Found problems: 85335
2024 Lusophon Mathematical Olympiad, 5
In a $9\times9$ board, the squares are labeled from 11 to 99, with the first digit indicating the row and the second digit indicating the column.
One would like to paint the squares in black or white in a way that each black square is adjacent to at most one other black square and each white square is adjacent to at most one other white square. Two squares are adjacent if they share a common side.
How many ways are there to paint the board such that the squares $44$ and $49$ are both black?
2023 CCA Math Bonanza, T4
Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Let $E$ be the foot from $B$ to $AC$ and $F$ be the foot from $C$ to $AB.$ Denote $M$ the midpoint of $BC.$ The circumcircles of $\triangle BMF$ and $\triangle CME$ meet at another point $G.$ Compute the length of $GC.$
[i]Team #4[/i]
2011 HMNT, 3
Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur?
MOAA Team Rounds, 2021.1
The value of
\[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
2004 ITAMO, 5
Decide if the following statement is true or false:
For every sequence $\{x_n\}_{n\in \mathbb{N}}$ of non-negative real numbers, there exist sequences $\{a_n\}_{n\in\mathbb{N}}$ and $\{b_n\}_{n\in\mathbb{N}}$ of non-negative real numbers such that:
(a) $x_n = a_n + b_n$ for all $n$;
(b) $a_1 + \cdots + a_n \le n$ for infinitely many values of $n$;
(c) $b_1 + \cdots + b_n \le n$ for infinitely many values of $n$.
Gheorghe Țițeica 2025, P2
Let $n\geq 2$ and consider the functions $f,g:\{1,2,\dots ,n\}\rightarrow\{1,2,\dots ,n\}$ such that $$g(k)=|\{i\mid f(i)\leq f(k)\}|$$ for all $1\leq k\leq n$.
[list=a]
[*] Show that $f$ is bijective if and only if $g$ is bijective.
[*] If $g$ is a given function, find how many functions $f$ (in terms of $g$) satisfy the hypothesis.
[/list]
[i]Silviu Cristea[/i]
2020 USOMO, 1
Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$. A variable point $X$ is chosen on minor arc $AB$ of $\omega$, and segments $CX$ and $AB$ meet at $D$. Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$, respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized.
[i]Proposed by Zuming Feng[/i]
2016 Indonesia TST, 6
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
LMT Speed Rounds, 2010.4
Determine the largest positive integer that is a divisor of all three of $A=2^{2010}\times3^{2010}, B=3^{2010}\times5^{2010},$ and $C=5^{2010}\times2^{2010}.$
2020 Greece Team Selection Test, 1
Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that
$f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$.
by Athanasios Kontogeorgis (aka socrates)
2021 HMNT, 6
Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying
$$3a + 5b \equiv 19 \,\,\, (mod \,\,\, n + 1)$$
$$4a + 2b \equiv 25 \,\,\, (mod \,\,\, n + 1)$$
Find $ 2a + 6b$.
2001 AMC 12/AHSME, 3
The state income tax where Kristin lives is levied at the rate of $ p \%$ of the first $ \$28000$ of annual income plus $ (p \plus{} 2) \%$ of any amount above $ \$28000$. Kristin noticed that the state income tax she paid amounted to $ (p \plus{} 0.25) \%$ of her annual income. What was her annual income?
$ \textbf{(A)} \ \$28000 \qquad \textbf{(B)} \ \$32000 \qquad \textbf{(C)} \ \$35000 \qquad \textbf{(D)} \ \$42000 \qquad \textbf{(E)} \ \$56000$
2016 NIMO Problems, 7
Suppose $a$, $b$, $c$, and $d$ are positive real numbers which satisfy the system of equations \[\begin{aligned} a^2+b^2+c^2+d^2 &= 762, \\ ab+cd &= 260, \\ ac+bd &= 365, \\ ad+bc &= 244. \end{aligned}\] Compute $abcd.$
[i]Proposed by Michael Tang[/i]
2010 ISI B.Math Entrance Exam, 1
Prove that in each year , the $13^{th}$ day of some month occurs on a Friday .
1995 Tournament Of Towns, (461) 6
Does there exist a nonconvex polyhedron such that not one of its vertices is visible from a point $M$ outside it? (The polyhedron is made out of an opaque material.)
(AY Belov, S Markelov)
1999 Bosnia and Herzegovina Team Selection Test, 5
For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that
$a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$
$b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$
where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$
1979 IMO Longlists, 19
For $k = 1, 2, \ldots$ consider the $k$-tuples $(a_1, a_2, \ldots, a_k)$ of positive integers such that
\[a_1 + 2a_2 + \cdots + ka_k = 1979.\]
Show that there are as many such $k$-tuples with odd $k$ as there are with even $k$.
2024 CMIMC Integration Bee, 15
\[\int_0^\infty 1+\cos\left(\tfrac 1{\sqrt x}\right)-2\cos\left(\tfrac 1{\sqrt {2x}}\right)\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2014 IFYM, Sozopol, 8
Prove that, if $a,b,c$ are sides of a triangle, then we have the following inequality:
$3(a^3 b+b^3 c+c^3 a)+2(ab^3+bc^3+ca^3 )\geq 5(a^2 b^2+a^2 c^2+b^2 c^2 )$.
2015 Kyiv Math Festival, P3
Is it true that every positive integer greater than $50$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?
2004 Harvard-MIT Mathematics Tournament, 10
A [i]lattice point[/i] is a point whose coordinates are both integers. Suppose Johann walks in a line from the point $(0, 2004)$ to a random lattice point in the interior (not on the boundary) of the square with vertices $(0, 0)$, $(0, 99)$, $(99,99)$, $(99, 0)$. What is the probability that his path, including the endpoints, contains an even number of lattice points?
2014 Peru Iberoamerican Team Selection Test, P1
Circles $C_1$ and $C_2$ intersect at different points $A$ and $B$. The straight lines tangents to $C_1$ that pass through $A$ and $B$ intersect at $T$. Let $M$ be a point on $C_1$ that is out of $C_2$. The $MT$ line intersects $C_1$ at $C$ again, the $MA$ line intersects again to $C_2$ in $K$ and the line $AC$ intersects again to the circumference $C_2$ in $L$. Prove that the $MC$ line passes through the midpoint of the $KL$ segment.
2002 USA Team Selection Test, 4
Let $n$ be a positive integer and let $S$ be a set of $2^n+1$ elements. Let $f$ be a function from the set of two-element subsets of $S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any elements $x, y, z$ of $S$, one of $f(\{x,y\}), f(\{y,z\}), f(\{z, x\})$ is equal to the sum of the other two. Show that there exist $a, b, c$ in $S$ such that $f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$ are all equal to 0.
1979 Czech And Slovak Olympiad IIIA, 5
Given a triangle $ABC$ with side sizes $a \ge b \ge c$. Among all pairs of points $X, Y$ on the boundary of triangle $ABC$, which this boundary divides into two parts of equal length, find all such for which the distance is $X Y$ maximum.
1949-56 Chisinau City MO, 13
Factor the polynomial $(a+b+c)^3- a^3 -b^3 -c^3$