Found problems: 85335
2010 Contests, 4
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
2016 Harvard-MIT Mathematics Tournament, 20
Let $ABC$ be a triangle with $AB=13$, $AC=14$, and $BC=15$. Let $G$ be the point on $AC$ such that the reflection of $BG$ over the angle bisector of $\angle B$ passes through the midpoint of $AC$. Let $Y$ be the midpoint of $GC$ and $X$ be a point on segment $AG$ such that $\frac{AX}{XG}=3$. Construct $F$ and $H$ on $AB$ and $BC$, respectively, such that $FX \parallel BG \parallel HY$. If $AH$ and $CF$ concur at $Z$ and $W$ is on $AC$ such that $WZ \parallel BG$, find $WZ$.
2019 Tournament Of Towns, 5
Let us say that the pair $(m, n)$ of two positive different integers m and n is [i]nice [/i] if $mn$ and $(m + 1)(n + 1)$ are perfect squares. Prove that for each positive integer m there exists at least one $n > m$ such that the pair $(m, n)$ is nice.
(Yury Markelov)
2004 AMC 10, 13
In the United States, coins have the following thicknesses: penny, $ 1.55$ mm; nickel, $ 1.95$ mm; dime, $ 1.35$ mm; quarter, $ 1.75$ mm. If a stack of these coins is exactly $ 14$ mm high, how many coins are in the stack?
$ \textbf{(A)}\ 7\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 11$
2017 MIG, 1
Solve for $x$: $2x+7=21$
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$
2012 JBMO ShortLists, 3
Let $a$ , $b$ , $c$ be positive real numbers such that $a+b+c=a^2+b^2+c^2$ . Prove that :
\[\frac{a^2}{a^2+ab}+\frac{b^2}{b^2+bc}+\frac{c^2}{c^2+ca} \geq \frac{a+b+c}{2}\]
1978 All Soviet Union Mathematical Olympiad, 264
Given $0 < a \le x_1\le x_2\le ... \le x_n \le b$. Prove that $$(x_1+x_2+...+x_n)\left ( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab}n^2$$
2010 Princeton University Math Competition, 2
Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.
2017 Iran Team Selection Test, 4
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$.
Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$.
Is there always a number $x$ that satisfies all the equations?
[i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]
1971 AMC 12/AHSME, 18
The current in a river is flowing steadily at $3$ miles per hour. A motor boat which travels at a constant rate in still water goes downstream $4$ miles and then returns to its starting point. The trip takes one hour, excluding the time spent in turning the boat around. The ratio of the downstream to the upstream rate is
$\textbf{(A) }4:3\qquad\textbf{(B) }3:2\qquad\textbf{(C) }5:3\qquad\textbf{(D) }2:1\qquad \textbf{(E) }5:2$
2018 ASDAN Math Tournament, 5
In the expansion of $(x + b)^{2018}$, the coefficients of $x^2$ and $x^3$ are equal. Compute $b$.
1992 Swedish Mathematical Competition, 3
Solve:
$$\begin{cases} 2x_1 - 5x_2 + 3x_3 \ge 0 \\
2x_2 - 5x_3 + 3x4 \ge 0 \\
...\\
2x_{23} - 5x_{24} + 3x_{25} \ge 0\\
2x_{24} - 5x_{25} + 3x_1 \ge 0\\
2x_{25} - 5x_1 + 3x_2 \ge 0 \end{cases}$$
2024 Sharygin Geometry Olympiad, 8.7
A convex quadrilateral $ABCD$ is given. A line $l \parallel AC$ meets the lines $AD$, $BC$, $AB$, $CD$ at points $X$, $Y$, $Z$, $T$ respectively. The circumcircles of triangles $XYB$ and $ZTB$ meet for the second time at point $R$. Prove that $R$ lies on $BD$.
2016 Finnish National High School Mathematics Comp, 4
How many pairs $(a, b)$ of positive integers $a,b$ solutions of the equation $(4a-b)(4b-a )=1770^n$ exist , if $n$ is a positive integer?
1994 Putnam, 3
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$
Kvant 2021, M2664
The point $O{}$ is given in the plane. Find all natural numbers $n{}$ for which $n{}$ points in the plane can be colored red, so that for any two red points $A{}$ and $B{}$ there is a third red point $C{}$ is such that $O{}$ lies strictly inside the triangle $ABC$.
[i]From the folklore[/i]
2020 Ukrainian Geometry Olympiad - April, 5
The plane shows $2020$ straight lines in general position, that is, there are none three intersecting at one point but no two parallel. Let's say, that the drawn line $a$ [i]detaches [/i] the drawn line $b$ if all intersection points of line $b$ with the other drawn lines lie in one half plane wrt to line $a$ (given the most straightforward $a$). Prove that you can be guaranteed find two drawn lines $a$ and $b$ that $a$ detaches $b$, but $b$ does not detach $a$.
1998 Tournament Of Towns, 2
For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result?
( G Galperin)
2015 Purple Comet Problems, 4
Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%
faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.
2023 Indonesia TST, N
Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.
2013 Princeton University Math Competition, 8
Eight all different sushis are placed evenly on the edge of a round table, whose surface can rotate around the center. Eight people also evenly sit around the table, each with one sushi in front. Each person has one favorite sushi among these eight, and they are all distinct. They find that no matter how they rotate the table, there are never more than three people who have their favorite sushis in front of them simultaneously. By this requirement, how many different possible arrangements of the eight sushis are there? Two arrangements that differ by a rotation are considered the same.
1994 Swedish Mathematical Competition, 4
Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$.
1980 IMO Longlists, 9
Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.
1967 IMO Shortlist, 2
Prove that
\[\frac{1}{3}n^2 + \frac{1}{2}n + \frac{1}{6} \geq (n!)^{\frac{2}{n}},\]
and let $n \geq 1$ be an integer. Prove that this inequality is only possible in the case $n = 1.$
2014-2015 SDML (High School), 5
Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square their numbers to get a new number and see that the sum of these new numbers is half of $169$. Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's by $5070$. What was the sum of their original numbers?
$\text{(A) }-4\qquad\text{(B) }-3\qquad\text{(C) }1\qquad\text{(D) }2\qquad\text{(E) }5$