Found problems: 85335
A positive integer is called a "double number" if its decimal representation consists of a block of digits, not commencing with $0$, followed immediately by an identical block. So, for instance, $360360$ is a double number, but $36036$ is not. Show that there are infinitely many double numbers which are perfect squares.
Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six?
Clarification: If $A$ is a friend of $B$ then $B$ is a friend of $A$.
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$ respectively. What is its area?
(A)$20\sqrt{3}$
(B)$20\sqrt{2}$
(C)$25\sqrt{3}$
(D)None of these
Cookie Monster says a positive integer $n$ is $crunchy$ if there exist $2n$ real numbers $x_1,x_2,\ldots,x_{2n}$, not all equal, such that the sum of any $n$ of the $x_i$'s is equal to the product of the other $n$ of the $x_i$'s. Help Cookie Monster determine all crunchy integers.
[i]Yannick Yao[/i]
suppose that $O$ is the circumcenter of acute triangle $ABC$.
we have circle with center $O$ that is tangent too $BC$ that named $w$
suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$)
$T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$.
$S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$.
prove that $ST$ is tangent $ABC$.
The localities $P_1, P_2, \dots, P_{1983}$ are served by ten international airlines $A_1,A_2, \dots , A_{10}$. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.
Find all solutions of the following functional equation: \[f(x^{2}+y+f(y))=2y+f(x)^{2}. \]
Let $a_0,b_0,c_0,a,b,c$ be integers such that $\gcd(a_0,b_0,c_0)=\gcd(a,b,c)=1$. Prove that there exists a positive integer $n$ and integers $a_1,a_2,\ldots,a_n=a,b_1,b_2,\ldots,b_n=b,c_1,c_2,\ldots,c_n=c$ such that for all $1\le i\le n$, $a_{i-1}a_i+b_{i-1}b_i+c_{i-1}c_i=1$.
[i]Proposed by Michael Ren.[/i]
Let $a > 2$ be a natural number. Show that there are infinitely many natural numbers n such that $a^n \equiv -1$ (mod $n^2$).
In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$.
[i]Proposed by Iman Maghsoudi[/i]
In a chess tournament participated $ 100 $ players. Every player played one game with every other player. For a win $1$ point is given, for loss $ 0 $ and for a draw both players get $0,5$ points. Ion got more points than every other player. Mihai lost only one game, but got less points than every other player. Find all possible values of the difference between the points accumulated by Ion and the points accumulated by Mihai.
Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively.
If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$.
Prove that line $AO$ passes through the midpoint of segment $BE$.
Let $AB$ be a segment of length $1$. Several particles start moving simultaneously at constant speeds from $A$ up to$ B$. As soon as a particle reaches $B$ , turns around and goes to $A$; when it reaches $A$, begins to move again towards $ B$ , and so on indefinitely.
Find all rational numbers$ r>1$ such that there exists an instant $t$ with the following property:
For each $n\ge 1$ , if $n+1$ particles with constant speeds $1,r,r^2,…,r^n$ move as described, at instant $t$, they all lie at the same interior point of segment $AB$.
Given that $m$ and $n$ are positive integers with property:
\[(mn)\mid(m^{2010}+n^{2010}+n)\]
Show that there exists a positive integer $k$ such that $n=k^{2010}$
[i]Nanang Susyanto, Yogyakarta[/i]
At least one of the coefficients of a polynomial $P(x)$ is negative. Can all of the coefficients of all of its powers $(P(x))^n$, $n > 1$, be positive?
(0 Kryzhanovskij)
Consider four real numbers $x$, $y$, $a$, and $b$, satisfying $x + y = a + b$ and $x^2 + y^2 = a^2 + b^2$. Prove that $x^n + y^n = a^n + b^n$, for all $n \in \mathbb{N}$.
In an isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$, points $L$ and $M$ lie on the sides $AB$ and $AC$, respectively such that $AL = CM$. Point $K$ lies on $AB$ such that $\angle AMK = 45^o$. If $\angle LMC = 75^o$, prove that $KM +ML = BC$.
[i]Proposed by Mahdi Etesamifard - Iran[/i]
Let $AB$ be a segment of length $1.$ Let $\odot A, \odot B$ be circles with radius $\overline{AB}$ centered at $A, B.$ Denote their intersection points $C, D.$ Draw circles $\odot C, \odot D$ with radius $\overline{CD}.$ Denote their intersection points $C, D.$ Draw circles $\odot C, \odot D$ with radius $\overline{CD}.$ Denote the intersection points of $\odot C$ and $\odot D$ by $E, F.$ Draw circles $\odot E, \odot F$ with radius $\overline{EF}$ and denote their intersection points $G, H.$
Compute the area of the pentagon $ACFHE.$
Let $t\in (1,2)$. Show that there exists a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ with the coefficients in $\{1,-1\}$ such that $\left|P(t)-2019\right| \leqslant 1.$
[i]Proposed by N. Safaei (Iran)[/i]
Let $S_1$ be a square of side length $3$. For $i=2,3,4,\dots$, inscribe a square $S_i$ inside $S_{i-1}$ such that the sides of the inner square form four $30^\circ-60^\circ-90^\circ$ triangles with the outer square. Compute the total sum
$$\sum_{i=1}^\infty\text{area}(S_i).$$
a) Prove that for any positive integer $m > 2$, the equation $$y^3 = x^3_1 + x^3_2 + ... + x^3_m$$ always has a positive integer solution.
b) Given a positive integer $n > 1$ and suppose $n \ne 3$. Prove that every rational number $x > 0$ can be expressed as
$$x =\frac{a^3_1 + a^3_2 + ... + a^3_n}{b^3_1 + b^3_2 + ... + b^3_n}$$ where $a_i, b_i$ $(i = 1, . . . , n)$ are positive integers.
Morteza Has $100$ sets. at each step Mahdi can choose two distinct sets of them and Morteza tells him the intersection and union of those two sets. Find the least steps that Mahdi can find all of the sets.
Proposed by Morteza Saghafian
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ is twice differentiable and satisfies $f(0)=2,f'(0)=-2,f(1)=1$.
Prove that there is a $\xi \in ]0,1[$ for which we have $f(\xi)\cdot f'(\xi)+f''(\xi)=0$.
A quadrilateral is circumscribed around a circle. Its diagonals intersect at the center of the circle. Prove that the quadrilateral is a rhombus.