Found problems: 85335
1982 AMC 12/AHSME, 1
When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is
$\textbf{(A)} \ 2 \qquad \textbf{(B)} \ -2 \qquad \textbf{(C)} \ -2x-2 \qquad \textbf{(D)} \ 2x+2 \qquad \textbf{(E)} \ 2x-2$
2023 Assara - South Russian Girl's MO, 5
In a $5 \times 5$ checkered square, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?
1983 IMO Shortlist, 25
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$
2019 Romania Team Selection Test, 2
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
1999 National Olympiad First Round, 1
Let $ ABC$ be a triangle with $ \left|AB\right| \equal{} 14$, $ \left|BC\right| \equal{} 12$, $ \left|AC\right| \equal{} 10$. Let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[BC\right]$ such that $ \left|AD\right| \equal{} 4$ and $ Area\left(ABC\right) \equal{} 2Area\left(CDE\right)$. Find $ Area\left(ABE\right)$.
$\textbf{(A)}\ 4\sqrt {6} \qquad\textbf{(B)}\ 6\sqrt {2} \qquad\textbf{(C)}\ 3\sqrt {6} \qquad\textbf{(D)}\ 4\sqrt {2} \qquad\textbf{(E)}\ 4\sqrt {5}$
Estonia Open Senior - geometry, 1997.2.3
The figure shows a square and three circles of equal radius tangent to each other and square passes. Find the radius of the circles if the square length is $1$.
[img]http://3.bp.blogspot.com/-iIjwupkz7DQ/XnrIRhKIJnI/AAAAAAAALhA/clERrIDqEtcujzvZk_qu975wsTjKaxCLQCK4BGAYYCw/s400/97%2Bestonia%2Bopen%2Bs2.3.png[/img]
2023 Korea National Olympiad, 2
Sets $A_0, A_1, \dots, A_{2023}$ satisfy the following conditions:
[list]
[*] $A_0 = \{ 3 \}$
[*] $A_n = \{ x + 2 \mid x \in A_{n - 1} \} \ \cup \{x(x+1) / 2 \mid x \in A_{n - 1} \}$ for each $n = 1, 2, \dots, 2023$.
[/list]
Find $|A_{2023}|$.
2019 IFYM, Sozopol, 5
Let $A$ be the number of 2019-digit numbers, that is made of 2 different digits (For example $10\underbrace{1...1}_{2016}0$ is such number). Determine the highest power of 3 that divides $A$.
Estonia Open Junior - geometry, 2014.2.5
In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?
2015 Korea Junior Math Olympiad, 5
Let $I$ be the incenter of an acute triangle $\triangle ABC$, and let the incircle be $\Gamma$.
Let the circumcircle of $\triangle IBC$ hit $\Gamma$ at $D, E$, where $D$ is closer to $B$ and $E$ is closer to $C$.
Let $\Gamma \cap BE = K (\not= E)$, $CD \cap BI = T$, and $CD \cap \Gamma = L (\not= D)$.
Let the line passing $T$ and perpendicular to $BI$ meet $\Gamma$ at $P$, where $P$ is inside $\triangle IBC$.
Prove that the tangent to $\Gamma$ at $P$, $KL$, $BI$ are concurrent.
2005 Iran Team Selection Test, 3
Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.
2022 BMT, 16
Let triangle $\vartriangle ABC$ be a triangle with $AB = 5$, $BC = 7$, and $CA = 8$, and let $I$ be the incenter of $\vartriangle ABC$. Let circle $C_A$ denote the circle with center $A$ and radius $\overline{AI}$, denote $C_B$ and circle $C_C$ similarly. Besides all intersecting at $I$, the circles $C_A$,$C_B$,$C_C$ also intersect pairwise at $F$, $G$, and $H$. Compute the area of triangle $\vartriangle FGH$.
1970 IMO Longlists, 13
Each side of an arbitrary $\triangle ABC$ is divided into equal parts, and lines parallel to $AB,BC,CA$ are drawn through each of these points, thus cutting $\triangle ABC$ into small triangles. Points are assigned a number in the following manner:
$(1)$ $A,B,C$ are assigned $1,2,3$ respectively
$(2)$ Points on $AB$ are assigned $1$ or $2$
$(3)$ Points on $BC$ are assigned $2$ or $3$
$(4)$ Points on $CA$ are assigned $3$ or $1$
Prove that there must exist a small triangle whose vertices are marked by $1,2,3$.
