Found problems: 85335
2017 AMC 12/AHSME, 21
Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?
$\textbf{(A)} \text{ 92} \qquad \textbf{(B)} \text{ 94} \qquad \textbf{(C)} \text{ 96} \qquad \textbf{(D)} \text{ 98} \qquad \textbf{(E)} \text{ 100}$
1958 November Putnam, B4
Let $C$ be a real number, and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function such that
$$ \lim_{x \to \infty} f(x)=C, \;\; \; \lim_{x \to \infty} f'''(x)=0.$$
Prove that
$$ \lim_{x \to \infty} f'(x) =0 \;\; \text{and} \;\; \lim_{x \to \infty} f''(x)=0.$$
2010 China Second Round Olympiad, 1
Given an acute triangle whose circumcenter is $O$.let $K$ be a point on $BC$,different from its midpoint.$D$ is on the extension of segment $AK,BD$ and $AC$,$CD$and$AB$intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic.
2010 LMT, 5
Big Welk writes the letters of the alphabet in order, and starts again at $A$ each time he gets to $Z.$ What is the $4^3$-rd letter that he writes down?
1996 All-Russian Olympiad, 3
Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon.
[i]N. Agakhanov, N. Tereshin[/i]
2019 Saudi Arabia JBMO TST, 3
Let $6$ pairwise different digits are given and all of them are different from $0$. Prove that there exist $2$ six-digit integers, such that their difference is equal to $9$ and each of them contains all given $6$ digits.
2018 Brazil Team Selection Test, 1
Let $n \ge 1$ be an integer. For each subset $S \subset \{1, 2, \ldots , 3n\}$, let $f(S)$ be the sum of the elements of $S$, with $f(\emptyset) = 0$. Determine, as a function of $n$, the sum $$\sum_{\mathclap{\substack{S \subset \{1,2,\ldots,3n\}\\
3 \mid f(S)}}} f(S)$$
where $S$ runs through all subsets of $\{1, 2,\ldots, 3n\}$ such that $f(S)$ is a multiple of $3$.
2014 Middle European Mathematical Olympiad, 8
Determine all quadruples $(x,y,z,t)$ of positive integers such that
\[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]
2020 Adygea Teachers' Geometry Olympiad, 4
A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.
Estonia Open Junior - geometry, 2015.1.5
Let $ABC$ be an acute triangle. The arcs $AB$ and $AC$ of the circumcircle of the triangle are reflected over the lines AB and $AC$, respectively. Prove that the two arcs obtained intersect in another point besides $A$.
2016 NIMO Problems, 5
For positive integers $n,$ let $s(n)$ be the sum of the digits of $n.$ Over all four-digit positive integers $n,$ which value of $n$ maximizes the ratio $\frac{s(n)}{n}$?
[i]Proposed by Michael Tang[/i]
2009 AMC 12/AHSME, 21
Let $ p(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$, where $ a$, $ b$, and $ c$ are complex numbers. Suppose that
\[ p(2009 \plus{} 9002\pi i) \equal{} p(2009) \equal{} p(9002) \equal{} 0
\]What is the number of nonreal zeros of $ x^{12} \plus{} ax^8 \plus{} bx^4 \plus{} c$?
$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$
2021 Greece Junior Math Olympiad, 3
Determine whether exists positive integer $n$ such that the number $A=8^n+47$ is prime.
1995 Bundeswettbewerb Mathematik, 4
Prove that every integer $k > 1$ has a multiple less than $k^4$ whose decimal expension has at most four distinct digits.
2016 239 Open Mathematical Olympiad, 4
Positive real numbers $a,b,c$ are given such that $abc=1$. Prove that$$a+b+c+\frac{3}{ab+bc+ca}\geq4.$$
2012 Stanford Mathematics Tournament, 6
There exist two triples of real numbers $(a,b,c)$ such that $a-\frac{1}{b}, b-\frac{1}{c}, c-\frac{1}{a}$ are the roots to the cubic equation $x^3-5x^2-15x+3$ listed in increasing order. Denote those $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$. If $a_1$, $b_1$, and $c_1$ are the roots to monic cubic polynomial $f$ and $a_2, b_2$, and $c_2$ are the roots to monic cubic polynomial $g$, find $f(0)^3+g(0)^3$
1996 Rioplatense Mathematical Olympiad, Level 3, 6
Find all integers $k$ for which, there is a function $f: N \to Z$ that satisfies:
(i) $f(1995) = 1996$
(ii) $f(xy) = f(x) + f(y) + kf(m_{xy})$ for all natural numbers $x, y$,where$ m_{xy}$ denotes the greatest common divisor of the numbers $x, y$.
Clarification: $N = \{1,2,3,...\}$ and $Z = \{...-2,-1,0,1,2,...\}$ .
2016 Indonesia TST, 4
In a non-isosceles triangle $ABC$, let $I$ be its incenter. The incircle of $ABC$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A line passing through $D$ and perpendicular to $AD$ intersects $IB$ and $IC$ at $A_b$ and $A_c$, respectively. Define the points $B_c$, $B_a$, $C_a$, and $C_b$ similarly. Let $G$ be the intersection of the cevians $AD$, $BE$, and $CF$. The points $O_1$ and $O_2$ are the circumcenter of the triangles $A_bB_cC_a$ and $A_cB_aC_b$, respectively. Prove that $IG$ is the perpendicular bisector of $O_1O_2$.
LMT Speed Rounds, 4
The numbers $1$, $2$, $3$, and $4$ are randomly arranged in a $2$ by $2$ grid with one number in each cell. Find the probability the sum of two numbers in the top row of the grid is even.
[i]Proposed by Muztaba Syed and Derek Zhao[/i]
[hide=Solution]
[i]Solution. [/i]$\boxed{\dfrac{1}{3}}$
Pick a number for the top-left. There is one number that makes the sum even no matter what we pick. Therefore, the
answer is $\boxed{\dfrac{1}{3}}$.[/hide]
2018 CMIMC Combinatorics, 1
Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?
2013 Stanford Mathematics Tournament, 24
Compute the square of the distance between the incenter (center of the inscribed circle) and circumcenter (center of the circumscribed circle) of a 30-60-90 right triangle with hypotenuse of length 2.
2024 AMC 8 -, 10
In January 1980 the Moana Loa Observation recorded carbon dioxide levels of 338 ppm (parts per million). Over the years the average carbon dioxide reading has increased by about 1.515 ppm each year. What is the expected carbon dioxide level in ppm in January 2030? Round your answer to the nearest integer.
$\textbf{(A) } 399\qquad\textbf{(B) } 414\qquad\textbf{(C) } 420\qquad\textbf{(D) } 444\qquad\textbf{(E) } 459$
IV Soros Olympiad 1997 - 98 (Russia), 10.7
Prove that the number $\left(\sqrt2+\sqrt3+\sqrt5\right)^{1997}$ can be represented as $$A\sqrt2+B\sqrt3+C\sqrt5+D\sqrt{30}$$ where $A$, $B$, $C$, $D$ are integers. Find with approximation to $10^{-10}$ the ratio $\frac{D}{A}$
2014 Contests, 3
Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.
2023 Yasinsky Geometry Olympiad, 1
Let $BD$ and $CE$ be the altitudes of triangle $ABC$ that intersect at point $H$. Let $F$ be a point on side $AC$ such that $FH\perp CE$. The segment $FE$ intersects the circumcircle of triangle $CDE$ at the point $K$. Prove that $HK\perp EF$ .
(Matthew Kurskyi)