Found problems: 85335
2017 ASDAN Math Tournament, 4
A ladder $10\text{ m}$ long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of $1\text{ m/s}$, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is $6\text{ m}$ from the wall?
1992 IMTS, 2
In how many ways can 1992 be expressed as the sum of one or more consecutive integers?
2007 AMC 12/AHSME, 12
A teacher gave a test to a class in which $ 10\%$ of the students are juniors and $ 90\%$ are seniors. The average score on the test was $ 84$. The juniors all received the same score, and the average score of the seniors was $ 83$. What score did each of the juniors receive on the test?
$ \textbf{(A)}\ 85 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 93 \qquad \textbf{(D)}\ 94 \qquad \textbf{(E)}\ 98$
1981 National High School Mathematics League, 1
Given two conditions:
A: Two triangles have the same area and two corresponding edge equal.
B: Two triangles are congruent.
Then, which one of the followings are true?
$(\text{A})$A is sufficient and necessary condition of B.
$(\text{B})$A is necessary but insufficient condition of B.
$(\text{C})$A is sufficient but unnecessary condition of B.
$(\text{D})$A is insufficient and unnecessary condition of B.
2014 IMS, 8
Is $\sum_{n=1}^{+\infty}\frac{\cos n}{n}(1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}})$ convergent? why?
2014 Miklós Schweitzer, 1
Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements.
Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating
Brazil L2 Finals (OBM) - geometry, 2012.4
The figure below shows a regular $ABCDE$ pentagon inscribed in an equilateral triangle $MNP$ . Determine the measure of the angle $CMD$.
[img]http://4.bp.blogspot.com/-LLT7hB7QwiA/Xp9fXOsihLI/AAAAAAAAL14/5lPsjXeKfYwIr5DyRAKRy0TbrX_zx1xHQCK4BGAYYCw/s200/2012%2Bobm%2Bl2.png[/img]
2011 Bosnia And Herzegovina - Regional Olympiad, 4
Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational
2003 AMC 12-AHSME, 8
Let $ \clubsuit(x)$ denote the sum of the digits of the positive integer $ x$. For example, $ \clubsuit(8)\equal{}8$ and $ \clubsuit(123)\equal{}1\plus{}2\plus{}3\equal{}6$. For how many two-digit values of $ x$ is $ \clubsuit(\clubsuit(x))\equal{}3$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ 10$
2019 VJIMC, 2
A triplet of polynomials $u,v,w \in \mathbb{R}[x,y,z]$ is called [i]smart[/i] if there exists polynomials $P,Q,R\in \mathbb{R}[x,y,z]$ such that the following polynomial identity holds :$$u^{2019}P +v^{2019 }Q+w^{2019} R=2019$$
a) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y+z$$ [i]smart[/i]?
b) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y-z$$ [i]smart[/i]?
[i]Proposed by Arturas Dubickas (Vilnius University).
[/i]
1995 IMO Shortlist, 5
For positive integers $ n,$ the numbers $ f(n)$ are defined inductively as follows: $ f(1) \equal{} 1,$ and for every positive integer $ n,$ $ f(n\plus{}1)$ is the greatest integer $ m$ such that there is an arithmetic progression of positive integers $ a_1 < a_2 < \ldots < a_m \equal{} n$ for which
\[ f(a_1) \equal{} f(a_2) \equal{} \ldots \equal{} f(a_m).\]
Prove that there are positive integers $ a$ and $ b$ such that $ f(an\plus{}b) \equal{} n\plus{}2$ for every positive integer $ n.$
1994 Bundeswettbewerb Mathematik, 3
Given a triangle $A_1 A_2 A_3$ and a point $P$ inside. Let $B_i$ be a point on the side opposite to $A_i$ for $i=1,2,3$, and let $C_i$ and $D_i$ be the midpoints of $A_i B_i$ and $P B_i$, respectively. Prove that the triangles $C_1 C_2 C_3$ and $D_1 D_2 D_3$ have equal area.
2018 MIG, 9
Define $f(x) = x^2 + 5$. Find the product of all $x$ such that $f(x) = 14$.
