Found problems: 85335
2020 Online Math Open Problems, 19
Compute the smallest positive integer $M$ such that there exists a positive integer $n$ such that
[list] [*] $M$ is the sum of the squares of some $n$ consecutive positive integers, and
[*] $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
[/list]
[i]Proposed by Jaedon Whyte[/i]
2002 AMC 10, 5
Circles of radius $ 2$ and $ 3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.
[asy]unitsize(3mm);
defaultpen(linewidth(0.7)+fontsize(8));
filldraw(Circle((0,0),5),grey,black);
filldraw(Circle((-2,0),3),white,black);
filldraw(Circle((3,0),2),white,black);
dot((-2,0));
dot((3,0));
draw((-2,0)--(1,0));
draw((3,0)--(5,0));
label("$3$",(-0.5,0),N);
label("$2$",(4,0),N);[/asy]
$ \textbf{(A)}\ 3\pi \qquad
\textbf{(B)}\ 4\pi \qquad
\textbf{(C)}\ 6\pi \qquad
\textbf{(D)}\ 9\pi \qquad
\textbf{(E)}\ 12\pi$
1996 Denmark MO - Mohr Contest, 4
Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?
2022 Taiwan TST Round 2, N
A positive integer is said to be [b]palindromic[/b] if it remains the same when its digits are reversed. For example, $1221$ or $74847$ are both palindromic numbers.
Let $k$ be a positive integer that can be expressed as an $n$-digit number $\overline{a_{n-1}a_{n-2} \cdots a_0}$. Prove that if $k$ is a palindromic number, then $k^2$ is also a palindromic number if and only if $a_0^2 + a^2_1 + \cdots + a^2_{n-1} < 10$.
[i]Proposed by Ho-Chien Chen[/i]
2017 Saudi Arabia IMO TST, 1
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.
1963 All Russian Mathematical Olympiad, 038
Find such real $p, q, a, b$, that for all $x$ an equality is held: $$(2x-1)^{20} - (ax+b)^{20} = (x^2+px+q)^{10}$$
2014-2015 SDML (High School), 6
Let $a$ and $b$ be positive reals such that $$a=1+\frac{a}{b}$$$$b=3+\frac{4+a}{b-2}$$ What is $a$?
$\text{(A) }\sqrt{2}\qquad\text{(B) }2+\sqrt{2}\qquad\text{(C) }2+\sqrt{2}+\sqrt[3]{2}\qquad\text{(D) }\sqrt{2}+\sqrt[3]{2}\qquad\text{(E) }\sqrt[3]{2}$
2017 ASDAN Math Tournament, 1
Compute
$$\int_0^6\frac{x-3}{x^2-6x-7}dx.$$
1988 Polish MO Finals, 1
$d$ is a positive integer and $f : [0,d] \rightarrow \mathbb{R}$ is a continuous function with $f(0) = f(d)$. Show that there exists $x \in [0,d-1]$ such that $f(x) = f(x+1)$.
1983 Polish MO Finals, 4
Prove that if natural numbers $a,b,c,d$ satisfy the equality $ab = cd$, then $\frac{gcd(a,c)gcd(a,d)}{gcd(a,b,c,d)}= a$
2022 MIG, 8
Let $ABC$ be a triangle and $D$ be a point on segment $BC$. If $\triangle ABD$ is equilateral and $\angle ACB = 14^{\circ}$, what is $\angle{DAC}$?
$\textbf{(A) }26^{\circ}\qquad\textbf{(B) }34^{\circ}\qquad\textbf{(C) }46^{\circ}\qquad\textbf{(D) }50^{\circ}\qquad\textbf{(E) }54^{\circ}$
2018 BMT Spring, 4
Consider a standard ($8$-by-$8$) chessboard. Bishops are only allowed to attack pieces that are along the same diagonal as them (but cannot attack along a row or column). If a piece can attack another piece, we say that the pieces threaten each other. How many bishops can you place a chessboard without any of them threatening each other?
2011 Today's Calculation Of Integral, 745
When real numbers $a,\ b$ move satisfying $\int_0^{\pi} (a\cos x+b\sin x)^2dx=1$, find the maximum value of $\int_0^{\pi} (e^x-a\cos x-b\sin x)^2dx.$
2011 Purple Comet Problems, 4
Five non-overlapping equilateral triangles meet at a common vertex so that the angles between adjacent triangles are all congruent. What is the degree measure of the angle between two adjacent triangles?
[asy]
size(100);
defaultpen(linewidth(0.7));
path equi=dir(300)--dir(240)--origin--cycle;
for(int i=0;i<=4;i=i+1)
draw(rotate(72*i,origin)*equi);
[/asy]
2010 NZMOC Camp Selection Problems, 4
Find all positive integer solutions $(a, b)$ to the equation $$\frac{1}{a}+\frac{1}{b}+ \frac{n}{lcm(a,b)}=\frac{1}{gcd(a, b)}$$ for
(i) $n = 2007$;
(ii) $n = 2010$.
2017 AIME Problems, 6
Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer.
2025 Macedonian Mathematical Olympiad, Problem 5
Let \(n>1\) be a natural number, and let \(K\) be the square of side length \(n\) subdivided into \(n^2\) unit squares. Determine for which values of \(n\) it is possible to dissect \(K\) into \(n\) connected regions of equal area using only the diagonals of those unit squares, subject to the condition that from each unit square at most one of its diagonals is used (some unit squares may have neither diagonal).
2010 F = Ma, 1
If the graph is a graph of POSITION vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?
(A) From A to B
(B) From B to C only
(C) From B to D
(D) From C to D only
(E) From D to E
2025 Bangladesh Mathematical Olympiad, P3
Let $ABC$ be a given triangle with circumcenter $O$ and orthocenter $H$. Let $D, E$ and $F$ be the feet of the perpendiculars from $A, B$ and $C$ to the opposite sides, respectively. Let $A'$ be the reflection of $A$ with respect to $EF$. Prove that $HOA'D$ is a cyclic quadrilateral.
[i]Proposed by Imad Uddin Ahmad Hasin[/i]
2002 AMC 12/AHSME, 6
For how many positive integers $ m$ does there exist at least one positive integer $ n$ such that $ m\cdot n \le m \plus{} n$?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}$ infinitely many
2018 Singapore MO Open, 5
Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$
Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$
[i]Proposed by Ma Zhao Yu
2006 Moldova Team Selection Test, 4
Let $A=\{1,2,\ldots,n\}$. Find the number of unordered triples $(X,Y,Z)$ that satisfy $X\bigcup Y \bigcup Z=A$
1983 Austrian-Polish Competition, 1
Nonnegative real numbers $a, b,x,y$ satisfy $a^5 + b^5 \le $1 and $x^5 + y^5 \le 1$. Prove that $a^2x^3 + b^2y^3 \le 1$.
2017 CCA Math Bonanza, I3
A sequence starts with $2017$ as its first term and each subsequent term is the sum of cubes of the digits in the previous number. What is the $2017$th term of this sequence?
[i]2017 CCA Math Bonanza Individual Round #3[/i]
2020 Brazil Team Selection Test, 3
Let $ABC$ be a triangle such that $AB > BC$ and let $D$ be a variable point on the line segment $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$, lying on the opposite side of $BC$ from $A$ such that $\angle BAE = \angle DAC$. Let $I$ be the incenter of triangle $ABD$ and let $J$ be the incenter of triangle $ACE$. Prove that the line $IJ$ passes through a fixed point, that is independent of $D$.
[i]Proposed by Merlijn Staps[/i]