Found problems: 15925
2010 Germany Team Selection Test, 2
We are given $m,n \in \mathbb{Z}^+.$ Show the number of solution $4-$tuples $(a,b,c,d)$ of the system
\begin{align*}
ab + bc + cd - (ca + ad + db) &= m\\
2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n
\end{align*}
is divisible by 10.
2011 Saudi Arabia Pre-TST, 2.3
Let $f = aX^2 + bX+ c \in Z[X]$ be a polynomial such that for every positive integer $n$,$ f(n )$ is a perfect square. Prove that $f = g^2$ for some polynomial $g \in Z[X]$.
2014 Israel National Olympiad, 5
Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.
2023 Durer Math Competition (First Round), 1
Find all positive integers $n$ such that $$\lfloor \sqrt{n} \rfloor +
\left\lfloor \frac{n}{\lfloor \sqrt{n} \rfloor} \right \rfloor> 2\sqrt{n}.$$
If $k$ is a real number, then $\lfloor k \rfloor$ means the floor of $k$, this is the greatest integer less than or equal to $k$.
2020 Kosovo Team Selection Test, 1
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$ satisfy,
$$f\left(x+yf(x+y)\right)=y^2+f(x)f(y)$$
[i]Proposed by Dorlir Ahmeti, Kosovo[/i]
2013 Korea National Olympiad, 3
Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that
(1) For all integer $m$, $f(m) \ne 0$.
(2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.
2021 Macedonian Team Selection Test, Problem 5
Determine all functions $f:\mathbb{N}\to \mathbb{N}$ such that for all $a, b \in \mathbb{N}$ the following conditions hold:
$(i)$ $f(f(a)+b) \mid b^a-1$;
$(ii)$ $f(f(a))\geq f(a)-1$.
2025 Czech-Polish-Slovak Junior Match., 5
For every integer $n\geq 1$ prove that
$$\frac{1}{n+1}-\frac{2}{n+2}+\frac{3}{n+3}-\frac{4}{n+4}+...+\frac{2n-1}{3n-1}>\frac{1}{3}.$$
1965 AMC 12/AHSME, 19
If $ x^4 \plus{} 4x^3 \plus{} 6px^2 \plus{} 4qx \plus{} r$ is exactly divisible by $ x^3 \plus{} 3x^2 \plus{} 9x \plus{} 3$, the value of $ (p \plus{} q)r$ is:
$ \textbf{(A)}\ \minus{} 18 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 45 \qquad$
2011 Dutch IMO TST, 4
Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.
2012 Bosnia And Herzegovina - Regional Olympiad, 1
For which real numbers $x$ and $\alpha$ inequality holds: $$\log _2 {x}+\log _x {2}+2\cos{\alpha} \leq 0$$
2021 Grand Duchy of Lithuania, 1
Prove that for any polynomial $f(x)$ (with real coefficients) there exist polynomials $g(x)$ and $h(x)$ (with real coefficients) such that $f(x) = g(h(x)) - h(g(x))$.
TNO 2023 Senior, 6
The points inside a circle \( \Gamma \) are painted with \( n \geq 1 \) colors. A color is said to be dense in a circle \( \Omega \) if every circle contained within \( \Omega \) has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within \( \Gamma \).
VII Soros Olympiad 2000 - 01, 10.2
Let $a$ and $ b$ be acute corners. Prove that if $\sin a$, $\sin b$, and $\sin (a + b)$ are rational numbers, then $\cos a$, $\cos b$, and $\cos (a + b)$ are also rational numbers.
2023 Romania JBMO TST, P1
Determine the real numbers $x$, $y$, $z > 0$ for which
$xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$
1985 IMO Shortlist, 18
Let $x_1, x_2, \cdots , x_n$ be positive numbers. Prove that
\[\frac{x_1^2}{x_1^2+x_2x_3} + \frac{x_2^2}{x_2^2+x_3x_4} + \cdots +\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1} +\frac{x_n^2}{x_n^2+x_1x_2} \leq n-1\]
2013 China Team Selection Test, 3
Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]
2021 Kosovo National Mathematical Olympiad, 3
Let $a,b$ and $c$ be positive real numbers such that $a^5+b^5+c^5=ab^2+bc^2+ca^2$.
Prove the inequality:
$$\frac{a^2+b^2}{b}+\frac{b^2+c^2}{c}+\frac{c^2+a^2}{a}\geq 2(ab+bc+ca).$$
1975 IMO Shortlist, 14
Let $x_0 = 5$ and $x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )$. Prove that
\[45 < x_{1000} < 45. 1.\]
2014 Regional Competition For Advanced Students, 2
You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\
bc + bd = 5c + 5d \\
ac + cd = 7a + 7d \\
ad + bd = 9a + 9b \end{cases} $
2004 Postal Coaching, 13
Find all functions $f,g : \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R}^{+}$ such that
\[ ( \sum_{j=1}^{n}a_{j}b_{j})^2 \leq (\sum_{j=1}^{n} f({a_{j},b_{j}))(\sum_{j=1}^{n} g({a_{j},b_{j})) \leq (\sum_{j=1}^{n} (a_j)^2 )(\sum_{j=1}^{n} (b_j)^2 ) }}\] for any two sets $a_j$ and $b_j$ of real numbers.
2024 Romania National Olympiad, 3
Find the functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$(f(x)-y)f(x+f(y))=f(x^2)-yf(y),$$ for all real numbers $x$ and $y.$
2001 India National Olympiad, 6
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(x +y) = f(x) f(y) f(xy)$ for all $x, y \in \mathbb{R}.$
2025 All-Russian Olympiad, 11.3
A pair of polynomials \(F(x, y)\) and \(G(x, y)\) with integer coefficients is called $\emph{important}$ if from the divisibility of both differences \(F(a, b) - F(c, d)\) and \(G(a, b) - G(c, d)\) by $100$, it follows that both \(a - c\) and \(b - d\) are divisible by 100. Does there exist such an important pair of polynomials \(P(x, y)\), \(Q(x, y)\), such that the pair \(P(x, y) - xy\) and \(Q(x, y) + xy\) is also important?
2008 China Team Selection Test, 3
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.