Found problems: 85335
2023 Indonesia TST, A
Let $a_1, a_2, a_3, a_4, a_5$ be non-negative real numbers satisfied
\[\sum_{k = 1}^{5} a_k = 20 \ \ \ \ \text{and} \ \ \ \ \sum_{k=1}^{5} a_k^2 = 100\]
Find the minimum and maximum of $\text{max} \{a_1, a_2, a_3, a_4, a_5\}$
1981 National High School Mathematics League, 6
In Cartesian coordinates, two areas $M,N$ are defined below:
$M:y\geq0,y\leq x,y\leq 2-x$;
$N:t\leq x\leq t+1$.
$t$ is a real number that $t\in[0,1]$.
Then the area of $M\cap N$ is
$\text{(A)}-t^2+t+\frac{1}{2}\qquad\text{(B)}-2t^2+2t\qquad\text{(C)}1-2t^2\qquad\text{(D)}\frac{1}{2}(t-2)^2$
1995 All-Russian Olympiad, 1
A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$.
[i]S. Tokarev[/i]
2010 LMT, 9
Let $ABC$ and $BCD$ be equilateral triangles, such that $AB=1,$ and $A \neq D.$ Find the area of triangle $ABD.$
1999 IMO Shortlist, 4
Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write
\[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \]
where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\]
a) Prove that $S$ is infinite.
b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$
2023 ELMO Shortlist, C3
Find all pairs of positive integers \((a,b)\) with the following property: there exists an integer \(N\) such that for any integers \(m\ge N\) and \(n\ge N\), every \(m\times n\) grid of unit squares may be partitioned into \(a\times b\) rectangles and fewer than \(ab\) unit squares.
[i]Proposed by Holden Mui[/i]
2018 MIG, 3
Solve for $x$ if $4x + 1 = 37$.
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }10$
1987 AMC 8, 17
Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:
"Bret is next to Carl."
"Abby is between Bret and Carl."
However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?
$\text{(A)}\ \text{Abby} \qquad \text{(B)}\ \text{Bret} \qquad \text{(C)}\ \text{Carl} \qquad \text{(D)}\ \text{Dana} \qquad \text{(E)}\ \text{There is not enough information to be sure.}$
2016 South East Mathematical Olympiad, 5
Let $n$ is positive integer, $D_n$ is a set of all positive divisor of $n$ and $f(n)=\sum_{d\in D_n}{\frac{1}{1+d}}$
Prove that for all positive integer $m$, $\sum_{i=1}^{m}{f(i)} <m$
TNO 2008 Senior, 4
Prove that the diagonals of a convex quadrilateral are perpendicular if and only if the sum of the squares of one pair of opposite sides is equal to the sum of the squares of the other pair.
2009 Sharygin Geometry Olympiad, 5
Rhombus $CKLN$ is inscribed into triangle $ABC$ in such way that point $L$ lies on side $AB$, point $N$ lies on side $AC$, point $K$ lies on side $BC$. $O_1, O_2$ and $O$ are the circumcenters of triangles $ACL, BCL$ and $ABC$ respectively. Let $P$ be the common point of circles $ANL$ and $BKL$, distinct from $L$. Prove that points $O_1, O_2, O$ and $P$ are concyclic.
(D.Prokopenko)
1961 AMC 12/AHSME, 11
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of triangle $APR$ is
${{ \textbf{(A)}\ 42\qquad\textbf{(B)}\ 40.5 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 39\frac{7}{8} }\qquad\textbf{(E)}\ \text{not determined by the given information} } $
2022 Latvia Baltic Way TST, P8
Call the intersection of two segments [i]almost perfect[/i] if for each of the segments the distance between the midpoint of the segment and the intersection is at least $2022$ times smaller than the length of the segment. Prove that there exists a closed broken line of segments such that every segment intersects at least one other segment, and every intersection of segments is [i]almost perfect[/i].
2005 Taiwan TST Round 3, 1
Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[
n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality.
