Found problems: 85335
2008 China Team Selection Test, 3
Let $ n>m>1$ be odd integers, let $ f(x)\equal{}x^n\plus{}x^m\plus{}x\plus{}1$. Prove that $ f(x)$ can't be expressed as the product of two polynomials having integer coefficients and positive degrees.
2017 NZMOC Camp Selection Problems, 8
Find all possible real values for $a, b$ and $c$ such that
(a) $a + b + c = 51$,
(b) $abc = 4000$,
(c) $0 < a \le 10$ and $c \ge 25$.
2021 Czech-Polish-Slovak Junior Match, 1
Consider a trapezoid $ABCD$ with bases $AB$ and $CD$ satisfying $| AB | > | CD |$. Let $M$ be the midpoint of $AB$. Let the point $P$ lie inside $ABCD$ such that $| AD | = | PC |$ and $| BC | = | PD |$. Prove that if $| \angle CMD | = 90^o$, then the quadrilaterals $AMPD$ and $BMPC$ have the same area.
2006 China Western Mathematical Olympiad, 1
Let $S=\{n|n-1,n,n+1$ can be expressed as the sum of the square of two positive integers.$\}$. Prove that if $n$ in $S$, $n^{2}$ is also in $S$.
2009 Postal Coaching, 2
Let $a > 2$ be a natural number. Show that there are infinitely many natural numbers n such that $a^n \equiv -1$ (mod $n^2$).
2012 Puerto Rico Team Selection Test, 1
Let $x, y$ and $z$ be consecutive integers such that
\[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\]
Find the maximum value of $x + y + z$.
1985 All Soviet Union Mathematical Olympiad, 412
One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle circumscribed around $AMD$ triangle has radius $R$.
2010 Purple Comet Problems, 15
In the number arrangement
\[\begin{array}{ccccc}
\texttt{1}&&&&\\
\texttt{2}&\texttt{3}&&&\\
\texttt{4}&\texttt{5}&\texttt{6}&&\\
\texttt{7}&\texttt{8}&\texttt{9}&\texttt{10}&\\
\texttt{11}&\texttt{12}&\texttt{13}&\texttt{14}&\texttt{15}\\
\vdots&&&&
\end{array}\]
what is the number that will appear directly below the number $2010$?
2020 Vietnam National Olympiad, 5
Let a system of equations:
$\left\{\begin{matrix}x-ay=yz\\y-az=zx\\z-ax=xy\end{matrix}\right.$
a)Find (x,y,z) if a=0
b)Prove that: the system have 5 distinct roots $\forall$a>1,a$\in\mathbb{R}.$
2010 Today's Calculation Of Integral, 644
For a constant $p$ such that $\int_1^p e^xdx=1$, prove that
\[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\]
Own
1976 IMO Longlists, 47
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.
2009 Pan African, 3
Let $x$ be a real number with the following property: for each positive integer $q$, there exists an integer $p$, such that
\[\left|x-\frac{p}{q} \right|<\frac{1}{3q}. \]
Prove that $x$ is an integer.
1996 Greece National Olympiad, 1
Let $a_n$ be a sequence of positive numbers such that:
i) $\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}$, for every $n\in\mathbb{N}^{\star}$
ii) $\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1$, for every $ k,n\in\mathbb{N}^{\star}$ with $|k-n|\neq 1$.
(a) Prove that $(a_n)$ is a geometric progression.
(n) Prove that exists $t>0$, such that $\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t$
2021 CCA Math Bonanza, L4.3
For a positive integer $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96)=1+2+3+6+9=21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a)=n$, where $f^k(a)$ denotes $f$ applied $k$ times to $a$.
[i]2021 CCA Math Bonanza Lightning Round #4.3[/i]
2003 India National Olympiad, 1
Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.
1980 All Soviet Union Mathematical Olympiad, 287
The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.
2025 China National Olympiad, 6
Let $a_1, a_2, \ldots, a_n$ be real numbers such that $\sum_{i=1}^n a_i = n$, $\sum_{i = 1}^n a_i^2 = 2n$, $\sum_{i=1}^n a_i^3 = 3n$.
(i) Find the largest constant $C$, such that for all $n \geqslant 4$, \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C. \]
(ii) Prove that there exists a positive constant $C_2$, such that \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C + C_2 n^{-\frac 32}, \]where $C$ is the constant determined in (i).
2017 Balkan MO Shortlist, A4
Let $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$ with $a+b+c=1 \}$ and $f: M\to R$ given as $$f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc}$$
Find the best (real) bounds $\alpha$ and $\beta$ such that $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$ and determine whether any of them is achievable.
II Soros Olympiad 1995 - 96 (Russia), 11.3
The math problem book contains $300$ problems. The teacher has cards with numbers. She pins these cards to a special stand and indicates the numbers of four problems that need to be solved during the lesson. What is the smallest number of cards that a teacher can use in order to be able to indicate the numbers of any four problems from the problem book?
2007 Stanford Mathematics Tournament, 8
A $13$-foot tall extraterrestrial is standing on a very small spherical planet with radius $156$ feet. It sees an ant crawling along the horizon. If the ant circles the extraterrestrial once, always staying on the horizon, how far will it travel (in feet)?
2023 Sharygin Geometry Olympiad, 8
A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines.
2015 Peru IMO TST, 4
Let $n\geq 2$ be an integer. The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called [i]quadratic[/i] if $a_ia_{i +1} + 1$ is a perfect square for all $1\leq i \leq n-1.$ The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called [i]cubic[/i] if $a_ia_{i + 1} + 1$ is a perfect cube for all $1\leq i \leq n - 1.$
a) Prove that for infinitely many values of $n$ is there at least one quadratic permutation of the numbers $1, 2,...,n.$
b) Prove that for no value of $n$ is there a cubic permutation of the numbers $1, 2,..., n.$
1992 Vietnam National Olympiad, 3
Label the squares of a $1991 \times 1992$ rectangle $(m, n)$ with $1 \leq m \leq 1991$ and $1 \leq n \leq 1992$. We wish to color all the squares red. The first move is to color red the squares $(m, n), (m+1, n+1), (m+2, n+1)$for some $m < 1990, n < 1992$. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?
2006 China Team Selection Test, 3
Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]
2021 Ukraine National Mathematical Olympiad, 6
Circles $w_1$ and $w_2$ intersect at points $P$ and $Q$ and touch a circle $w$ with center at point $O$ internally at points $A$ and $B$, respectively. It is known that the points $A,B$ and $Q$ lie on one line. Prove that the point $O$ lies on the external bisector $\angle APB$.
(Nazar Serdyuk)