Found problems: 85335
2024 Indonesia TST, 3
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
2020 USMCA, 23
Let $f_n$ be a sequence defined by $f_0=2020$ and
\[f_{n+1} = \frac{f_n + 2020}{2020f_n + 1}\]
for all $n \geq 0$. Determine $f_{2020}$.
2018 Centroamerican and Caribbean Math Olympiad, 4
Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$.
1981 IMO Shortlist, 14
Prove that a convex pentagon (a five-sided polygon) $ABCDE$ with equal sides and for which the interior angles satisfy the condition $\angle A \geq \angle B \geq \angle C \geq \angle D \geq \angle E$ is a regular pentagon.
PEN H Problems, 8
Show that the equation \[x^{3}+y^{3}+z^{3}+t^{3}=1999\] has infinitely many integral solutions.
2023 Regional Olympiad of Mexico West, 3
Let $x>1$ be a real number that is not an integer. Denote $\{x\}$ as its decimal part and $\lfloor x\rfloor$ the floor function. Prove that
$$ \left(\frac{x+\{x\}}{\lfloor x\rfloor}-\frac{\lfloor x\rfloor}{x+\{x\}}\right)+\left(\frac{x+\lfloor x\rfloor}{\{x\}}-\frac{\{x\}}{x+\lfloor x\rfloor}\right)>\frac{16}{3}$$
1998 Belarusian National Olympiad, 8
a) Prove that for no real a such that $0 \le a <1$ there exists a function defined on the set of all positive numbers and taking values in the same set, satisfying for all positive $x$ the equality $$f\left(f(x)+\frac{1}{f(x)}\right)=x+a \,\,\,\,\,\,\, (*) $$
b) Prove that for any $a>1$ there are infinitely many functions defined on the set of all positive numbers, with values in the same set, satisfying ($*$) for all positive x.
2003 China Girls Math Olympiad, 8
Let $ n$ be a positive integer, and $ S_n,$ be the set of all positive integer divisors of $ n$ (including 1 and itself). Prove that at most half of the elements in $ S_n$ have their last digits equal to 3.
2016 ASMT, 6
Let $ABC$ be a triangle with $AB = 5$ and $AC = 4$. Let $D$ be the reflection of $C$ across $AB$, and let $E$ be the reflection of $B$ across $AC$. $D$ and $E$ have the special property that $D, A, E$ are collinear. Finally, suppose that lines $DB$ and $EC$ intersect at a point $F$. Compute the area of $\vartriangle BCF$.
2000 National Olympiad First Round, 2
Discriminant of a second degree polynomial with integer coefficients cannot be
$ \textbf{(A)}\ 23
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 25
\qquad\textbf{(D)}\ 28
\qquad\textbf{(E)}\ 33
$
2023 CCA Math Bonanza, L5.1
Estimate the number of ordered pairs $(a,b)$ of relatively prime positive integers such that $a+b<1412.$ Your score is determined by the function $max\{0, 20 - \lfloor \frac{|A - E|}{500}\rfloor\}$where $A$ is the actual answer, and $E$ is your estimate.
[i]Lightning 5.1[/i]
2020 Poland - Second Round, 6.
Let $(a_0,a_1,a_2,...)$ and $(b_0,b_1,b_2,...)$ be such sequences of non-negative real numbers, that for every integer $i\geqslant 1$ holds $a_i^2\leqslant a_{i-1}a_{i+1}$ and $b_i^2\leqslant b_{i-1}b_{i+1}$.
Define sequence $c_0,c_1,c_2,...$ as
$$c_0=a_0b_0, \; c_n=\sum_{i=0}^{n} {{n}\choose{i}} a_ib_{n-i}.$$
Prove that for every integer $k\geqslant 1$ holds $c_{k}^2\leqslant c_{k-1}c_{k+1}$.
1997 Tournament Of Towns, (548) 2
Prove that the equation $x^2 + y^2 - z^2 = 1997$ has infinitely many solutions in integers $x$, $y$ and $z$.
(N Vassiliev)
2008 Harvard-MIT Mathematics Tournament, 8
Trodgor the dragon is burning down a village consisting of $ 90$ cottages. At time $ t \equal{} 0$ an angry peasant arises from each cottage, and every $ 8$ minutes ($ 480$ seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor $ 5$ seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many [b]seconds[/b] does it take Trodgor to burn down the entire village?
2005 National High School Mathematics League, 10
In tetrahedron $ABCD$, the volume of tetrahedron $ABCD$ is $\frac{1}{6}$, and $\angle ACB=45^{\circ},AD+BC+\frac{AC}{\sqrt2}=3$, then $CD=$________.
2010 China Team Selection Test, 3
Fine all positive integers $m,n\geq 2$, such that
(1) $m+1$ is a prime number of type $4k-1$;
(2) there is a (positive) prime number $p$ and nonnegative integer $a$, such that
\[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]
2023 AMC 8, 18
Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump $5$ pads to the right or $3$ pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located $2023$ pads to the right of her starting point?
$\textbf{(A)}~405\qquad\textbf{(B)}~407\qquad\textbf{(C)}~409\qquad\textbf{(D)}~411\qquad\textbf{(E)}~413$
2006 APMO, 1
Let $n$ be a positive integer. Find the largest nonnegative real number $f(n)$ (depending on $n$) with the following property: whenever $a_1,a_2,...,a_n$ are real numbers such that $a_1+a_2+\cdots +a_n$ is an integer, there exists some $i$ such that $\left|a_i-\frac{1}{2}\right|\ge f(n)$.
1990 AMC 12/AHSME, 29
A subset of the integers $1, 2, ..., 100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
$ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 66 \qquad\textbf{(C)}\ 67 \qquad\textbf{(D)}\ 76 \qquad\textbf{(E)}\ 78 $
1989 Brazil National Olympiad, 5
A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.
Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.
2017 India Regional Mathematical Olympiad, 3
Let \(P(x)=x^2+\dfrac x 2 +b\) and \(Q(x)=x^2+cx+d\) be two polynomials with real coefficients such that \(P(x)Q(x)=Q(P(x))\) for all real \(x\). Find all real roots of \(P(Q(x))=0\).
1997 National High School Mathematics League, 3
The first item and common difference of an arithmetic sequence are nonnegative intengers. The number of items is not less than $3$, and the sum of all items is $97^2$. Then the number of such sequences is
$\text{(A)}2\qquad\text{(B)}3\qquad\text{(C)}4\qquad\text{(D)}5$
1999 Rioplatense Mathematical Olympiad, Level 3, 4
Prove the following inequality:
$$ \frac{1}{\sqrt[3]{1^2}+\sqrt[3]{1 \cdot 2}+\sqrt[3]{2^2} }+\frac{1}{\sqrt[3]{3^2}+\sqrt[3]{3 \cdot 4}+\sqrt[3]{4^2} }+...+ \frac{1}{\sqrt[3]{999^2}+\sqrt[3]{999 \cdot 1000}+\sqrt[3]{1000^2} }> \frac{9}{2}$$
(The member on the left has 500 fractions.)
2020 CMIMC Team, 2
Find all sets of five positive integers whose mode, mean, median, and range are all equal to $5$.
2021 HMNT, 9
$ABCDE$ is a cyclic convex pentagon, and $AC = BD = CE$. $AC$ and $BD$ intersect at $X$, and $BD$ and $CE$ intersect at $Y$ . If $AX = 6$, $XY = 4$, and $Y E = 7$, then the area of pentagon $ABCDE$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a$, $ b$, $c$ are integers, $c$ is positive, $b$ is square-free, and gcd$(a, c) = 1$. Find $100a + 10b + c$.