Found problems: 85335
2008 China National Olympiad, 2
Find the smallest integer $n$ satisfying the following condition: regardless of how one colour the vertices of a regular $n$-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour.
Kvant 2019, M2565
We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich exactly), and it shows an arbitrary result on every turn. What is the smallest number of turns required to find the heaviest coin?
[hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]
1991 Greece National Olympiad, 1
Find all polynomials $P(x)$ , such that $$P(x^3+1)=\left(P (x+1)\right)^3$$
2022 HMNT, 1
Compute $\sqrt{2022^2-12^6}.$
1993 IMO Shortlist, 2
A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$
a.) Show that every prime number $n$ has property $P.$
b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$
2023 Assara - South Russian Girl's MO, 3
In equality
$$1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023$$
Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used?
2002 BAMO, 4
For $n \ge 1$, let $a_n$ be the largest odd divisor of $n$, and let $b_n = a_1+a_2+...+a_n$.
Prove that $b_n \ge \frac{ n^2 + 2}{3}$, and determine for which $n$ equality holds.
For example, $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3$, thus $b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23$
.
1998 Czech and Slovak Match, 1
Let $P$ be an interior point of the parallelogram $ABCD$. Prove that $\angle APB+ \angle CPD = 180^\circ$ if and only if $\angle PDC = \angle PBC$.
1986 Traian Lălescu, 2.1
Consider the numbers $ a_n=1-\binom{n}{3} +\binom{n}{6} -\cdots, b_n= -\binom{n}{1} +\binom{n}{4}-\binom{n}{7} +\cdots $ and $ c_n=\binom{n}{2} -\binom{n}{5} +\binom{n}{8} -\cdots , $ for a natural number $ n\ge 2. $ Prove that
$$ a_n^2+b_n^2+c_n^2-a_nb_n-b_nc_n-c_na_n =3^{n-1}. $$
2021 Argentina National Olympiad, 4
The sum of several positive integers, not necessarily different, all of them less than or equal to $10$, is equal to $S$. We want to distribute all these numbers into two groups such that the sum of the numbers in each group is less than or equal to $80.$ Determine all values of $S$ for which this is possible.
2005 Today's Calculation Of Integral, 72
Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$
Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.
2007 F = Ma, 10
Two wheels with fixed hubs, each having a mass of $1 \text{ kg}$, start from rest, and forces are applied as shown. Assume the hubs and spokes are massless, so that the rotational inertia is $I = mR^2$. In order to impart identical angular accelerations about their respective hubs, how large must $F_2$ be?
[asy]
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw(circle((0,0),0.5));
draw((1, 0.5)--(0,0.5)--(0,-0.5),BeginArrow);
draw((-0.5,0)--(0.5,0));
draw((-0.5*sqrt(2)/2, 0.5*sqrt(2)/2)--(0.5*sqrt(2)/2,-0.5*sqrt(2)/2));
draw((0.5*sqrt(2)/2, 0.5*sqrt(2)/2)--(-0.5*sqrt(2)/2,-0.5*sqrt(2)/2));
label("$R$ = 0.5 m", (0, -0.5),S);
label("$F_1$ = 1 N",(1,0.5),N);
draw(circle((3,0.5),1));
draw((4.5,1.5)--(3,1.5)--(3,-0.5),BeginArrow);
draw((2,0.5)--(4,0.5));
draw((3-sqrt(2)/2, 0.5+sqrt(2)/2)--(3+sqrt(2)/2, 0.5-sqrt(2)/2));
draw((3+sqrt(2)/2, 0.5+sqrt(2)/2)--(3-sqrt(2)/2,0.5-sqrt(2)/2));
label("$F_2$", (4.5, 1.5), N);
label("$R$ = 1 m",(3, -0.5),S);
[/asy]
$ \textbf{(A)}\ 0.25\text{ N}\qquad\textbf{(B)}\ 0.5\text{ N}\qquad\textbf{(C)}\ 1\text{ N}\qquad\textbf{(D)}\ 2\text{ N}\qquad\textbf{(E)}\ 4\text{ N} $
1967 Polish MO Finals, 4
Prove that the polynomial $ x^3 + x + 1 $ is a factor of the polynomial $ P_n(x) = x^{n + 2} + (x+1)^{2n+1} $ for every integer $ n \geq 0 $.
2012 CentroAmerican, 1
Trilandia is a very unusual city. The city has the shape of an equilateral triangle of side lenght 2012. The streets divide the city into several blocks that are shaped like equilateral triangles of side lenght 1. There are streets at the border of Trilandia too. There are 6036 streets in total. The mayor wants to put sentinel sites at some intersections of the city to monitor the streets. A sentinel site can monitor every street on which it is located. What is the smallest number of sentinel sites that are required to monitor every street of Trilandia?
2020 CMIMC Combinatorics & Computer Science, 9
Let $\Gamma = \{\varepsilon,0,00,\ldots\}$ be the set of all finite strings consisting of only zeroes. We consider $\textit{six-state unary DFAs}$ $D = (F,q_0,\delta)$ where $F$ is a subset of $Q = \{1,2,3,4,5,6\}$, not necessarily strict and possibly empty; $q_0\in Q$ is some $\textit{start state}$; and $\delta: Q\rightarrow Q$ is the $\textit{transition function}$.
