Found problems: 85335
We have 19 triminos (2 x 2 squares without one unit square) and infinite amount of 2 x 2 squares. Find the greatest odd number $n$ for which a square $n$ x $n$ can be covered with the given figures.
Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$ such as $\{3, 31\}$.
Evaluate the sum $\sqrt{1-\frac{1}{2}\cdot\sqrt{1\cdot3}}+\sqrt{2-\frac{1}{2}\cdot\sqrt{3\cdot5}}+\sqrt{3-\frac{1}{2}\cdot\sqrt{5\cdot7}}+\dots+\sqrt{40-\frac{1}{2}\cdot\sqrt{79\cdot81}}$.
Prove that for every integer $n \geq 2$ there exist $n$ different positive integers such that for any two of these integers $a$ and $b$ their sum $a+b$ is divisible by their difference $a - b.$
Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.
In a tournament every team plays every other team just once. Each game is won by one of the teams (there are no draws). Each team loses at least once. Show that there must be three teams $A$, $B$, $C$ such that $A$ beat $B$, $B$ beat $C$ and $C$ beat $A$.
Prove that there are infinitely many positive $n$ that for all prime divisors $p$ of $n^2 + 3, \exists 0 \leq k \leq \sqrt{n}$ and $p \mid k^2+3$
A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.
Find the smallest term of the sequence $a_1, a_2, a_3, \ldots$ defined by $a_1=2014^{2015^{2016}}$ and
$$
a_{n+1}=
\begin{cases}
\frac{a_n}{2} & \text{ if } a_n \text{ is even} \\
a_n + 7 & \text{ if } a_n \text{ is odd} \\
\end{cases}
$$
A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$. Is this possible to make the number $(20^{17}+2016)$ on the board? \\
(Explain your answer) \\
[hide=Note]This type of the question is well known but I am going to make a collection so, :blush: [/hide]
Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $8,12,\cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $17,19\cdots$. Assume $n\ne 0$. Then $s_1=s_2$ for:
$\text{(A)} \ \text{no value of n} \qquad \text{(B)} \ \text{one value of n} \qquad \text{(C)} \ \text{two values of n}$
$\text{(D)} \ \text{four values of n} \qquad \text{(E)} \ \text{more than four values of n}$
Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .
Let $ABCD$ be a quadrilateral such that line $AB$ intersects $CD$ at $X$. Denote circles with inradius $r_1$ and centers $A, B$ as $w_a$ and $w_b$ with inradius $r_2$ and centers $C, D$ as $w_c$ and $w_d$. $w_a$ intersects $w_d$ at $P, Q$. $w_b$ intersects $w_c$ at $R, S$. Prove that if $XA.XB+r_2^2=XC.XD+r_1^2$, then $P,Q,R,S$ are cyclic.
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that:
$$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$
for all reals $x, y $.
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]
What is the imaginary part of the complex number $\frac{-4+7i}{1+2i}$?
$\text{(A) }-\frac{1}{2}\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }\frac{7}{2}\qquad\text{(E) }-\frac{18}{5}$
Find all positive integers $ a$ and $ b$ for which
\[ \left \lfloor \frac{a^2}{b} \right \rfloor \plus{} \left \lfloor \frac{b^2}{a} \right \rfloor \equal{} \left \lfloor \frac{a^2 \plus{} b^2}{ab} \right \rfloor \plus{} ab.\]
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
$$Problem 2 :$$If ABCD is as convex quadrilateral with $\angle ADC = 30$
and $BD = AB+BC+CA$,
prove that $BD$ bisects $\angle ABC$.
A scalene acute triangle $ABC$ is drawn on the plane, in which $BC$ is the longest side. Points $P$ and $D$ are constructed, the first inside $ABC$ and the second outside, so that $\angle ABC = \angle CBD$, $\angle ACP = \angle BCD$ and that the area of triangle $ABC$ is equal to the area of quadrilateral $BPCD$. Prove that triangles $BCD$ and $ACP$ are similar.
What is $3(2 \log_4 (2(2 \log_3 9)))$ ?
Let $ f : \mathbf{R} \to \mathbf{R} $ be a smooth function such that $f'(x)=f(1-x)$ for all $x$ and $f(0)=1$. Find $f(1)$.
Positive integers $m, n$ are given such that $\sqrt{2}<\frac{m}{n}<\sqrt{2}+\frac{1}{2}$ and $m$ is even. Prove that there exist positive integers $k<m$ and $l<n$ such that
$$|\frac{k}{l}-\sqrt{2}|<\frac{m}{n}-\sqrt{2}$$
In a convex quadrilateral $ABCD$, $P$ and $Q$ are points on sides $BC$ and $DC$ such that $B\hat{A}P = D\hat{A}Q$. If the line that passes through the orthocenters of $\triangle ABP$ and $\triangle ADQ$ is perpendicular to $AC$, prove that the area of these triangles are equals.