$\textbf{(A) }{-}9\qquad\textbf{(B) }{-}3\qquad\textbf{(C) }0\qquad\textbf{(D) }3\qquad\textbf{(E) }9$
2019 Saudi Arabia Pre-TST + Training Tests, 2.2
Let be given a positive integer $n > 1$. Find all polynomials $P(x)$ non constant, with real coefficients such that
$$P(x)P(x^2) ... P(x^n) = P\left( x^{\frac{n(n+1)}{2}}\right)$$ for all $x \in R$
1934 Eotvos Mathematical Competition, 1
Let $n$ be a given positive integer and
$$A =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}$$
Prove that at least one term of the sequence $A, 2A,4A,8A,...,2^kA, ... $ is an integer.
2001 Poland - Second Round, 2
Points $A,B,C$ with $AB<BC$ lie in this order on a line. Let $ABDE$ be a square. The circle with diameter $AC$ intersects the line $DE$ at points $P$ and $Q$ with $P$ between $D$ and $E$. The lines $AQ$ and $BD$ intersect at $R$. Prove that $DP=DR$.
1999 Tournament Of Towns, 4
Is it possible to divide the integers from $1$ to $100$ inclusive into $50$ pairs such that for $1\le k\le 50$, the difference between the two numbers in the $k$-th pair is $k$?
(V Proizvolov)
2013 NIMO Problems, 8
For a finite set $X$ define \[
S(X) = \sum_{x \in X} x \text{ and }
P(x) = \prod_{x \in X} x.
\] Let $A$ and $B$ be two finite sets of positive integers such that $\left\lvert A \right\rvert = \left\lvert B \right\rvert$, $P(A) = P(B)$ and $S(A) \neq S(B)$. Suppose for any $n \in A \cup B$ and prime $p$ dividing $n$, we have $p^{36} \mid n$ and $p^{37} \nmid n$. Prove that \[ \left\lvert S(A) - S(B) \right\rvert > 1.9 \cdot 10^{6}. \][i]Proposed by Evan Chen[/i]
2023 Moldova Team Selection Test, 10
Let $ABC$ be a triangle with $\angle ACB=90$ and $AC>BC.$ Let $\Omega$ be the circumcircle of $ABC.$ Point $ D $ is the midpoint of small arc $AC$ of $\Omega.$ Point $ M $ is symmetric with $ A$ with respect to $D.$ Point $ N$ is the midpoint of $MC.$ Line $AN$ intersects $\Omega$ in point $ P $ and line $BP$ intersects line $DN$ in point $Q.$ Prove that line $QM$ passes through the midpoint of $AC.$
2023 Math Prize for Girls Problems, 17
Let $C$ be a unit cube. Let $D$ be a translate of $C$ such that one corner of $D$ is located at the center of $C$ and one corner of $C$ is located at the center of $D$. Let $D^\prime$ be the image of $D$ under a $60^\circ$ clockwise rotation about the line that passes through both cube centers when looking from the center of $D$ to the center of $C$. What is the volume of the intersection of $C$ with $D^\prime$?
2018 Saudi Arabia GMO TST, 2
Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.
2010 Romanian Masters In Mathematics, 5
Let $n$ be a given positive integer. Say that a set $K$ of points with integer coordinates in the plane is connected if for every pair of points $R, S\in K$, there exists a positive integer $\ell$ and a sequence $R=T_0,T_1, T_2,\ldots ,T_{\ell}=S$ of points in $K$, where each $T_i$ is distance $1$ away from $T_{i+1}$. For such a set $K$, we define the set of vectors
\[\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}\]
What is the maximum value of $|\Delta(K)|$ over all connected sets $K$ of $2n+1$ points with integer coordinates in the plane?
[i]Grigory Chelnokov, Russia[/i]
2020 Dutch Mathematical Olympiad, 4
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.
1991 Tournament Of Towns, (302) 3
Prove that $$\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}}=1$$
This means
$1/(2+ (1/(3+ (1/(4+(...+1/1991))))))
+1/(1 + (1/(1 + (1/(3 + (1/(4 + (...+ 1/1991...)))))))) = 1.)$
(G. Galperin, Moscow-Tel Aviv)
2016 Purple Comet Problems, 5
Julius has a set of five positive integers whose mean is 100. If Julius removes the median of the set of five
numbers, the mean of the set increases by 5, and the median of the set decreases by 5. Find the maximum
possible value of the largest of the five numbers Julius has.