Found problems: 85335
2013 Dutch BxMO/EGMO TST, 1
In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.
2009 Iran MO (3rd Round), 2
2-There is given a trapezoid $ ABCD$.We have the following properties:$ AD\parallel{}BC,DA \equal{} DB \equal{} DC,\angle BCD \equal{} 72^\circ$. A point $ K$ is taken on $ BD$ such that $ AD \equal{} AK,K \neq D$.Let $ M$ be the midpoint of $ CD$.$ AM$ intersects $ BD$ at $ N$.PROVE $ BK \equal{} ND$.
2012 CHMMC Spring, 7
A positive integer $x$ is $k$-[i]equivocal [/i] if there exists two positive integers $b$, $b'$ such that when $x$ is represented in base $b$ and base $b'$, the two representations have digit sequences of length $k$ that are permutations of each other. The smallest $2$-equivocal number is $7$, since $7$ is $21$ in base $3$ and $12$ in base $5$. Find the smallest $3$-equivocal number.
2018 AMC 12/AHSME, 14
The solution to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\tfrac{1}{3}$ or $\tfrac{1}{2}$, can be written as $\tfrac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
$\textbf{(A) } 5 \qquad
\textbf{(B) } 13 \qquad
\textbf{(C) } 17 \qquad
\textbf{(D) } 31 \qquad
\textbf{(E) } 35 $
1981 Putnam, A2
Two distinct squares of the $8\times8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common.
Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$ there exist twoadjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.
2020 Moldova Team Selection Test, 2
Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \geq 3+\sqrt{3}$
2018 Nepal National Olympiad, 2a
[b]Problem Section #2
a) If $$ax+by=7$$ $$ax^2+by^2=49$$ $$ax^3+by^3=133$$ $$ax^4+by^4=406$$ ,
find the value of $2014(x+y-xy)-100(a+b).$
2015 China Team Selection Test, 5
Set $S$ to be a subset of size $68$ of $\{1,2,...,2015\}$. Prove that there exist $3$ pairwise disjoint, non-empty subsets $A,B,C$ such that $|A|=|B|=|C|$ and $\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c$
2008 Argentina Iberoamerican TST, 3
Show that exists a sequence of $ 100$ terms such that:
1)Every term is a perfect square
2) every term is greater than the one before it ( it is strictly increasing)
3)Every two terms of the sequence are relative prime
4) The average between two consecutive terms is also a perfect square
Daniel
2016 AMC 8, 16
Annie and Bonnie are running laps around a 400-meter oval track. They started together, but Annie has pulled ahead because she is $25 \%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?
$\textbf{(A) }1 \frac{1}{4}\qquad\textbf{(B) }3 \frac{1}{3}\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }25$
LMT Team Rounds 2021+, 7
Let $n = 6901$. There are $6732$ positive integers less than or equal to $n$ that are also relatively prime to $n$. Find the sum of the distinct prime factors of $n$.
2018 IMC, 5
Let $p$ and $q$ be prime numbers with $p<q$. Suppose that in a convex polygon $P_1,P_2,…,P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that
$$P_1P_2+P_2P_3+\cdots +P_kP_{k+1}\geqslant \frac{k^3+k}{2}$$holds for every integer $k$ with $1\leqslant k\leqslant p$.
[i]Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin[/i]
2014-2015 SDML (Middle School), 4
Shannon, Laura, and Tasha found a shirt which came in five colors at their favorite store, and they each bought one of each color of that shirt. On Monday, they all wear one of their new shirts to work. What is the probability that Shannon, Laura, and Tasha will not all be wearing the same color shirt that day?
$\text{(A) }\frac{12}{25}\qquad\text{(B) }\frac{16}{25}\qquad\text{(C) }\frac{21}{25}\qquad\text{(D) }\frac{22}{25}\qquad\text{(E) }\frac{24}{25}$
1966 IMO Shortlist, 63
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
2002 Bulgaria National Olympiad, 6
Find the smallest number $k$, such that $ \frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively.
[i]Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov[/i]
2003 AMC 12-AHSME, 24
If $ a\ge b>1$, what is the largest possible value of $ \log_a(a/b)\plus{}\log_b(b/a)$?
$ \textbf{(A)}\ \minus{}2 \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
2001 China Team Selection Test, 1
Given any odd integer $n>3$ that is not divisible by $3$, determine whether it is possible to fill an $n \times n$ grid with $n^2$ integers such that (each cell filled with a number, the number at the intersection of the $i$-th row and $j$-th column is denoted as $a_{ij}$):
$\cdot$ Each row and each column contains a permutation of the numbers $0,1,2, \cdots, n-1$.
$\cdot$ The pairs $(a_{ij},a_{ji})$ for $i<j$ are all distinct.
2019 Iran Team Selection Test, 1
Find all polynomials $P(x,y)$ with real coefficients such that for all real numbers $x,y$ and $z$:
$$P(x,2yz)+P(y,2zx)+P(z,2xy)=P(x+y+z,xy+yz+zx).$$
[i]Proposed by Sina Saleh[/i]
2008 AMC 12/AHSME, 14
What is the area of the region defined by the inequality $ |3x\minus{}18|\plus{}|2y\plus{}7|\le 3$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ \frac{9}{2} \qquad
\textbf{(E)}\ 5$
2021 Princeton University Math Competition, A3 / B5
Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$. Find the value of the following expression: $$f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4).$$
2003 Purple Comet Problems, 2
What is the smallest number that could be the date of the first Saturday after the second Monday following the second Thursday of a month?
1980 All Soviet Union Mathematical Olympiad, 302
The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$. Prove that the cosine of the angle between lines $(AC)$ and $(BD)$ is less than $|CD|/|AB|$.
2023 Purple Comet Problems, 2
There are positive real numbers $a$, $b$, $c$, $d$, and $p$ such that $a$ is $62.5\%$ of $b$, $b$ is $64\%$ of $c$, c is $125\%$ of $d$, and $d$ is $p\%$ of $a$. Find $p$.
2024 China Team Selection Test, 12
Given positive odd number $m$ and integer ${a}.$ Proof: For any real number $c,$
$$\#\left\{x\in\mathbb Z\cap [c,c+\sqrt m]\mid x^2\equiv a\pmod m\right\}\le 2+\log_2m.$$
[i]Proposed by Yinghua Ai[/i]
2015 Albania JBMO TST, 1
For every positive integer $n{}$ denote $a_n$ as the last digit of the sum of the number from $1$ to $n{}$. For example $a_5=5, a_6=1.$
a) Find $a_{21}.$
b) Compute the sum $a_1+a_2+\ldots+a_{2015}.$