Found problems: 85335
2008 Saint Petersburg Mathematical Olympiad, 1
Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point.
EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet.
Note: fresh translation
MathLinks Contest 7th, 1.3
We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n \minus{} 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called [i]nice[/i], if $ f_i(x_i) \equal{} f_i(x_{i \plus{} 1})$, for each $ i \equal{} 1,2,\dots,n \minus{} 1$. Prove that the number of nice vectors is at least
\[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}.
\]
2023 LMT Fall, 7
How many $2$-digit factors does $555555$ have?
2021 Purple Comet Problems, 23
The sum $$\sum_{k=3}^{\infty} \frac{1}{k(k^4-5k^2+4)^2}$$ is equal to $\frac{m^2}{2n^2}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2010 Indonesia TST, 1
Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and
$$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$ $\forall n=1,2,...$
Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$
2023 Romania National Olympiad, 1
We consider the equation $x^2 + (a + b - 1)x + ab - a - b = 0$, where $a$ and $b$ are positive integers with $a \leq b$.
a) Show that the equation has $2$ distinct real solutions.
b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and $b < 2a.$
2001 China Western Mathematical Olympiad, 1
Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.
2019 India PRMO, 9
Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$. What is the value of $p - 3q$?
2004 All-Russian Olympiad, 3
The natural numbers from 1 to 100 are arranged on a circle with the characteristic that each number is either larger as their two neighbours or smaller than their two neighbours. A pair of neighbouring numbers is called "good", if you cancel such a pair, the above property remains still valid. What is the smallest possible number of good pairs?
2021 AMC 10 Fall, 7
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form [i]special[/i] if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\
12 \qquad\textbf{(E)}\ 13$
2019 Durer Math Competition Finals, 3
Determine all triples $(p, q, r)$ of prime numbers for which $p^q + p^r$ is a perfect square.
2000 Czech And Slovak Olympiad IIIA, 1
Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.
2016 CMIMC, 6
Suppose integers $a < b < c$ satisfy \[ a + b + c = 95\qquad\text{and}\qquad a^2 + b^2 + c^2 = 3083.\] Find $c$.
1961 Czech and Slovak Olympiad III A, 1
Consider an infinite sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, \ldots, \underbrace{n,\ldots,n}_{n\text{ times}},\ldots.$$
Find the 1000th term of the sequence.
2008 Mediterranean Mathematics Olympiad, 1
Let $ABCDEF$ be a convex hexagon such that all of its vertices are on a circle. Prove that $AD$, $BE$ and $CF$ are concurrent if and only if $\frac {AB}{BC}\cdot\frac {CD}{DE}\cdot\frac {EF}{FA}= 1$.
2018 Nordic, 2
A sequence of primes $p_1, p_2, \dots$ is given by two initial primes $p_1$ and $p_2$, and $p_{n+2}$ being the greatest prime divisor of $p_n + p_{n+1} + 2018$ for all $n \ge 1$. Prove that the sequence only contains finitely many primes for all possible values of $p_1$ and $p_2$.
1991 Baltic Way, 7
If $\alpha,\beta,\gamma$ are the angles of an acute-angled triangle, prove that
\[\sin \alpha + \sin \beta > \cos \alpha + \cos\beta + \cos\gamma.\]
2020 Malaysia IMONST 1, 12
A football is made by sewing together some black and white leather
patches. The black patches are regular pentagons of the same size. The white
patches are regular hexagons of the same size. Each pentagon is bordered by 5
hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football.
How many hexagons are needed to make one football?
1970 AMC 12/AHSME, 30
In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively. Then the measure of $AB$ is equal to
$\textbf{(A) }\dfrac{1}{2}a+2b\qquad\textbf{(B) }\dfrac{3}{2}b+\dfrac{3}{4}a\qquad\textbf{(C) }2a-b\qquad\textbf{(D) }4b-\dfrac{1}{2}a\qquad \textbf{(E) }a+b$
[asy]
size(175);
defaultpen(linewidth(0.8));
real r=50, a=4,b=2.5,c=6.25;
pair A=origin,B=c*dir(r),D=(a,0),C=shift(b*dir(r))*D;
draw(A--B--C--D--cycle);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,E);
label("$D$",D,S);
label("$a$",D/2,N);
label("$b$",(C+D)/2,NW);
//Credit to djmathman for the diagram[/asy]
2022 Durer Math Competition (First Round), 5
Let $a_1 \le a_2 \le ... \le a_n$ be real numbers for which $$\sum_{i=1}^{n} a_i^{2k+1} = 0$$ holds for all integers $0 \le k < n$. Show that in this case, $a_i = -a_{n+1-i}$ holds for all $1 \le i \le n$.
2024 HMNT, 9
Let $ABCDEF$ be a regular hexagon with center $O$ and side length $1.$ Point $X$ is placed in the interior of the hexagon such that $\angle BXC = \angle AXE = 90^\circ.$ Compute all possible values of $OX.$
2018 Junior Regional Olympiad - FBH, 3
In some primary school there were $94$ students in $7$th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies $40$ students outside school program, german $27$ students and $60$ students do sports. Out of the students doing sports, $24$ of them also goes to spanish language. $10$ students who study spanish also study german. $12$ students who study german also do sports. Only $4$ students go to all three activities. How many of them does only one of the activities, and how much of them do not go to any activity?
2016 Latvia National Olympiad, 3
Is it possible to insert numbers $1, \ldots, 16$ into a table $4 \times 4$ (each cell should have a different number) so that every two adjacent cells (i.e. cells sharing a common side) have numbers $a$ and $b$ satisfying\\
(a) $|a-b| \geq 6$\\
(b) $|a-b| \geq 7$
2025 Czech-Polish-Slovak Junior Match., 2
Find all triangles that can be divided into congruent right-angled isosceles triangles with side lengths $1, 1, \sqrt{2}$.
2006 AMC 12/AHSME, 24
The expression
\[ (x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006}
\]is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
$ \textbf{(A) } 6018 \qquad \textbf{(B) } 671,676 \qquad \textbf{(C) } 1,007,514 \qquad \textbf{(D) } 1,008,016 \qquad \textbf{(E) } 2,015,028$