You should use hide for answers.
The precent ages in years of two brothers $A$ and $B$,and their father $C$ are three distinct positive integers $a ,b$ and $c$ respectively .Suppose $\frac{b-1}{a-1}$ and $\frac{b+1}{a+1}$ are two consecutive integers , and $\frac{c-1}{b-1}$ and $\frac{c+1}{b+1}$ are two consecutive integers . If $a+b+c\le 150$ , determine $a,b$ and $c$.