Found problems: 85335
1992 IMO Shortlist, 16
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.
1986 National High School Mathematics League, 10
$x,y,z$ are nonnegative real numbers, and $4^{\sqrt{5x+9y+4z}}-68\times2^{\sqrt{5x+9y+4z}}+256=0$. Then, the product of the maximum and minimum value of $x+y+z$ is________.
1997 Singapore MO Open, 3
Find all the natural numbers $N$ which satisfy the following properties:
(i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and
(ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$.
Justify your answers.
2005 Silk Road, 3
Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$
are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$
2017 Online Math Open Problems, 15
Find the number of integers $1\leq k\leq1336$ such that $\binom{1337}{k}$ divides $\binom{1337}{k-1}\binom{1337}{k+1}$.
[i]Proposed by Tristan Shin[/i]
2022 AMC 12/AHSME, 11
Let $ f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n $, where $i = \sqrt{-1}$. What is $f(2022)$
$ \textbf{(A)}\ -2 \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ \sqrt{3} \qquad
\textbf{(E)}\ 2$
2008 Iran MO (3rd Round), 4
Let $ u$ be an odd number. Prove that $ \frac{3^{3u}\minus{}1}{3^u\minus{}1}$ can be written as sum of two squares.
2012 Postal Coaching, 2
Let $a_1, a_2,\cdots ,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible
by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product
$(a + a_1 - 1)(a + a_2 - 1) \cdots (a + a_n - 1)$.
2016 Brazil National Olympiad, 5
Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials
\(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\).
[list='a']
[*]Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\).
[*]Find how many integers \(m\) are such that \(P_n(m)<0\) for infinite positive integers \(n\).
[/list]
1958 AMC 12/AHSME, 46
For values of $ x$ less than $ 1$ but greater than $ \minus{}4$, the expression
\[ \frac{x^2 \minus{} 2x \plus{} 2}{2x \minus{} 2}
\]
has:
$ \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\
\textbf{(B)}\ \text{a minimum value of }{\plus{}1}\qquad \\
\textbf{(C)}\ \text{a maximum value of }{\plus{}1}\qquad \\
\textbf{(D)}\ \text{a minimum value of }{\minus{}1}\qquad \\
\textbf{(E)}\ \text{a maximum value of }{\minus{}1}$
2017 Miklós Schweitzer, 8
Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$. (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,:\, k,n\in\mathbb{Z}\right\}$, then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by
$$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and
$$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty\, ?$$
2018 Ecuador Juniors, 4
Given a positive integer $n > 1$ and an angle $\alpha < 90^o$, Jaime draws a spiral $OP_0P_1...P_n$ of the following form (the figure shows the first steps):
$\bullet$ First draw a triangle $OP_0P_1$ with $OP_0 = 1$, $\angle P_1OP_0 = \alpha$ and $P_1P_0O = 90^o$
$\bullet$ then for every integer $1 \le i \le n$ draw the point $P_{i+1}$ so that $\angle P_{i+1}OP_i = \alpha$, $\angle P_{i+1}P_iO = 90^o$ and $P_{i-1}$ and $P_{i+1}$ are in different half-planes with respect to the line $OP_i$
[img]https://cdn.artofproblemsolving.com/attachments/f/2/aa3913989dac1cf04f2b42b5d630b2e096dcb6.png[/img]
a) If $n = 6$ and $\alpha = 30^o$, find the length of $P_0P_n$.
b) If $n = 2018$ and $\alpha= 45^o$, find the length of $P_0P_n$.
2023 Hong Kong Team Selection Test, Problem 1
Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.
2012 Indonesia TST, 3
Let $S$ be a subset of $\{1,2,3,4,5,6,7,8,9,10\}$. If $S$ has the property that the sums of three elements of $S$ are all different, find the maximum number of elements of $S$.
2023 Durer Math Competition Finals, 4
Benedek wrote down the following numbers: $1$ piece of one, $2$ pieces of twos, $3$ pieces of threes, $... $, $50$ piecies of fifties. How many digits did Benedek write down?
2005 AMC 12/AHSME, 6
In $ \triangle ABC$, we have $ AC \equal{} BC \equal{} 7$ and $ AB \equal{} 2$. Suppose that $ D$ is a point on line $ AB$ such that $ B$ lies between $ A$ and $ D$ and $ CD \equal{} 8$. What is $ BD$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 2 \sqrt {3}\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 4 \sqrt {2}$
1998 India National Olympiad, 4
Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.
2025 AIME, 8
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collection to exceed $N.$ For example, to get 42 cents, Silas will choose a 25-cent coin, then a 10-cent coin, then 7 1-cent coins. However, this collection of 9 coins uses more coins than necessary to get a total of 42 cents; indeed, choosing 4 10-cent coins and 2 1-cent coins achieves the same total value with only 6 coins. In general, the greedy algorithm succeeds for a given $N$ if no other collection of 1-cent, 10-cent, and 25-cent coins gives a total value of $N$ cents using strictly fewer coins than the collection given by the greedy algorithm. Find the number of values of $N$ between $1$ and $1000$ inclusive for which the greedy algorithm succeeds.
2014 Purple Comet Problems, 28
Find the number of ordered triples of positive integers $(a, b, c)$ such that $abc$ divides $(ab + 1)(bc + 1)(ca + 1)$.
1999 Taiwan National Olympiad, 2
Let $a_{1},a_{2},...,a_{1999}$ be a sequence of nonnegative integers such that for any $i,j$ with $i+j\leq 1999$ , $a_{i}+a_{j}\leq a_{i+j}\leq a_{i}+a_{j}+1$. Prove that there exists a real number $x$ such that $a_{n}=[nx]\forall n$.
Kvant 2023, M2772
7. There are 100 chess bishops on white squares of a $100 \times 100$ chess board. Some of them are white and some of them are black. They can move in any order and capture the bishops of opposing color. What number of moves is sufficient for sure to retain only one bishop on the chess board?
2013 Romania National Olympiad, 3
Find all injective functions$f:\mathbb{Z}\to \mathbb{Z}$ that satisfy:
$\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$.
2016 Latvia National Olympiad, 3
Prove that for every integer $n$ ($n > 1$) there exist two positive integers $x$ and $y$ ($x \leq y$) such that
$$\frac{1}{n} = \frac{1}{x(x+1)} + \frac{1}{(x+1)(x+2)} + \cdots + \frac{1}{y(y+1)}$$
2006 IMO Shortlist, 5
In triangle $ABC$, let $J$ be the center of the excircle tangent to side $BC$ at $A_{1}$ and to the extensions of the sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$ respectively. Suppose that the lines $A_{1}B_{1}$ and $AB$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular from $C_{1}$ to line $DJ$. Determine the angles $\angle{BEA_{1}}$ and $\angle{AEB_{1}}$.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
2009 May Olympiad, 3
In the following sum: $1 + 2 + 3 + 4 + 5 + 6$, if we remove the first two “+” signs, we obtain the new sum $123 + 4 + 5 + 6 = 138$. By removing three “$+$” signs, we can obtain $1 + 23 + 456 = 480$. Let us now consider the sum $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13$, in which some “$+$” signs are to be removed. What are the three smallest multiples of $100$ that we can get in this way?