Found problems: 85335
2013 IFYM, Sozopol, 8
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\]
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.
[i]Proposed by Witold Szczechla, Poland[/i]
2015 Baltic Way, 10
A subset $S$ of $ {1,2,...,n}$ is called balanced if for every $a $ from $S $ there exists some $ b $from $S$, $b\neq a$, such that $ \frac{(a+b)}{2}$ is in $S$ as well.
(a) Let $k > 1 $be an integer and let $n = 2k$. Show that every subset $ S$ of ${1,2,...,n} $ with $|S| > \frac{3n}{4}$ is balanced.
(b) Does there exist an $n =2k$, with $ k > 1 $ an integer, for which every subset $ S$ of ${1,2,...,n} $ with $ |S| >\frac{2n}{3} $ is balanced?
2024 Junior Balkan Team Selection Tests - Romania, P1
For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that
$$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$
2011 Today's Calculation Of Integral, 689
Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis.
Proposed by kunny
2021 Bulgaria EGMO TST, 3
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
2017 Macedonia JBMO TST, Source
[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url]
[url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.
[url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that
$$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$
Show that $TB_2AC_2$ is a rectangle.
[url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that
$$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$
When does equality happen?
[url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$
[url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.
1996 Argentina National Olympiad, 2
Decide if there exists any number of $10$ digits such that rearranging $10,000$ times its digits results in $10,000$ different numbers that are multiples of $7$.
KoMaL A Problems 2022/2023, A. 839
We are given a finite, simple, non-directed graph. Ann writes positive real numbers on each edge of the graph such that for all vertices the following is true: the sum of the numbers written on the edges incident to a given vertex is less than one. Bob wants to write non-negative real numbers on the vertices in the following way: if the number written at vertex $v$ is $v_0$, and Ann's numbers on the edges incident to $v$ are $e_1,e_2,\ldots,e_k$, and the numbers on the other endpoints of these edges are $v_1,v_2,\ldots,v_k$, then $v_0=\sum_{i=1}^k e_iv_i+2022$. Prove that Bob can always number the vertices in this way regardless of the graph and the numbers chosen by Ann.
Proposed by [i]Boldizsár Varga[/i], Verőce
2000 Romania National Olympiad, 2a
Knowing that $1 < y < 2$ and $x - y + 1 = 0,$ calculate the value of the expression:
$$E = \sqrt{4x^2 +4y-3} + 2\sqrt{y^2 - 6x - 2y +10}.$$
1998 IberoAmerican Olympiad For University Students, 1
The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$.
Prove that there is a real number $c$ such that
\[f(c)+g(c)\leq 2\]
2024 USAMTS Problems, 1
Shade some squares in the grid so that:
[list=1]
[*]Squares with numbers are unshaded.
[*]Each number is equal to the product of the number of unshaded squares it can “see” in its row and column. (A square can see another square if they’re in the same row or column and the sight line between them doesn’t have any shaded squares. Each square can see itself.)
[*]The shaded squares must make one connected group. Two squares are considered to be connected if they share an edge.
[/list]
[asy]
size(10cm);
int n = 10; // Grid size
pair start = (0,0);
// Draw the grid
for (int i = 0; i <= n; ++i) {
draw((start.x + i, start.y) -- (start.x + i, start.y + n), black); // Vertical lines
draw((start.x, start.y + i) -- (start.x + n, start.y + i), black); // Horizontal lines
}
// List of locations and corresponding labels
pair locations[] = {(9.5,9.5), (3.5, 8.5), (6.5, 8.5), (2.5, 6.5), (3.5, 6.5),
(4.5, 5.5), (5.5,4.5), (6.5,3.5), (7.5,3.5), (3.5,1.5),
(6.5,1.5), (0.5,0.5)};
string labels[] = {"4", "6", "6", "36", "24", "16", "24", "36", "18", "6", "12", "36"};
// Add labels using a loop
for (int i = 0; i < locations.length; ++i) {
label(labels[i], locations[i], fontsize(16pt));
}
[/asy]
2014 Purple Comet Problems, 16
Start with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubling $B$. Obtain $D$ by subtracting $500$ from $C$. Given that $A + B + C + D = 2014$, fi nd $A$.
