Found problems: 85335
2020 CHMMC Winter (2020-21), 6
Anna and Bob are playing a game on a rectangular board with $i$ rows and $j$ columns. Anna and Bob alternate turns with Anna going first. On each turn, a player places a penny in a square and then all squares in the same row and column of that square are marked. A player cannot place a penny in any marked square. When a player cannot place a penny in any square, they lose and the other player wins. How many ordered pairs of integers $(i, j)$ with $1 \le i \le 2020, 1 \le j \le 2020$ are there such that Anna wins?
2014 Contests, 3
Give are the integers $a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111$( with $2n$ digits) with $n > 8$ .
Let $q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n$ the remainder of the division of $a_{i}$ by$ 11$ .
Prove that the sum of nine consecutive quotients: $s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}$ is a multiple of $9$ for any $i= 1,2,3, ... , (n-8)$
1950 AMC 12/AHSME, 4
Reduced to lowest terms, $ \frac {a^2\minus{}b^2}{ab}\minus{} \frac {ab\minus{}b^2}{ab\minus{}a^2}$ is equal to:
$\textbf{(A)}\ \dfrac{a}{b} \qquad
\textbf{(B)}\ \dfrac{a^2-2b^2}{ab} \qquad
\textbf{(C)}\ a^2 \qquad
\textbf{(D)}\ a-2b \qquad
\textbf{(E)}\ \text{None of these}$
2019 Brazil Team Selection Test, 2
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
1991 Irish Math Olympiad, 2
Let $$a_n=\frac{n^2+1}{\sqrt{n^4+4}}, \quad n=1,2,3,\dots$$ and let $b_n$ be the product of $a_1,a_2,a_3,\dots ,a_n$. Prove that $$\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^2+1}}{\sqrt{n^2+2n+2}},$$ and deduce that $$\frac{1}{n^3+1}<\frac{b_n}{\sqrt{2}}-\frac{n}{n+1}<\frac{1}{n^3}$$ for all positive integers $n$.
2021 AMC 10 Spring, 17
Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given $2$ cards out of a set of $10$ cards numbered $1,2,3, \dots,10.$ The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon--$11,$ Oscar--$4,$ Aditi--$7,$ Tyrone--$16,$ Kim--$17.$ Which of the following statements is true?
$\textbf{(A) }\text{Ravon was given card 3.}$
$\textbf{(B) }\text{Aditi was given card 3.}$
$\textbf{(C) }\text{Ravon was given card 4.}$
$\textbf{(D) }\text{Aditi was given card 4.}$
$\textbf{(E) }\text{Tyrone was given card 7.}$
2016 VJIMC, 2
Find all positive integers $n$ such that $\varphi(n)$ divides $n^2 + 3$.
Ukrainian TYM Qualifying - geometry, 2016.15
A non isosceles triangle $ABC$ is given, in which $\angle A = 120^o$. Let $AL$ be its angle bisector, $AK$ be it's median, drawn from vertex $A$, point $O$ be the center of the circumcircle of this triangle, $F$ be the point of intersection of the lines $OL$ and $AK$. Prove that $\angle BFC = 60^o$.
1972 IMO Longlists, 35
$(a)$ Prove that for $a, b, c, d \in\mathbb{R}, m \in [1,+\infty)$ with $am + b =-cm + d = m$,
\[(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and}\]
\[(ii) 2 \le \frac{4m^2}{1+m^2} < 4.\]
$(b)$ Express $a, b, c, d$ as functions of $m$ so that there is equality in $(i).$
1994 IMO Shortlist, 1
$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$
2021 MOAA, 5
Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?
[asy]
size(4cm);
fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3));
draw((0,0)--(0,12)--(-5,0)--cycle);
draw((0,0)--(8,0)--(0,6));
label("5", (-2.5,0), S);
label("13", (-2.5,6), dir(140));
label("6", (0,3), E);
label("8", (4,0), S);
[/asy]
[i]Proposed by Nathan Xiong[/i]
2000 France Team Selection Test, 2
$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.
