Found problems: 85335
2000 Stanford Mathematics Tournament, 14
The author of this question was born on April 24, 1977. What day of the week was that?
2005 AMC 10, 6
The average (mean) of $ 20$ numbers is $ 30$, and the average of $ 30$ other numbers is $ 20$. What is the average of all $ 50$ numbers?
$ \textbf{(A)}\ 23 \qquad
\textbf{(B)}\ 24 \qquad
\textbf{(C)}\ 25 \qquad
\textbf{(D)}\ 26 \qquad
\textbf{(E)}\ 27$
1932 Eotvos Mathematical Competition, 1
Let $a, b$ and $n$ be positive integers such that $ b$ is divisible by $a^n$. Prove that $(a+1)^b-1$ is divisible by $a^{n+1}$.
1999 Czech and Slovak Match, 3
Find all natural numbers $k$ for which there exists a set $M$ of ten real numbers such that there are exactly $k$ pairwise non-congruent triangles whose side lengths are three (not necessarily distinct) elements of $M$.
LMT Team Rounds 2021+, B7
Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x}=\frac{y}{13}=\frac{x}{y}$, find the value of $x^3 + y^3$.
Proposed by Ephram Chun
2024 Sharygin Geometry Olympiad, 20
Lines $a_1, b_1, c_1$ pass through the vertices $A, B, C$ respectively of a triange $ABC$; $a_2, b_2, c_2$ are the reflections of $a_1, b_1, c_1$ about the corresponding bisectors of $ABC$; $A_1 = b_1 \cap c_1, B_1 = a_1 \cap c_1, C_1 = a_1 \cap b_1$, and $A_2, B_2, C_2$ are defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ have the same ratios of the area and circumradius (i.e. $\frac{S_1}{R_1} = \frac{S_2}{R_2}$, where $S_i = S(\triangle A_iB_iC_i)$, $R_i = R(\triangle A_iB_iC_i)$)
1988 IMO Longlists, 51
The positive integer $n$ has the property that, in any set of $n$ integers, chosen from the integers $1,2, \ldots, 1988,$ twenty-nine of them form an arithmetic progression. Prove that $n > 1788.$
1982 Brazil National Olympiad, 1
The angles of the triangle $ABC$ satisfy $\angle A / \angle C = \angle B / \angle A = 2$. The incenter is $O. K, L$ are the excenters of the excircles opposite $B$ and $A$ respectively. Show that triangles $ABC$ and $OKL$ are similar.
2011 IMO Shortlist, 4
For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences
\[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4.
[i]Proposed by Gerhard Wöginger, Austria[/i]
1979 Austrian-Polish Competition, 9
Find the greatest power of $2$ that divides $a_n = [(3+\sqrt{11} )^{2n+1}]$, where $n$ is a given positive integer.
2010 Contests, 4
Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously:
(1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$;
(2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$;
(3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$.
Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.
1996 National High School Mathematics League, 3
For a prime number $p$, there exists $n\in\mathbb{Z}_+$, $\sqrt{p+n}+\sqrt{n}$ is an integer, then
$\text{(A)}$ there is no such $p$
$\text{(B)}$ there in only one such $p$
$\text{(C)}$ there is more than one such $p$, but finitely many
$\text{(D)}$ there are infinitely many such $p$
2019 Federal Competition For Advanced Students, P2, 6
Find the smallest possible positive integer n with the following property:
For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$.
(Gerhard J. Woeginger)
2017 Balkan MO Shortlist, N3
Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.
2023 Thailand TST, 2
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
2025 Kosovo National Mathematical Olympiad`, P2
Find all real numbers $a$ and $b$ that satisfy the system of equations:
$$\begin{cases}
a &= \frac{2}{a+b} \\
\\
b &= \frac{2}{3a-b} \\
\end{cases}$$
MOAA Individual Speed General Rounds, 2023.2
In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$.
[i]Proposed by Andy Xu[/i]
2021-IMOC, A4
Find all functions f : R-->R such that
f (f (x) + y^2) = x −1 + (y + 1)f (y)
holds for all real numbers x, y
2023 Austrian MO Regional Competition, 3
Determine all natural numbers $n \ge 2$ with the property that there are two permutations $(a_1, a_2,... , a_n) $ and $(b_1, b_2,... , b_n)$ of the numbers $1, 2,..., n$ such that $(a_1 + b_1, a_2 +b_2,..., a_n + b_n)$ are consecutive natural numbers.
[i](Walther Janous)[/i]
2021 Novosibirsk Oral Olympiad in Geometry, 7
A circle concentric with the inscribed circle of $ABC$ intersects the sides of the triangle at six points forming a convex hexagon $A_1A_2B_1B_2C_1C_2$ (points $C_1$ and $C_2$ on the $AB$ side, $A_1$ and $A_2$ on $BC$, $B_1$ and $B_2$ on $AC$). Prove that if line $A_1B_1$ is parallel to the bisector of angle $B$, then line $A_2C_2$ is parallel to the bisector of angle $C$.
2012 Today's Calculation Of Integral, 811
Let $a$ be real number. Evaluate $\int_a^{a+\pi} |x|\cos x\ dx.$
2024 pOMA, 5
Prove that there do not exist positive integers $a,b,c$ such that the polynomial
\[
P(x) = x^3 - 2^ax^2 + 3^bx - 6^c
\]
has three integer roots.
2012 NIMO Summer Contest, 10
A [i]triangulation[/i] of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
[i]Proposed by Lewis Chen[/i]
2022 Junior Balkan Team Selection Tests - Romania, P2
Let $ABC$ be an acute scalene triangle. Let $D$ be the foot of the $A$-bisectrix and $E$ be the foot of the $A$-altitude. The perpendicular bisector of the segment $AD$ intersects the semicircles of diameter $AB$ and $AC$ which lie on the outside of triangle $ABC$ at $X$ and $Y$ respectively. Prove that the points $X,Y,D$ and $E$ lie on a circle.
1982 Brazil National Olympiad, 5
Show how to construct a line segment length $(a^4 + b^4)^{1/4}$ given segments lengths $a$ and $b$.