Found problems: 85335
2010 LMT, 17
Al wishes to label the faces of his cube with the integers $2009,2010,$ and $2011,$ with one integer per face, such that adjacent faces (faces that share an edge) have integers that differ by at most $1.$ Determine the number of distinct ways in which he can label the cube, given that two configurations that can be rotated on to each other are considered the same, and that we disregard the orientation in which each number is written on to the cube.
2022 JHMT HS, 2
Erica intends to construct a subset $T$ of $S=\{ I,J,K,L,M,N \}$, but if she is unsure about including an element $x$ of $S$ in $T$, she will write $x$ in bold and include it in $T$. For example, $\{ I,J \},$ $\{ J,\mathbf{K},L \},$ and $\{ \mathbf{I},\mathbf{J},\mathbf{M},\mathbf{N} \}$ are valid examples of $T$, while $\{ I,J,\mathbf{J},K \}$ is not. Find the total number of such subsets $T$ that Erica can construct.
1990 Vietnam Team Selection Test, 3
There are $n\geq 3$ pupils standing in a circle, and always facing the teacher that stands at the centre of the circle. Each time the teacher whistles, two arbitrary pupils that stand next to each other switch their seats, while the others stands still. Find the least number $M$ such that after $M$ times of whistling, by appropriate switchings, the pupils stand in such a way that any two pupils, initially standing beside each other, will finally also stand beside each other; call these two pupils $ A$ and $ B$, and if $ A$ initially stands on the left side of $ B$ then $ A$ will finally stand on the right side of $ B$.
2018 Azerbaijan IZhO TST, 4
There are $10$ cities in each of the three countries. Each road connects two cities from two different countries (there is at most one road between any two cities.) There are more than $200$ roads between these three countries. Prove that three cities, one city from each country, can be chosen such that there is a road between any two of these cities.
2013 AMC 8, 4
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \$2.50 to cover her portion of the total bill. What was the total bill?
$\textbf{(A)}\ \$120 \qquad \textbf{(B)}\ \$128 \qquad \textbf{(C)}\ \$140 \qquad \textbf{(D)}\ \$144 \qquad \textbf{(E)}\ \$160$
2018 Harvard-MIT Mathematics Tournament, 5
A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. He notices that none of the marbles he drew were ugly. Given this information, what is the expected value of the number of total marbles he drew?
2008 ISI B.Math Entrance Exam, 7
Let
$C=\{ (i,j)|i,j$ integers such that $0\leq i,j\leq 24\}$
How many squares can be formed in the plane all of whose vertices are in $C$ and whose sides are parallel to the $X-$axis and $Y-$ axis?
1992 Vietnam Team Selection Test, 3
In a scientific conference, all participants can speak in total $2 \cdot n$ languages ($n \geq 2$). Each participant can speak exactly two languages and each pair of two participants can have at most one common language. It is known that for every integer $k$, $1 \leq k \leq n-1$ there are at most $k-1$ languages such that each of these languages is spoken by at most $k$ participants. Show that we can choose a group from $2 \cdot n$ participants which in total can speak $2 \cdot n$ languages and each language is spoken by exactly two participants from this group.
1998 India National Olympiad, 5
Suppose $a,b,c$ are three rela numbers such that the quadratic equation \[ x^2 - (a +b +c )x + (ab +bc +ca) = 0 \] has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that
(i) The numbers $a,b,c$ are all positive.
(ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.
2011 Bulgaria National Olympiad, 1
Prove whether or not there exist natural numbers $n,k$ where $1\le k\le n-2$ such that
\[\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 \]
2004 Federal Competition For Advanced Students, P2, 1
Prove without using advanced (differential) calculus:
(a) For any real numbers a,b,c,d it holds that $a^6+b^6+c^6+d^6-6abcd \ge -2$.
When does equality hold?
(b) For which natural numbers $k$ does some inequality of the form $a^k +b^k +c^k +d^k -kabcd \ge M_k$ hold for all real $a,b,c,d$? For each such $k$,
2018 Federal Competition For Advanced Students, P1, 2
Let $ABC$ be a triangle with incenter $I$. The incircle of the triangle is tangent to the sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let $P$ denote the common point of lines $AI$ and $DE$, and let $M$ and $N$ denote the midpoints of sides $BC$ and $AB$, respectively. Prove that points $M, N$ and $P$ are collinear.
[i](Proposed by Karl Czakler)[/i]
2017 Peru IMO TST, 5
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
2021 BMT, 21
There exist integers $a$ and $b$ such that $(1 +\sqrt2)^{12}= a + b\sqrt2$. Compute the remainder when $ab$ is divided by $13$.
2024 AIME, 10
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$.
2013 Polish MO Finals, 5
Let k,m and n be three different positive integers. Prove that \[
\left( k-\frac{1}{k} \right)\left( m-\frac{1}{m} \right)\left( n-\frac{1}{n} \right) \le kmn-(k+m+n). \]
2015 China Second Round Olympiad, 3
Prove that there exist infinitely many positive integer triples $(a,b,c)(a,b,c>2015)$ such that
$$ a|bc-1, b|ac+1, c|ab+1.$$
2011 Singapore Senior Math Olympiad, 2
Determine if there is a set $S$ of 2011 positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.
2017 AMC 8, 4
When 0.000315 is multiplied by 7,928,564 the product is closest to which of the following?
$\textbf{(A) }210\qquad\textbf{(B) }240\qquad\textbf{(C) }2100\qquad\textbf{(D) }2400\qquad\textbf{(E) }24000$
2010 India IMO Training Camp, 1
Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.
2015 Kosovo Team Selection Test, 4
Let $P_1,P_2,...,P_{2556}$ be distinct points inside a regular hexagon $ABCDEF$ of side $1$. If any three points from the set $S=\{A,B,C,D,E,F,P_1,P_2...,P_{2556}\}$ aren't collinear, prove that there exists a triangle with area smaller than $\frac{1}{1700}$, with vertices from the set $S$.
2021 USMCA, 9
For how many two-digit integers $n$ is $13 \mid 1 - 2^n - 3^n + 5^n$?
1990 IMO Longlists, 73
Let $\mathbb Q$ be the set of all rational numbers and $\mathbb R$ be the set of real numbers. Function $f: \mathbb Q \to \mathbb R$ satisfies the following conditions:
(i) $f(0) = 0$, and for any nonzero $a \in Q, f(a) > 0.$
(ii) $f(x + y) = f(x)f(y) \qquad \forall x,y \in \mathbb Q.$
(iii) $f(x + y) \leq \max\{f(x), f(y)\} \qquad \forall x,y \in \mathbb Q , x,y \neq 0.$
Let $x$ be an integer and $f(x) \neq 1$. Prove that $f(1 + x + x^2+ \cdots + x^n) = 1$ for any positive integer $n.$
2013 District Olympiad, 3
Let be the regular hexagonal prism $ABCDEFA'B C'D'E'F'$ with the base edge of $12$ and the height of $12 \sqrt{3}$. We denote by $N$ the middle of the edge $CC'$.
a) Prove that the lines $BF'$ and $ND$ are perpendicular
b) Calculate the distance between the lines $BF'$ and $ND$.
2023 LMT Spring, 5
How many ways are there to place the integers from $1$ to $8$ on the vertices of a regular octagon such that the sum of the numbers on any $4$ vertices forming a rectangle is even? Rotations and reflections of the same arrangement are considered distinct