2022 District Olympiad, P2
$a)$ Prove that $2x^3-3x^2+1\geq 0,~(\forall)x\geq0.$
$b)$ Let $x,y,z\geq 0$ such that $\frac{2}{1+x^3}+\frac{2}{1+y^3}+\frac{2}{1+z^3}=3.$ Prove that $\frac{1-x}{1-x+x^2}+\frac{1-y}{1-y+y^2}+\frac{1-z}{1-z+z^2}\geq 0.$
2019 India Regional Mathematical Olympiad, 4
Consider the following $3\times 2$ array formed by using the numbers $1,2,3,4,5,6$,
$$\begin{pmatrix} a_{11}& a_{12}\\a_{21}& a_{22}\\ a_{31}& a_{32}\end{pmatrix}=\begin{pmatrix}1& 6\\2& 5\\ 3& 4\end{pmatrix}.$$
Observe that all row sums are equal, but the sum of the square of the squares is not the same for each row. Extend the above array to a $3\times k$ array $(a_{ij})_{3\times k}$ for a suitable $k$, adding more columns, using the numbers $7,8,9,\dots ,3k$ such that $$\sum_{j=1}^k a_{1j}=\sum_{j=1}^k a_{2j}=\sum_{j=1}^k a_{3j}~~\text{and}~~\sum_{j=1}^k (a_{1j})^2=\sum_{j=1}^k (a_{2j})^2=\sum_{j=1}^k (a_{3j})^2$$
1983 Miklós Schweitzer, 11
Let $ M^n \subset \mathbb{R}^{n\plus{}1}$ be a complete, connected hypersurface embedded into the Euclidean space. Show that $ M^n$ as a Riemannian manifold decomposes to a nontrivial global metric direct product if and only if it is a real cylinder, that is, $ M^n$ can be decomposed to a direct product of the form $ M^n\equal{}M^k \times \mathbb{R}^{n\minus{}k} \;(k<n)$ as well, where $ M^k$ is a hypersurface in some $ (k\plus{}1)$-dimensional subspace $ E^{k\plus{}1} \subset \mathbb{R}^{n\plus{}1} , \mathbb{R}^{n\minus{}k}$ is the orthogonal complement of $ E^{k\plus{}1}$.
[i]Z. Szabo[/i]
2013 ELMO Shortlist, 4
Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$.
[i]Proposed by Evan Chen[/i]
2016 Latvia Baltic Way TST, 4
Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds:
$$f(2^x+2y) =2^yf(f(x))f(y).$$
1993 All-Russian Olympiad Regional Round, 10.1
Point $D$ is chosen on the side $AC$ of an acute-angled triangle $ABC$. The median $AM$ intersects the altitude $CH$ and the segment $BD$ at points $N$ and $K$ respectively. Prove that if $AK = BK$, then $AN = 2KM$.
2017 Turkey MO (2nd round), 5
Let $x_0,\dots,x_{2017}$ are positive integers and $x_{2017}\geq\dots\geq x_0=1$ such that $A=\{x_1,\dots,x_{2017}\}$ consists of exactly $25$ different numbers. Prove that $\sum_{i=2}^{2017}(x_i-x_{i-2})x_i\geq 623$, and find the number of sequences that holds the case of equality.
2018 Dutch IMO TST, 2
Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.
2017 AMC 8, 3
What is the value of the expression $\sqrt{16\sqrt{8\sqrt{4}}}$?
$\textbf{(A) }4\qquad\textbf{(B) }4\sqrt{2}\qquad\textbf{(C) }8\qquad\textbf{(D) }8\sqrt{2}\qquad\textbf{(E) }16$
JOM 2015 Shortlist, A3
Let $ a, b, c $ be positive real numbers less than or equal to $ \sqrt{2} $ such that $ abc = 2 $, prove that $$ \sqrt{2}\displaystyle\sum_{cyc}\frac{ab + 3c}{3ab + c} \ge a + b + c $$
1989 IMO Longlists, 33
Find all square numbers $ S_1$ and $ S_2$ such that $ S_1 \minus{} S_2 \equal{} 1989.$
2016 Costa Rica - Final Round, G3
Let $\vartriangle ABC$ be acute, with incircle $\Gamma$ and incenter $ I$. $\Gamma$ touches sides $AB$, $BC$ and $AC$ at $Z$, $X$ and $Y$, respectively. Let $D$ be the intersection of $XZ$ with $CI$ and $L$ the intersection of $BI$ with $XY$. Suppose $D$ and $L$ are outside of $\vartriangle ABC$. Prove that $A$, $D$, $Z$, $I$, $Y$, and $ L$ lie on a circle.
2007 Stanford Mathematics Tournament, 7
Daniel counts the number of ways he can form a positive integer using the digits $1, 2, 2, 3$, and $4$ (each digit at most once). Edward counts the number of ways you can use the letters in the word "$BANANAS$" to form a six-letter word (it doesn't have to be a real English word). Fernando counts the number of ways you can distribute nine identical pieces of candy to three children. By their powers combined, they add each of their values to form the number that represents the meaning of life. What is the meaning of life? (Hint: it's not $42$.)