Found problems: 85335
Russian TST 2022, P1
Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible.
[i]Carl Schildkraut, USA[/i]
2018 PUMaC Geometry B, 2
Let a right cone of the base radius $r=3$ and height greater than $6$ be inscribed in a sphere of radius $R=6$. The volume of the cone can be expressed as $\pi(a\sqrt{b}+c)$, where $b$ is square free. Find $a+b+c$.
2020 Stanford Mathematics Tournament, 9
Let $ABC$ be a right triangle with hypotenuse $AC$. Let $G$ be the centroid of this triangle and suppose that we have $AG^2 + BG^2 + CG^2 = 156$. Find $AC^2$.
2019 Spain Mathematical Olympiad, 4
Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$
2007 Bulgaria Team Selection Test, 1
Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$
2024 5th Memorial "Aleksandar Blazhevski-Cane", P5
For a given integer $k \geq 1$, find all $k$-tuples of positive integers $(n_1,n_2,...,n_k)$ with $\text{GCD}(n_1,n_2,...,n_k) = 1$ and $n_2|(n_1+1)^{n_1}-1$, $n_3|(n_2+1)^{n_2}-1$, ... , $n_1|(n_k+1)^{n_k}-1$.
[i]Authored by Pavel Dimovski[/i]
1961 Putnam, B7
Given a sequence $(a_n)$ of non-negative real numbers such that $a_{n+m}\leq a_{n} a_{m} $ for all pairs of positive integers $m$ and $n,$ prove that the sequence $(\sqrt[n]{a_n })$ converges.
2009 Canadian Mathematical Olympiad Qualification Repechage, 9
Suppose that $m$ and $k$ are positive integers. Determine the number of sequences $x_1, x_2, x_3, \dots , x_{m-1}, x_m$ with
[list]
[*]$x_i$ an integer for $i = 1, 2, 3, \dots , m$,
[*]$1\le x_i \le k$ for $i = 1, 2, 3, \dots , m$,
[*]$x_1\neq x_m$, and
[*]no two consecutive terms equal.[/list]
1997 Singapore Senior Math Olympiad, 3
Find the smallest positive integer $x$ such that $x^2$ ends with the four digits $9009$.
2010 Today's Calculation Of Integral, 585
Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.
2015 Middle European Mathematical Olympiad, 5
Let $ABC$ be an acute triangle with $AB>AC$. Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$, $C$, $X$, and $Y$ lie on a circle and
$$\angle AXB-\angle ACB=\angle CYA-\angle CBA$$
holds, the line $XY$ passes through $D$.
2025 Harvard-MIT Mathematics Tournament, 10
Determine, with proof, all possible values of $\gcd(a^2+b^2+c^2,abc)$ across all triples of positive integers $(a,b,c).$
1989 IMO Longlists, 34
Prove the identity
\[ 1 \plus{} \frac{1}{2} \minus{} \frac{2}{3} \plus{} \frac{1}{4} \plus{} \frac{1}{5} \minus{} \frac{2}{6} \plus{} \ldots \plus{} \frac{1}{478} \plus{} \frac{1}{479} \minus{} \frac{2}{480}
\equal{} 2 \cdot \sum^{159}_{k\equal{}0} \frac{641}{(161\plus{}k) \cdot (480\minus{}k)}.\]
1995 Poland - First Round, 2
A number is called a palindromic number if its decimal representation read from the left to the right is the same as read from the right to the left. Let $(x_n)$ be the increasing sequence of all palindromic numbers. Determine all primes, which are divisors of at least one of the differences $x_{n+1} - x_n$.
2020 ASDAN Math Tournament, 4
There are $2$ ways to write $2020$ as a sum of $2$ squares: $2020 = a^2 + b^2$ and $2020 = c^2 + d^2$, where $a$, $b$, $c$, and $d$ are distinct positive integers with $a < b$ and $c < d$. Compute $a+b+c+d$.
2010 China Northern MO, 4
As shown in the figure, chess pieces are placed at the intersection points of the $64$ grid lines of the $7\times 7$ grid table. At most $1$ piece is placed at each point, and a total of $k$ left chess pieces are placed. No matter how they are placed, there will always be $4$ chess pieces, and the grid in which they are located the points form the four vertices of a rectangle (the sides of the rectangle are parallel to the grid lines). Try to find the minimum value of $k$.
[img]https://cdn.artofproblemsolving.com/attachments/5/b/23a79f43d3f4c9aade1ba9eaa7a282c3b3b86f.png[/img]
2012 Morocco TST, 4
$ABC$ is a non-isosceles triangle. $O, I, H$ are respectively the center of its circumscribed circle, the inscribed circle and its orthocenter.
prove that $\widehat{OIH}$ is obtuse.
1952 Moscow Mathematical Olympiad, 209
Prove the identity:
a) $(ax + by + cz)^2 + (bx + cy + az)^2 + (cx + ay + bz)^2 =(cx + by + az)^2 + (bx + ay + cz)^2 + (ax + cy + bz)^2$
b) $(ax + by + cz + du)^2+(bx + cy + dz + au)^2 +(cx + dy + az + bu)^2 + (dx + ay + bz + cu)^2 =$
$(dx + cy + bz + au)^2+(cx + by + az + du)^2 +(bx + ay + dz + cu)^2 + (ax + dy + cz + bu)^2$.
1961 Putnam, A7
Let $S$ be a nonempty closed set in the euclidean plane for which there is a closed disk $D$ containing $S$ such that $D$ is a subset of every closed disk that contains $S$. Prove that every point inside $D$ is the midpoint of a segment joining two points of $S.$
2018 Singapore Senior Math Olympiad, 5
Starting with any $n$-tuple $R_0$, $n\ge 1$, of symbols from $A,B,C$, we define a sequence $R_0, R_1, R_2,\ldots,$ according to the following rule: If $R_j= (x_1,x_2,\ldots,x_n)$, then $R_{j+1}= (y_1,y_2,\ldots,y_n)$, where $y_i=x_i$ if $x_i=x_{i+1}$ (taking $x_{n+1}=x_1$) and $y_i$ is the symbol other than $x_i, x_{i+1}$ if $x_i\neq x_{i+1}$. Find all positive integers $n>1$ for which there exists some integer $m>0$ such that $R_m=R_0$.
1997 Baltic Way, 17
A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$.
2001 National Olympiad First Round, 35
How many ordered pairs $(p,n)$ are there such that $(1+p)^n = 1+pn + n^p$ where $p$ is a prime and $n$ is a positive integer?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2006 AMC 10, 22
Two farmers agree that pigs are worth $ \$300$ and that goats are worth $ \$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $ \$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
$ \textbf{(A) } \$5\qquad \textbf{(B) } \$10\qquad \textbf{(C) } \$30\qquad \textbf{(D) } \$90\qquad \textbf{(E) } \$210$
2021 AMC 10 Fall, 8
A two-digit positive integer is said to be [i]cuddly[/i] if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$
2018 CMIMC Algebra, 5
Suppose $a$, $b$, and $c$ are nonzero real numbers such that \[bc+\frac1a = ca+\frac2b = ab+\frac7c = \frac1{a+b+c}.\] Find $a+b+c$.