Found problems: 85335
2019 Taiwan TST Round 2, 2
There are $ n \ge 3 $ puddings in a room. If a pudding $ A $ hates a pudding $ B $, then $ B $ hates $ A $ as well. Suppose the following two conditions holds:
1. Given any four puddings, there are two puddings who like each other.
2. For any positive integer $ m $, if there are $ m $ puddings who like each other, then there exists $ 3 $ puddings (from the other $ n-m $ puddings) that hate each other.
Find the smallest possible value of $ n $.
2019 Czech and Slovak Olympiad III A, 4
Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$. Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$. Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$, respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$. Prove that $AF\perp FJ$.
1947 Kurschak Competition, 1
Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.
2006 Harvard-MIT Mathematics Tournament, 8
Compute $\displaystyle\int_0^{\pi/3}x\tan^2(x)dx$.
India EGMO 2024 TST, 6
Let $ABC$ be an acute angled triangle with orthocentre $H$. Let $E = BH \cap AC$ and $F= CH \cap AB$. Let $D, M, N$ denote the midpoints of segments $AH, BD, CD$ respectively, and $T = FM \cap EN$. Suppose $D, E, T, F$ are concylic. Prove that $DT$ passes through the circumcentre of $ABC$.
[i]Proposed by Pranjal Srivastava[/i]
2014 AMC 12/AHSME, 18
The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A) }19\qquad
\textbf{(B) }31\qquad
\textbf{(C) }271\qquad
\textbf{(D) }319\qquad
\textbf{(E) }511\qquad$
2015 Dutch IMO TST, 5
Let $N$ be the set of positive integers.
Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
2017 Princeton University Math Competition, 2
Let $a\%b$ denote the remainder when $a$ is divided by $b$. Find $\Sigma_{i=1}^{100}(100\%i)$.
2017 Bulgaria JBMO TST, 2
Solve the following equation over the integers
$$ 25x^2y^2+10x^2y+25xy^2+x^2+30xy+2y^2+5x+7y+6= 0.$$
2005 Alexandru Myller, 1
[b]1)[/b] Prove that there are finite sequences, of any length, of nonegative integers having the property that the arithmetic mean of any choice of its elements is natural.
[b]2)[/b] Study if there is an increasing infinite sequence of nonegative integers having the property that the arithmetic mean of any finite choice of its elements is natural.
Kyiv City MO 1984-93 - geometry, 1991.7.4
Given a circle, point $C$ on it and point $A$ outside the circle. The equilateral triangle $ACP$ is constructed on the segment $AC$. Point $C$ moves along the circle. What trajectory will the point $P$ describe?
2011 China Team Selection Test, 3
Let $G$ be a simple graph with $3n^2$ vertices ($n\geq 2$). It is known that the degree of each vertex of $G$ is not greater than $4n$, there exists at least a vertex of degree one, and between any two vertices, there is a path of length $\leq 3$. Prove that the minimum number of edges that $G$ might have is equal to $\frac{(7n^2- 3n)}{2}$.
2022 Cyprus TST, 1
Find all pairs of real numbers $(x,y)$ for which
\[
\begin{aligned}
x^2+y^2+xy&=133 \\
x+y+\sqrt{xy}&=19
\end{aligned}
\]
2011 AMC 12/AHSME, 1
A cell phone plan costs $\$20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?
$ \textbf{(A)}\ \$ 24.00 \qquad
\textbf{(B)}\ \$ 24.50\qquad
\textbf{(C)}\ \$ 25.50\qquad
\textbf{(D)}\ \$ 28.00\qquad
\textbf{(E)}\ \$ 30.00$
2016 Korea Winter Program Practice Test, 2
Let $a_i, b_i$ ($1 \le i \le n$, $n \ge 2$) be positive real numbers such that $\sum_{i=1}^n a_i = \sum_{i=1}^n b_i$.
Prove that $\sum_{i=1}^n \frac{(a_{i+1}+b_{i+1})^2}{n(a_i-b_i)^2+4(n-1)\sum_{j=1}^n a_jb_j} \ge \frac{1}{n-1}$
2023 VN Math Olympiad For High School Students, Problem 7
Given a polynomial with integer coefficents$$P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0,n\ge 1$$satisfying these conditions:
i) $|a_0|$ is not a perfect square.
ii) $P(x)$ is irreducible in $\mathbb{Q}[x].$
Prove that: $P(x^2)$ is irreducible in $\mathbb{Q}[x].$
2014 AMC 10, 2
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
${ \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$
MathLinks Contest 4th, 1.1
Let $a \ge 2$ be an integer. Find all polynomials $f$ with real coefficients such that
$$A = \{a^{n^2} | n \ge 1, n \in Z\} \subset \{f(n) | n \ge 1, n \in Z\} = B.$$
2011 Singapore Senior Math Olympiad, 4
Let $n$ and $k$ be positive integers with $n\geq k\geq 2$. For $i=1,\dots,n$, let $S_i$ be a nonempty set of consecutive integers such that among any $k$ of them, there are two with nonempty intersection. Prove that there is a set $X$ of $k-1$ integers such that each $S_i$, $i=1,\dots,n$ contains at least one integer in $X$.
2002 AMC 10, 6
Cindy was asked by her teacher to subtract $ 3$ from a certain number and then divide the result by $ 9$. Instead, she subtracted $ 9$ and then divided the result by $ 3$, giving an answer of $ 43$. What would her answer have been had she worked the problem correctly?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 34 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 51 \qquad \textbf{(E)}\ 138$
2022 China Team Selection Test, 3
Given a positive integer $n \ge 2$. Find all $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$, such that $1<a_1 \le a_2 \le a_3 \le \cdots \le a_n$, $a_1$ is odd, and
(1) $M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n$ is a positive integer;
(2) One can pick $n$-tuples of integers $(k_{i,1},k_{i,2},\ldots,k_{i,n})$ for $i=1,2,\ldots,M$ such that for any $1 \le i_1 <i_2 \le M$, there exists $j \in \{1,2,\ldots,n\}$ such that $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$.
Indonesia MO Shortlist - geometry, g3
Given a quadrilateral $ABCD$ inscribed in circle $\Gamma$.From a point P outside $\Gamma$, draw tangents $PA$ and $PB$ with $A$ and $B$ as touspoints. The line $PC$ intersects $\Gamma$ at point $D$. Draw a line through $B$ parallel to $PA$, this line intersects $AC$ and $AD$ at points $E$ and $F$ respectively. Prove that $BE = BF$.
1981 AMC 12/AHSME, 6
If $\frac{x}{x-1}=\frac{y^2+2y-1}{y^2-2y-2}$, then $x$ equals
$\text{(A)}\ y^2+2y-1 \qquad \text{(B)}\ y^2+2y-2 \qquad \text{(C)}\ y^2+2y+2$
$\text{(D)}\ y^2+2y+1 \qquad \text{(E)}\ -y^2-2y+1$
2015 Baltic Way, 18
Let $f(x)=x^n + a_{n-1}x^{n-1} + ...+ a_0 $ be a polynomial of degree $ n\ge 1 $ with $ n$ (not necessarily distinct) integer roots. Assume that there exist distinct primes $p_0,p_1,..,p_{n-1}$ such that $a_i > 1$ is a power of $p_i$, for all $ i=0,1,..,n-1$. Find all possible values of $ n$.
2011 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a triangle, $I_a$ the center of the excircle at side $BC$, and $M$ its reflection across $BC$. Prove that $AM$ is parallel to the Euler line of the triangle $BCI_a$.