For each such DFA $D$, we associate a set $F_D\subseteq\Gamma$ as the set of all strings $w\in\Gamma$ such that
\[\underbrace{\delta(\cdots(\delta(q_0))\cdots)}_{|w|\text{ applications}}\in F,\]
We say a set $\mathcal D$ of DFAs is $\textit{diverse}$ if for all $D_1,D_2\in\mathcal D$ we have $F_{D_1}\neq F_{D_2}$. What is the maximum size of a diverse set?
2012 Online Math Open Problems, 46
If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$.
[i]Author: Alex Zhu[/i]
2023 Chile Junior Math Olympiad, 5
$1600$ bananas are distributed among $100$ monkeys (it is possible that some monkeys do not receive bananas). Prvove that at least four monkeys receive the same amount of bananas.
1998 All-Russian Olympiad, 7
Let n be an integer at least 4. In a convex n-gon, there is NO four vertices lie on a same circle. A circle is called circumscribed if it passes through 3 vertices of the n-gon and contains all other vertices. A circumscribed circle is called boundary if it passes through 3 consecutive vertices, a circumscribed circle is called inner if it passes through 3 pairwise non-consecutive points. Prove the number of boundary circles is 2 more than the number of inner circles.
2014 AMC 10, 24
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is [i]bad[/i] if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
$ \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 $
2023 Taiwan TST Round 2, N
Let $f_n$ be a polynomial with real coefficients for all $n \in \mathbb{Z}$. Suppose that
\[f_n(k) = f_{n+k}(k) \quad n, k \in \mathbb{Z}.\]
(a) Does $f_n = f_m$ necessarily hold for all $m,n \in \mathbb{Z}$?
(b) If furthermore $f_n$ is a polynomial with integer coefficients for all $n \in\mathbb{Z}$, does $f_n = f_m$ necessarily hold for all $m, n \in\mathbb{Z}$?
[i]Proposed by usjl[/i]
2007 Tuymaada Olympiad, 1
Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.
2004 Tournament Of Towns, 2
A box contains red, green, blue, and white balls, 111 balls in all. If you take out 100 balls without looking, then there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to guarantee that among them there will always be balls of at least 3 different colors?
2016 USAMTS Problems, 1:
Fill in each cell of the grid with one of the numbers 1, 2, or 3. After all numbers are filled in, if a row, column, or any diagonal has a number of cells equal to a multiple of 3, then it must have the same amount of 1’s, 2’s, and 3’s. (There are 10 such diagonals, and they are all marked in the grid by a gray dashed line.) Some numbers have been given to you.
[asy]
defaultpen(linewidth(0.45));
real[][] arr = {
{0, 2, 1, 0, 0, 0, 0, 0, 0},
{3, 0, 0, 2, 0, 0, 0, 0, 0},
{0, 0, 0, 2, 0, 0, 3, 2, 0},
{0, 2, 1, 0, 0, 1, 0, 0, 3},
{3, 0, 0, 0, 0, 3, 0, 0, 3},
{2, 0, 0, 0, 0, 0, 2, 3, 0},
{3, 2, 3, 2, 0, 2, 0, 0, 3},
{0, 0, 0, 0, 0, 3, 0, 0, 1},
{0, 0, 0, 0, 0, 0, 1, 3, 0}};
for (int i=0; i<9; ++i){
for (int j=0; j<9; ++j){
draw((i,j)--(i+1,j)--(i+1, j+1)--(i,j+1)--cycle);
if(arr[8-j][i] != 0){
label((string) arr[8-j][i], (i+0.5, j+0.5));
}
}
}
draw((3,0)--(0,3), linetype(new real[] {4,4})+grey);
draw((6,0)--(0,6), linetype(new real[] {4,4})+grey);
draw((9,0)--(0,9), linetype(new real[] {4,4})+grey);
draw((3,9)--(9,3), linetype(new real[] {4,4})+grey);
draw((6,9)--(9,6), linetype(new real[] {4,4})+grey);
draw((6,0)--(9,3), linetype(new real[] {4,4})+grey);
draw((3,0)--(9,6), linetype(new real[] {4,4})+grey);
draw((0,0)--(9,9), linetype(new real[] {4,4})+grey);
draw((0,3)--(6,9), linetype(new real[] {4,4})+grey);
draw((0,6)--(3,9), linetype(new real[] {4,4})+grey);
[/asy]
You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
2024 IFYM, Sozopol, 8
Let \( ABC \) and \( A_1B_1C_1 \) be two triangles such that the segments \( AA_1 \) and \( BC \) intersect, the segments \( BB_1 \) and \( AC \) intersect, and the segments \( CC_1 \) and \( AB \) intersect. If it is known that there exists a point \( X \) inside both triangles such that
\[
\begin{aligned}
\angle XAB &= \angle XA_1B_1, &\angle XBC &= \angle XC_1A_1, &\angle XCA &= \angle XB_1C_1,\\
\angle XAC &= \angle XB_1A_1, &\angle XBA &= \angle XA_1C_1, &\angle XCB &= \angle XC_1B_1.
\end{aligned}
\]
Prove that the lines \( AC_1 \), \( BB_1 \), and \( CA_1 \) are concurrent or parallel.
2009 Brazil Team Selection Test, 1
Let $r$ be a positive real number. Prove that the number of right triangles with prime positive integer sides that have an inradius equal to $r$ are zero or a power of $2$.
[hide=original wording]Seja r um numero real positivo. Prove que o numero de triangulos retangulos com lados inteiros positivos primos entre si que possuem inraio igual a r e zero ou uma potencia de 2.[/hide]