Found problems: 15925
2023 China National Olympiad, 1
Define the sequences $(a_n),(b_n)$ by
\begin{align*}
& a_n, b_n > 0, \forall n\in\mathbb{N_+} \\
& a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\
& b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}}
\end{align*}
1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$;
2) If $a_{100} = b_{99}$, determine which is larger between $a_{100}+b_{100}$ and $a_{101}+b_{101}$.
2019 Iran MO (2nd Round), 5
Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$.
Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!
2002 Czech and Slovak Olympiad III A, 6
Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equality
\[f(xf(y))=f(xy)+x\]
2007 Bulgaria Team Selection Test, 2
Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$
2023 ISI Entrance UGB, 7
(a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta= XY+YZ+ZX$ and $\gamma = XYZ$.
(b) Let $G_n=x^n \sin(nA)+y^n \sin(nB)+z^n \sin(nC)$, where $x,y,z, A,B,C$ are real numbers such that $A+B+C$ is an integral multiple of $\pi$. Using (a) or otherwise show that if $G_1=G_2=0$, then $G_n=0$ for all positive integers $n$.
1989 IMO Shortlist, 3
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?
2023 LMT Fall, 20
The remainder when $x^{100} -x^{99} +... -x +1$ is divided by $x^2 -1$ can be written in the form $ax +b$. Find $2a +b$.
[i]Proposed by Calvin Garces[/i]
2015 Brazil Team Selection Test, 1
Let's call a function $f : R \to R$ [i]cool[/i] if there are real numbers $a$ and $b$ such that $f(x + a)$ is an even function and $f(x + b)$ is an odd function.
(a) Prove that every cool function is periodic.
(b) Give an example of a periodic function that is not cool.
2013 Korea National Olympiad, 4
$\{a_n\}$ is a positive integer sequence such that $ a_{i+2} = a_{i+1} + a_{i} (i \ge 1) $. For positive integer $n$, define $\{b_n\}$ as
\[ b_n = \frac{1}{a_{2n+1}} \sum_{i=1}^{4n-2} { a_i } \]
Prove that $b_n$ is positive integer, and find the general form of $b_n$.
2019 Estonia Team Selection Test, 10
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.
DMM Individual Rounds, 2007 Tie
[b]p1.[/b] Let $p_b(m)$ be the sum of digits of $m$ when $m$ is written in base $b$. (So, for example, $p_2(5) = 2$). Let $f(0) = 2007^{2007}$, and for $n \ge 0$ let $f(n + 1) = p_7(f(n))$. What is $f(10^{10000})$?
[b]p2.[/b] Compute:
$$\sum^{\infty}_{n=1}\frac{(-1)^{n+1}4n}{n^4 - 8n^2 + 4}.$$
[b]p3.[/b] $ABCDEFGH$ is an octagon whose eight interior angles all have the same measure. The lengths of the eight sides of this octagon are, in some order, $$2, 2\sqrt2, 4, 4\sqrt2, 6, 7, 7, \,\,\, and \,\,\, 8.$$
Find the area of $ABCDEFGH$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2014 International Zhautykov Olympiad, 1
Does there exist a polynomial $P(x)$ with integral coefficients such that $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5 $?
[i]Proposed by Alexander S. Golovanov, Russia[/i]
2024 Chile TST Ibero., 1
Determine all integers \( x \) for which the expression \( x^2 + 10x + 160 \) is a perfect square.
1954 Moscow Mathematical Olympiad, 266
Find all solutions of the system consisting of $3$ equations:
$x \left(1 - \frac{1}{2^n}\right) +y \left(1 - \frac{1}{2^{n+1}}\right) +z \left(1 - \frac{1}{2^{n+2}}\right) = 0$ for $n = 1, 2, 3$.
2025 CMIMC Algebra/NT, 4
Consider the system of equations $$\log_x y +\log_y z + \log_z x =8$$ $$\log_{\log_y x}z = -3$$ $$\log_z y + \log_x z = 16$$
Find $z.$
2016 India PRMO, 9
Let $a$ and $b$ be the roots of the equation $x^2 + x - 3 = 0$. Find the value of the expression $4 b^2 -a^3$.
2012 Saint Petersburg Mathematical Olympiad, 1
Find all integer $b$ such that $[x^2]-2012x+b=0$ has odd number of roots.
2010 Romania National Olympiad, 2
We say that a ring $A$ has property $(P)$ if any non-zero element can be written uniquely as the sum of an invertible element and a non-invertible element.
a) If in $A$, $1+1=0$, prove that $A$ has property $(P)$ if and only if $A$ is a field.
b) Give an example of a ring that is not a field, containing at least two elements, and having property $(P)$.
[i]Dan Schwarz[/i]
2006 Putnam, A2
Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)
2015 Baltic Way, 3
Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]
IV Soros Olympiad 1997 - 98 (Russia), 11.12
Find how many different solutions depending on $a$ has the system of equations :
$$\begin{cases} x+z=2a
\\ y+u+xz=a-3
\\ yz+xu=2a
\\ yu=1
\end{cases}$$
2022 Kyiv City MO Round 2, Problem 2
Initially memory of computer contained a single polynomial $x^2-1$. Every minute computer chooses any polynomial $f(x)$ from its memory and writes $f(x^2-1)$ and $f(x)^2-1$ to it, or chooses any two distinct polynomials $g(x), h(x)$ from its memory and writes polynomial $\frac{g(x) + h(x)}{2}$ to it (no polynomial is ever erased from its memory). Can it happen that after some time, memory of computer contains $P(x) = \frac{1}{1024}(x^2-1)^{2048} - 1$?
[i](Proposed by Arsenii Nikolaiev)[/i]
1991 IMO Shortlist, 17
Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$
2023 Princeton University Math Competition, A5 / B7
Compute $\left\lfloor \sum_{k=0}^{10}\left(3+2\cos\left(\frac{2k\pi}{11}\right)\right)^{10}\right\rfloor \pmod{100}.$
2010 Purple Comet Problems, 9
Find positive integer $n$ so that $\tfrac{80-6\sqrt{n}}{n}$ is the reciprocal of $\tfrac{80+6\sqrt{n}}{n}.$