1986 Iran MO (2nd round), 1
Let $f$ be a function such that
\[f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.\]
Find the limit of $f$ in the point $x_0=1.$
2024 USA TSTST, 2
Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0)=Q(0)=1$, there is no nonconstant polynomial dividing both $P$ and $Q$, and
\[
1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 +
(p-1)x}}}=\frac{P(x)}{Q(x)}.
\]
Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$, and all coefficients of $Q$ are [i]not[/i] divisible by $p$.
[i]Andrew Gu[/i]
2013 Moldova Team Selection Test, 2
Find all pairs of real numbers $(x,y)$ satisfying
$\left\{\begin{array}{rl}
2x^2+xy &=1 \\
\frac{9x^2}{2(1-x)^4}&=1+\frac{3xy}{2(1-x)^2}
\end{array}\right.$
2021 Cono Sur Olympiad, 6
Let $ABC$ be a scalene triangle with circle $\Gamma$. Let $P,Q,R,S$ distinct points on the $BC$ side, in that order, such that $\angle BAP = \angle CAS$ and $\angle BAQ = \angle CAR$. Let $U, V, W, Z$ be the intersections, distinct from $A$, of the $AP, AQ, AR$ and $AS$ with $\Gamma$, respectively. Let $X = UQ \cap SW$, $Y = PV \cap ZR$, $T = UR \cap VS$ and $K = PW \cap ZQ$. Suppose that the points $M$ and $N$ are well determined, such that $M = KX \cap TY$ and $N = TX \cap KY$. Show that $M, N, A$ are collinear.
2002 HKIMO Preliminary Selection Contest, 5
A positive integer is said to be a “palindrome” if it reads the same from left to right as from right to left. For example 2002 is a palindrome. Find the sum of all 4-digit palindromes.
2015 Online Math Open Problems, 28
Let $N$ be the number of $2015$-tuples of (not necessarily distinct) subsets $(S_1, S_2, \dots, S_{2015})$ of $\{1, 2, \dots, 2015 \}$ such that the number of permutations $\sigma$ of $\{1, 2, \dots, 2015 \}$ satisfying $\sigma(i) \in S_i$ for all $1 \le i \le 2015$ is odd. Let $k_2, k_3$ be the largest integers such that $2^{k_2} | N$ and $3^{k_3} | N$ respectively. Find $k_2 + k_3.$
[i]Proposed by Yang Liu[/i]
2019 Thailand Mathematical Olympiad, 2
Let $a,b$ be two different positive integers. Suppose that $a,b$ are relatively prime. Prove that $\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is not an integer.
2016 NIMO Problems, 3
Convex pentagon $ABCDE$ satisfies $AB \parallel DE$, $BE \parallel CD$, $BC \parallel AE$, $AB = 30$, $BC = 18$, $CD = 17$, and $DE = 20$. Find its area.
[i] Proposed by Michael Tang [/i]
1994 Poland - First Round, 5
Given positive numbers $a,b$. Prove that the following sentences are equivalent:
($1$) $ \sqrt{a} + 1 > \sqrt{b} $;
($2$) for every $ x > 1, ax + \frac{x}{x - 1} > b$.
2013 China Northern MO, 4
For positive integers $n,a,b$, if $n=a^2 +b^2$, and $a$ and $b$ are coprime, then the number pair $(a,b)$ is called a [i]square split[/i] of $n$ (the order of $a, b$ does not count). Prove that for any positive $k$, there are only two square splits of the integer $13^k$.
2005 Taiwan TST Round 1, 1
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
2023 Harvard-MIT Mathematics Tournament, 5
Let $ABC$ be a triangle with $AB = 13, BC = 14, $and$ CA = 15$. Suppose $PQRS$ is a square such that $P$ and $R$ lie on line $BC, Q$ lies on line $CA$, and $S$ lies on line $AB$. Compute the side length of this square.