Found problems: 85335
2009 Indonesia TST, 1
Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.
2020 IMO Shortlist, A5
A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?
2011 IMC, 2
Does there exist a real $3\times 3$ matrix $A$ such that $\text{tr}(A)=0$ and $A^2+A^t=I?$ ($\text{tr}(A)$ denotes the trace of $A,\ A^t$ the transpose of $A,$ and $I$ is the identity matrix.)
[i]Proposed by Moubinool Omarjee, Paris[/i]
2014 Turkey EGMO TST, 3
Denote by $d(n)$ be the biggest prime divisor of $|n|>1$. Find all polynomials with integer coefficients satisfy;
$$P(n+d(n))=n+d(P(n)) $$
for the all $|n|>1$ integers such that $P(n)>1$ and $d(P(n))$ can be defined.
2019 Tournament Of Towns, 1
The polynomial P(x,y) is such that for every integer n >= 0 each of the polynomials P(x,n) and P(n,y) either is a constant zero or has a degree not greater than n. Is it possible that P(x,x) has an odd degree?
2021 MMATHS, 1
Let $a,b,c$ be the roots of the polynomial $x^3 - 20x^2 + 22.$ Find \[\frac{bc}{a^2}+\frac{ac}{b^2}+\frac{ab}{c^2}.\]
[i]Proposed by Deyuan Li and Andrew Milas[/i]
1975 Poland - Second Round, 6
Let $ f(x) $ and $ g(x) $ be polynomials with integer coefficients. Prove that if for every integer value $ n $ the number $ g(n) $ is divisible by the number $ f(n) $, then $ g(x) = f(x)\cdot h(x) $, where $ h(x) $ is a polynomial,. Show with an example that the coefficients of the polynomial $ h(x) $ do not have to be integer.
1952 AMC 12/AHSME, 18
$ \log p \plus{} \log q \equal{} \log (p \plus{} q)$ only if:
$ \textbf{(A)}\ p \equal{} q \equal{} 0 \qquad\textbf{(B)}\ p \equal{} \frac {q^2}{1 \minus{} q} \qquad\textbf{(C)}\ p \equal{} q \equal{} 1$
$ \textbf{(D)}\ p \equal{} \frac {q}{q \minus{} 1} \qquad\textbf{(E)}\ p \equal{} \frac {q}{q \plus{} 1}$
2024 MMATHS, 10
In acute $\triangle{ABC},$ $AB=11$ and $CB=10.$ Points $E$ and $D$ are constructed such that $\angle{CBE}$ and $\angle{ABD}$ are right, and $ACEBD$ is a non-degenerate pentagon. Additionally, $\angle{AEB} \cong \angle{DCB}, AE=CD,$ and $ED=20.$ Given that $EA$ and $CD$ intersect at $P$ and $AP=4,$ find $CP^2.$
2023 BMT, 4
Let f$(x)$ be a continuous function over the real numbers such that for every integer $n$, $f(n) = n^2$ and $f(x) $ is linear over the interval $[n, n + 1]$. There exists a unique two-variable polynomial $g$ such that $g(x, \lfloor x \rfloor) = f(x)$ for all $x$. Compute $g(20, 23)$. (Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. For example, $\lfloor 2\rfloor = 2$ and $\lfloor -3.5 \rfloor = -4$.)
2013 Czech-Polish-Slovak Junior Match, 5
Let $a, b, c$ be positive real numbers for which $ab + ac + bc \ge a + b + c$. Prove that $a + b + c \ge 3$.
IV Soros Olympiad 1997 - 98 (Russia), 10.4
Solve the equation $$ \sqrt{\sqrt{2x^2+x-3}+2x^2-3}=x.$$
2010 APMO, 5
Find all functions $f$ from the set $\mathbb{R}$ of real numbers into $\mathbb{R}$ which satisfy for all $x, y, z \in \mathbb{R}$ the identity \[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\]
2017 Online Math Open Problems, 18
Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks a integer $c \in \{2,3,\dots,p-1\}$.
Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper.
Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$, Granite writes $f(f(f(x)))$, and so on, with the players taking turns writing.
The game ends when two numbers appear on the paper whose difference is a multiple of $p$, and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy.
[i]Proposed by Yang Liu[/i]
2001 Mediterranean Mathematics Olympiad, 2
Find all integers $n$ for which the polynomial $p(x) = x^5 -nx -n -2$ can be represented as a product of two non-constant polynomials with integer coefficients.
2020 MBMT, 39
Let $f(x) = \sqrt{4x^2 - 4x^4}$. Let $A$ be the number of real numbers $x$ that satisfy
$$f(f(f(\dots f(x)\dots ))) = x,$$ where the function $f$ is applied to $x$ 2020 times. Compute $A \pmod {1000}$.
[i]Proposed by Timothy Qian[/i]
1956 AMC 12/AHSME, 45
A wheel with a rubber tire has an outside diameter of $ 25$ in. When the radius has been decreased a quarter of an inch, the number of revolutions in one mile will:
$ \textbf{(A)}\ \text{be increased about }2\% \qquad\textbf{(B)}\ \text{be increased about }1\%$
$ \textbf{(C)}\ \text{be increased about }20\% \qquad\textbf{(D)}\ \text{be increased about }\frac {1}{2}\% \qquad\textbf{(E)}\ \text{remain the same}$
2019 Junior Balkan Team Selection Tests - Romania, 1
If $a, b, c$ are real numbers such that a$b + bc + ca = 0$, prove the inequality $$2(a^2 + b^2 + c^2)(a^2b^2 + b^2c^2 + c^2a^2) \ge 27a^2b^2c^2$$
When does the equality hold ?
Leonard Giugiuc
2022 MMATHS, 10
Suppose that $A_1A_2A_3$ is a triangle with $A_1A_2 = 16$ and $A_1A_3 = A_2A_3 = 10$. For each integer $n \ge 4$, set An to be the circumcenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. There exists a unique point $Z$ lying in the interiors of the circumcircles of triangles $A_kA_{k+1}A_{k+2}$ for all integers $k \ge 1$. If $ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$.
2012 ELMO Problems, 4
Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$.
[i]David Yang.[/i]
2025 Vietnam National Olympiad, 6
Let $a,b,c$ be non-negative numbers such that $a+b+c=3.$ Prove that
\[\sqrt{3a^3+4bc+b+c}+\sqrt{3b^3+4ca+c+a}+\sqrt{3c^3+4ab+a+b} \geqslant 9.\]
2012 South East Mathematical Olympiad, 3
For composite number $n$, let $f(n)$ denote the sum of the least three divisors of $n$, and $g(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$, such that $g(n)=(f(n))^m$ ($m\in N^*$).
2019 ASDAN Math Tournament, 8
Let triangle $\vartriangle AEF$ be inscribed in a square $ABCD$ such that $E$ lies on $BC$ and $F$ lies on $CD$. If $\angle EAF = 45^o$ and $\angle BEA = 70^o$, compute $\angle CF E$.
1988 Iran MO (2nd round), 2
In a cyclic quadrilateral $ABCD$, let $I,J$ be the midpoints of diagonals $AC, BD$ respectively and let $O$ be the center of the circle inscribed in $ABCD.$ Prove that $I, J$ and $O$ are collinear.
2020 Yasinsky Geometry Olympiad, 2
An equilateral triangle $BDE$ is constructed on the diagonal $BD$ of the square $ABCD$, and the point $C$ is located inside the triangle $BDE$. Let $M$ be the midpoint of $BE$. Find the angle between the lines $MC$ and $DE$.
(Dmitry Shvetsov)