2011 AMC 10, 2
A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
$ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15 $
2006 Putnam, A3
Let $1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots$ be a sequence defined by $x_{k}=k$ for $k=1,2\dots,2006$ and $x_{k+1}=x_{k}+x_{k-2005}$ for $k\ge 2006.$ Show that the sequence has 2005 consecutive terms each divisible by 2006.
PEN E Problems, 28
Show that $n^{\pi(2n)-\pi(n)}<4^{n}$ for all positive integer $n$.
1985 AMC 8, 10
The fraction halfway between $ \frac{1}{5}$ and $ \frac{1}{3}$ (on the number line) is
\[ \textbf{(A)}\ \frac{1}{4} \qquad
\textbf{(B)}\ \frac{2}{15} \qquad
\textbf{(C)}\ \frac{4}{15} \qquad
\textbf{(D)}\ \frac{53}{200} \qquad
\textbf{(E)}\ \frac{8}{15}
\]
2018 VJIMC, 2
Let $n$ be a positive integer and let $a_1\le a_2 \le \dots \le a_n$ be real numbers such that
\[a_1+2a_2+\dots+na_n=0.\]
Prove that
\[a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0\]
for every real number $x$. (Here $[t]$ denotes the integer satisfying $[t] \le t<[t]+1$.)
2006 Chile National Olympiad, 1
Juana and Juan have to write each one an ordered list of fractions so that the two lists have the same number of fractions and that the difference between the sum of all the fractions from Juana's list and the sum of all fractions from Juan's list is greater than $123$.
The fractions in Juana's list are $$\frac{1^2}{1}, \frac{2^2}{3},\frac{3^2}{5},\frac{4^2}{7},\frac{5^2}{9},...$$
And the fractions in John's list are $$\frac{1^2}{3}, \frac{2^2}{5},\frac{3^2}{7},\frac{4^2}{9},\frac{5^2}{11},...$$
Find the least amount of fractions that each one must write to achieve the objective.
2017 Korea National Olympiad, problem 4
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function as
\[ f(x) = \begin{cases} \frac{1}{x-1}& (x > 1)\\ 1& (x=1)\\ \frac{x}{1-x} & (x<1) \end{cases} \]
Let $x_1$ be a positive irrational number which is a zero of a quadratic polynomial with integer coefficients. For every positive integer $n$, let $x_{n+1} = f(x_n)$. Prove that there exists different positive integers $k$ and $\ell$ such that $x_k = x_\ell$.
2007 Hong Kong TST, 2
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 2
Let $A$, $B$ and $C$ be real numbers such that
(i) $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$,
(ii) $\tan C$ and $\cot C$ are defined.
Find the minimum value of $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$.
2018 NZMOC Camp Selection Problems, 9
Let $x, y, p, n, k$ be positive integers such that $$x^n + y^n = p^k.$$
Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.
2019 District Olympiad, 2
Let $n$ be a positive integer and $f:[0,1] \to \mathbb{R}$ be an integrable function. Prove that there exists a point $c \in \left[0,1- \frac{1}{n} \right],$ such that [center] $ \int\limits_c^{c+\frac{1}{n}}f(x)\mathrm{d}x=0$ or $\int\limits_0^c f(x) \mathrm{d}x=\int\limits_{c+\frac{1}{n}}^1f(x)\mathrm{d}x.$ [/center]
2014 ELMO Shortlist, 9
Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]
2021 China Second Round Olympiad, Problem 14
Define the set $P=\{a_1, a_2, a_3, \cdots, a_n\}$ and its arithmetic mean $$C_p = \frac{a_1+a_2+\cdots+a_m}m.$$ If we divide $S = \{1, 2, 3, \cdots, n\}$ into two disjoint subsets $A, B$, compute the greatest possible value of $|C_A-C_B|$. For how many $(A, B)$ is equality attained?
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 14)[/i]
2003 ITAMO, 6
Every of $n$ guests invited to a dinner has got an invitation denoted by a number from $1$ to $n$. The guests will be sitting around a round table with $n$ seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there.
Find all $n$ for which he can select the guests in such an order to serve all the guests.