Found problems: 85335
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.
1974 Czech and Slovak Olympiad III A, 4
Let $\mathcal M$ be the set of all polynomial functions $f$ of degree at most 3 such that \[\forall x\in[-1,1]:\ |f(x)|\le 1.\] Denote $a$ the (possibly zero) coefficient of $f$ at $x^3.$ Show that there is a positive number $k$ such that \[\forall f\in\mathcal M:\ |a|\le k\] and find the least $k$ with this property.
2015 Dutch Mathematical Olympiad, 1
We make groups of numbers. Each group consists of [i]fi ve[/i] distinct numbers. A number may occur in multiple groups. For any two groups, there are exactly four numbers that occur in both groups.
(a) Determine whether it is possible to make $2015$ groups.
(b) If all groups together must contain exactly [i]six [/i] distinct numbers, what is the greatest number of groups that you can make?
(c) If all groups together must contain exactly [i]seven [/i] distinct numbers, what is the greatest number of groups that you can make?
2022 HMNT, 22
Find the number of pairs of integers $(a,b)$ with $1 \le a < b \le 57$ such that $a^2$ has a smaller remainder than $b^2$ when divided by $57.$
1999 Estonia National Olympiad, 1
Find all pairs of integers $(m, n)$ such that $(m - n)^2 =\frac{4mn}{m + n - 1}$
2011 Puerto Rico Team Selection Test, 4
Let $P$ be a point inside the triangle $ABC$, such that the angles $\angle CBP$ and $\angle PAC$ are equal. Denote the intersection of the line $AP$ and the segment $BC$ by $D$, and the intersection of the line $BP$ with the segment $AC$ by $E$. The circumcircles of the triangles $ADC$ and $BEC$ meet at $C$ and $F$. Show that the line $CP$ bisects the angle $DFE$.
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