Found problems: 85335
2018 Estonia Team Selection Test, 7
Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.
2008 Balkan MO Shortlist, G2
Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.
2011 Romanian Masters In Mathematics, 2
For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$.
(We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.)
[i](United Kingdom) Luke Betts[/i]
2010 Contests, 1
Consider points $D,E$ and $F$ on sides $BC,AC$ and $AB$, respectively, of a triangle $ABC$, such that $AD, BE$ and $CF$ concurr at a point $G$. The parallel through $G$ to $BC$ cuts $DF$ and $DE$ at $H$ and $I$, respectively. Show that triangles $AHG$ and $AIG$ have the same areas.
2006 IMO Shortlist, 2
For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.
[i]Proposed by J.P. Grossman, Canada[/i]
2021 China Second Round Olympiad, Problem 13
Let $n$ be a given positive integer. The sequence of real numbers $a_1, a_2, a_3, \cdots, a_n$ satisfy for each $m \leq n$, $$\left|\sum_{k=1}^m\frac{a_k}k\right| \leq 1.$$ Given this information, find the greatest possible value of $\left|\sum_{k=1}^n a_k\right|$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 13)[/i]
2006 National Olympiad First Round, 2
If $p$ and $p^2+2$ are prime numbers, at most how many prime divisors can $p^3+3$ have?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
2019 Girls in Mathematics Tournament, 3
We say that a positive integer N is [i]nice[/i] if it satisfies the following conditions:
$\bullet$ All of its digits are $1$ or $2$
$\bullet$ All numbers formed by $3$ consecutive digits of $N$ are distinct.
For example, $121222$ is nice, because the $4$ numbers formed by $3$ consecutive digits of $121222$, which are $121,212,122$ and $222$, are distinct. However, $12121$ is not nice. What is the largest quantity possible number of numbers that a nice number can have? What is the greatest nice number there is?
2024 CCA Math Bonanza, I8
Each vertex of a regular heptagon ($7$-gon) is colored either red or blue. Find the number of distinct colorings such that no three consecutive vertices have the same color. Two colorings are considered distinct if one cannot be obtained from the other by a rotation of the heptagon.
[i]Individual #8[/i]
2020 India National Olympiad, 3
Let $S$ be a subset of $\{0,1,2,\dots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.
[i]Proposed by Sutanay Bhattacharya[/i]
[hide=Original Wording]
As pointed out by Wizard_32, the original wording is:
Let $X=\{0,1,2,\dots,9\}.$ Let $S \subset X$ be such that any positive integer $n$ can be written as $p+q$ where the non-negative integers $p, q$ have all their digits in $S.$ Find the smallest possible number of elements in $S.$
[/hide]
2013 Moldova Team Selection Test, 2
We call a triangle $\triangle ABC$, $Q$-angled if $\tan\angle A,\tan\angle B,\tan\angle C \in \mathbb{Q}$, where $\angle A,\angle B ,\angle C$ are the interior angles of the triangle $\triangle ABC$.
$a)$ Prove that $Q$-angled triangles exist;
$b)$ Let triangle $\triangle ABC$ be $Q$-angled. Prove that for any non-negative integer $n$, numbers of the form
$E_n=\sin^n\angle A \sin^n\angle B \sin^n\angle C + \cos^n\angle A\cos^n\angle B\cos^n\angle C$ are rational.
2003 AMC 8, 5
If $20\%$ of a number is $12$, what is $30\%$ of the same number?
$\textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 20 \qquad
\textbf{(D)}\ 24 \qquad
\textbf{(E)}\ 30$
2023 IMC, 4
Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?
1975 Miklós Schweitzer, 10
Prove that an idempotent linear operator of a Hilbert space is self-adjoint if and only if it has norm $ 0$ or $ 1$.
[i]J. Szucs[/i]
2008 International Zhautykov Olympiad, 1
For each positive integer $ n$,denote by $ S(n)$ the sum of all digits in decimal representation of $ n$.
Find all positive integers $ n$,such that $ n\equal{}2S(n)^3\plus{}8$.
Kvant 2024, M2824
There are $15$ boys and $15$ girls in the class. The first girl is friends with $4$ boys, the second with $5$, the third with $6$, . . . , the $11$th with $14$, and each of the other four girls is friends with all the boys. It turned out that there are exactly $3 \cdot 2^{25}$ ways to split the entire class into pairs, so that each pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends with all the other girls too.
[i]G.M.Sharafetdinova[/i]
2004 AMC 12/AHSME, 17
Let $ f$ be a function with the following properties:
(i) $f(1) \equal{} 1$, and
(ii) $ f(2n) \equal{} n\times f(n)$, for any positive integer $ n$.
What is the value of $ f(2^{100})$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2^{99} \qquad \textbf{(C)}\ 2^{100} \qquad \textbf{(D)}\ 2^{4950} \qquad \textbf{(E)}\ 2^{9999}$
2007 ITest, 27
The face diagonal of a cube has length $4$. Find the value of $n$ given that $n\sqrt2$ is the $\textit{volume}$ of the cube.
2003 Abels Math Contest (Norwegian MO), 2b
Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$
2009 Tournament Of Towns, 1
A rectangle is dissected into several smaller rectangles. Is it possible that for each pair of these rectangles, the line segment connecting their centers intersects some third rectangle?
2023 IFYM, Sozopol, 2
Given a triangle $ABC$, a line in its plane is called a [i]cool[/i] if it divides the triangle into two parts with equal areas and perimeters.
a) Does there exist a triangle $ABC$ with at least seven [i]cool[/i] lines?
b) Prove that all [i]cool[/i] lines intersect at a point $X$. If $\angle AXB = 126^\circ$, prove that $(8\sin^2 \angle ACB - 5)^2$ is an integer.
2010 German National Olympiad, 6
Let $A,B,C,D,E,F,G$ and $H$ be eight pairwise distinct points on the surface of a sphere. The quadruples $(A,B,C,D), (A,B,F,E),(B,C,G,F),(C,D,H,G)$ and $(D,A,E,H)$ of points are coplanar.
Prove that the quadruple $(E,F,G,H)$ is coplanar aswell.
2018 India IMO Training Camp, 1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
2004 All-Russian Olympiad Regional Round, 8.7
A set of five-digit numbers $\{N_1,... ,N_k\}$ is such that any five-digit a number whose digits are all in ascending order is the same in at least one digit with at least one of the numbers $N_1$,$...$ ,$N_k$. Find the smallest possible value of $k$.
1969 Miklós Schweitzer, 12
Let $ A$ and $ B$ be nonsingular matrices of order $ p$, and let $ \xi$ and $ \eta$ be independent random vectors of dimension $ p$. Show that if $ \xi,\eta$ and $ \xi A\plus{} \eta B$ have the same distribution, if their first and second moments exist, and if their covariance matrix is the identity matrix, then these random vectors are normally distributed.
[i]B. Gyires[/i]