Found problems: 85335
1990 Balkan MO, 2
The polynomial $P(X)$ is defined by $P(X)=(X+2X^{2}+\ldots +nX^{n})^{2}=a_{0}+a_{1}X+\ldots +a_{2n}X^{2n}$. Prove that $a_{n+1}+a_{n+2}+\ldots +a_{2n}=\frac{n(n+1)(5n^{2}+5n+2)}{24}$.
1955 AMC 12/AHSME, 35
Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles:
$ \textbf{(A)}\ \text{is none of the following} \qquad
\textbf{(B)}\ \text{cannot be determined from the given data}\\
\textbf{(C)}\ \text{is 20 or 26} \qquad
\textbf{(D)}\ \text{is 14 or 32} \qquad
\textbf{(E)}\ \text{is 8 or 38}$
2017 Middle European Mathematical Olympiad, 1
Determine all pairs of polynomials $(P, Q)$ with real coefficients satisfying
$$P(x + Q(y)) = Q(x + P(y))$$
for all real numbers $x$ and $y$.
2015 South Africa National Olympiad, 4
Let $ABC$ be an acute-angled triangle with $AB < AC$, and let points $D$ and $E$ be chosen on the side $AC$ and $BC$ respectively in such a way that $AD = AE = AB$. The circumcircle of $ABE$ intersects the line $AC$ at $A$ and $F$ and the line $DE$ at $E$ and $P$. Prove that $P$ is the circumcentre of $BDF$.
Kvant 2024, M2809
Given is a triangle $ABC$ and the points $M, P$ lie on the segments $AB, BC$, respectively, such that $AM=BC$ and $CP=BM$. If $AP$ and $CM$ meet at $O$ and $2\angle AOM=\angle ABC$, find the measure of $\angle ABC$.
2023 International Zhautykov Olympiad, 3
Let $a_1, a_2, \cdots, a_k$ be natural numbers. Let $S(n)$ be the number of solutions in nonnegative integers to $a_1x_1 + a_2x_2 + \cdots + a_kx_k = n$. Suppose $S(n) \neq 0$ for all big enough $n$. Show that for all sufficiently large $n$, we have $S(n+1) < 2S(n)$.
1988 China Team Selection Test, 4
Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are undistinguishing in $S_5$. Considering the subset $A$ of $S_k$ such that the elements of $A$ are pairwise distinguishing. Let $r_k$ be the maximum possible number of elements of $A$.
(i) Find $r_5$.
(ii) Find $r_7$.
(iii) Find $r_k$ for $k \in \mathbb{N}$.
1982 IMO Longlists, 22
Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$
2012 Sharygin Geometry Olympiad, 4
Determine all integer $n > 3$ for which a regular $n$-gon can be divided into equal triangles by several (possibly intersecting) diagonals.
(B.Frenkin)
1966 IMO Longlists, 25
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]
2017 CMIMC Combinatorics, 1
Robert colors each square in an empty 3 by 3 grid either red or green. Find the number of colorings such that no row or column contains more than one green square.
2016 Mathematical Talent Reward Programme, MCQ: P 14
Let $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Find $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor \rfloor = 88$
[list=1]
[*] $\pi$
[*] 3.14
[*] $\frac{22}{7}$
[*] All of these
[/list]
1994 Balkan MO, 2
Let $n$ be an integer. Prove that the polynomial $f(x)$ has at most one zero, where \[ f(x) = x^4 - 1994 x^3 + (1993+n)x^2 - 11x + n . \]
[i]Greece[/i]
1979 Poland - Second Round, 5
Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.
1958 November Putnam, B6
Let a complete oriented graph on $n$ points be given. Show that the vertices can be enumerated as $v_1 , v_2 ,\ldots, v_n$ such that $v_1 \rightarrow v_2 \rightarrow \cdots \rightarrow v_n.$
Ukraine Correspondence MO - geometry, 2016.7
The circle $\omega$ inscribed in an isosceles triangle $ABC$ ($AC = BC$) touches the side $BC$ at point $D$ .On the extensions of the segment $AB$ beyond points $A$ and $B$, respectively mark the points $K$ and $L$ so that $AK = BL$, The lines $KD$ and $LD$ intersect the circle $\omega$ for second time at points $G$ and $H$, respectively. Prove that point $A$ belongs to the line $GH$.
2005 Czech-Polish-Slovak Match, 2
A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$ and circumscribed to a circle with center $I$. Its diagonals meet at $P$. Prove that points $O, I$ and $P$ lie on a line.
1992 Romania Team Selection Test, 4
Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$
2005 Abels Math Contest (Norwegian MO), 4b
Let $a, b$ and $c$ be real numbers such that $ab + bc + ca> a + b + c> 0$. Show then that $a+b+c>3$
1992 IMO Longlists, 76
Given any triangle $ABC$ and any positive integer $n$, we say that $n$ is a [i]decomposable[/i] number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to $\triangle ABC$. Determine the positive integers that are decomposable numbers for every triangle.
2015 Purple Comet Problems, 3
The repeating decimal $2.0151515\ldots$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2011 Dutch IMO TST, 1
Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.
2007 Balkan MO, 1
Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.
2019 China Northern MO, 8
For positive intenger $n$, define $f(n)$: the smallest positive intenger that does not divide $n$.
Consider sequence $(a_n): a_1=a_2=1, a_n=a_{f(n)}+1(n\geq3)$.
For example, $a_3=a_2+1=2,a_4=a_3+1=3$.
[b](a)[/b] Prove that there exists a positive intenger $C$, for any positive intenger $n$, $a_n\leq C$.
[b](b)[/b] Are there positive intengers $M$ and $T$, satisfying that for any positive intenger $n\geq M$, $a_n=a_{n+T}$.
2024 AMC 10, 13
Two transformations are said to [i]commute[/i] if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:
- A translation $2$ units to the right
- A $90^\circ$- rotation counterclockwise about the origin.
- A reflection across the $x$-axis, and
- A dilation centered at the origin with scale factor $2$.
Of the $6$ pairs of distinct transformations from this list, how many commute?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) }5 \qquad
$