[i]Proposed by Finbarr Holland, Ireland[/i]
2021 Federal Competition For Advanced Students, P2, 4
Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$.
(Walther Janous)
2010 India National Olympiad, 6
Define a sequence $ < a_n > _{n\geq0}$ by $ a_0 \equal{} 0$, $ a_1 \equal{} 1$ and
\[ a_n \equal{} 2a_{n \minus{} 1} \plus{} a_{n \minus{} 2},\]
for $ n\geq2.$
$ (a)$ For every $ m > 0$ and $ 0\leq j\leq m,$ prove that $ 2a_m$ divides
$ a_{m \plus{} j} \plus{} ( \minus{} 1)^ja_{m \minus{} j}$.
$ (b)$ Suppose $ 2^k$ divides $ n$ for some natural numbers $ n$ and $ k$. Prove that $ 2^k$ divides $ a_n.$
2002 China Team Selection Test, 3
Let \[ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3. \] For any reals $r,s,t$, we denote \[ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|. \] Find the minimum value of $g(r,s,t)$.
2010 Stars Of Mathematics, 3
Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$.
(Dan Schwarz)
2013 Saudi Arabia BMO TST, 2
Define Fibonacci sequence $\{F\}_{n=0}^{\infty}$ as $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n +F_{n-1}$ for every integer $n > 1$. Determine all quadruples $(a, b, c,n)$ of positive integers with a $< b < c$ such that each of $a, b,c,a + n, b + n,c + 2n$ is a term of the Fibonacci sequence.
2022 Girls in Math at Yale, R3
[b]p7[/b] Cindy cuts regular hexagon $ABCDEF$ out of a sheet of paper. She folds $B$ over $AC$, resulting in a pentagon. Then, she folds $A$ over $CF$, resulting in a quadrilateral. The area of $ABCDEF$ is $k$ times the area of the resulting folded shape. Find $k$.
[b]p8[/b] Call a sequence $\{a_n\} = a_1, a_2, a_3, . . .$ of positive integers [i]Fib-o’nacci[/i] if it satisfies $a_n = a_{n-1}+a_{n-2}$ for all $n \ge 3$. Suppose that $m$ is the largest even positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = m$, and suppose that $n$ is the largest odd positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = n$. Find $mn$.
[b]p9[/b] Compute the number of ways there are to pick three non-empty subsets $A$, $B$, and $C$ of $\{1, 2, 3, 4, 5, 6\}$, such that $|A| = |B| = |C|$ and the following property holds:
$$A \cap B \cap C = A \cap B = B \cap C = C \cap A.$$
2005 Tournament of Towns, 6
Karlsson-on-the-Roof has $1000$ jars of jam. The jars are not necessarily identical; each contains no more than $\dfrac{1}{100}$-th of the total amount of the jam. Every morning, Karlsson chooses any $100$ jars and eats the same amount of the jam from each of them. Prove that Karlsson can eat all the jam.
[i](8 points)[/i]
2004 India National Olympiad, 6
Show that the number of 5-tuples ($a$, $b$, $c$, $d$, $e$) such that $abcde = 5(bcde + acde + abde + abce + abcd)$ is odd
Estonia Open Senior - geometry, 2002.1.2
The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.
1977 IMO Longlists, 26
Let $p$ be a prime number greater than $5.$ Let $V$ be the collection of all positive integers $n$ that can be written in the form $n = kp + 1$ or $n = kp - 1 \ (k = 1, 2, \ldots).$ A number $n \in V$ is called [i]indecomposable[/i] in $V$ if it is impossible to find $k, l \in V$ such that $n = kl.$ Prove that there exists a number $N \in V$ that can be factorized into indecomposable factors in $V$ in more than one way.
2015 District Olympiad, 2
At a math contest there were $ 50 $ participants, where they were given $ 3 $ problems each to solve. The results have shown that every candidate has solved correctly at least one problem, and that a total of $ 100 $ problems have been evaluated by the jury as correct.
Show that there were, at most, $ 25 $ winners who got the maximum score.