Found problems: 85335
2006 Thailand Mathematical Olympiad, 10
Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.
2014 Iran Team Selection Test, 6
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC$ at $D$.
let $X$ is a point on arc $BC$ from circumcircle of triangle $ABC$ such that if $E,F$ are feet of perpendicular from $X$ on $BI,CI$ and $M$ is midpoint of $EF$ we have $MB=MC$.
prove that $\widehat{BAD}=\widehat{CAX}$
2014 ASDAN Math Tournament, 1
Alex gets $8$ points on an exam, while his friend gets $3$ times as many points as Alex. What is the average of their scores?
1986 AMC 8, 13
[asy]draw((0,0)--(0,6)--(8,6)--(8,3)--(4,3)--(4,0)--cycle);
label("6",(0,3),W);
label("8",(4,6),N);[/asy]
Given that all angles shown are marked, the perimeter of the polygon shown is
\[ \textbf{(A)}\ 14 \qquad
\textbf{(B)}\ 20 \qquad
\textbf{(C)}\ 28 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ \text{cannot be determined from the information given} \qquad
\]
2022 Moldova EGMO TST, 3
Find the smallest nonnegative integer $n$ such that in every set of $n$ numbers there are always two distinct numbers such that their sum or difference is divisible by $2022$.
2024 Azerbaijan National Mathematical Olympiad, 2
Find all the real number triples $(x, y, z)$ satisfying the following system of inequalities under the condition $0 < x, y, z < \sqrt{2}$:
$$y\sqrt{4-x^2y^2}\ge \frac{2}{\sqrt{xz}}$$
$$x\sqrt{4-x^2z^2}\ge \frac{2}{\sqrt{yz}}$$
$$z\sqrt{4-y^2z^2}\ge \frac{2}{\sqrt{xy}}$$.
2024 Korea Junior Math Olympiad (First Round), 11.
There is a square $ ABCD. $
$ P $ is on $\bar{AB}$ , and $Q$ is on $ \bar{AD} $
They follow $ \bar{AP}=\bar{AQ}=\frac{\bar{AB}}{5} $
Let $ H $ be the foot of the perpendicular point from $ A $ to $ \bar{PD} $
If $ |\triangle APH|=20 $, Find the area of $ \triangle HCQ $.
2003 Romania National Olympiad, 1
[b]a)[/b] Determine the center of the ring of square matrices of a certain dimensions with elements in a given field, and prove that it is isomorphic with the given field.
[b]b)[/b] Prove that
$$ \left(\mathcal{M}_n\left( \mathbb{R} \right) ,+, \cdot\right)\not\cong \left(\mathcal{M}_n\left( \mathbb{C} \right) ,+,\cdot\right) , $$
for any natural number $ n\ge 2. $
[i]Marian Andronache, Ion Sava[/i]
1990 All Soviet Union Mathematical Olympiad, 531
For which positive integers $n$ is $3^{2n+1} - 2^{2n+1} - 6^n$ composite?
2012 Online Math Open Problems, 1
The average of two positive real numbers is equal to their difference. What is the ratio of the larger number to the smaller one?
[i]Author: Ray Li[/i]
Kyiv City MO Juniors 2003+ geometry, 2013.8.5
Let $ABCD$ be a convex quadrilateral. Prove that the circles inscribed in the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a common point if and only if $ABCD$ is a rhombus.
2017 South East Mathematical Olympiad, 8
Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$
Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, x_3, y_1, y_2, y_3$ such that $x_1 < x_2 < x_3, y_1 < y_2 < y_3$ and
$$(x_1, y_2), (x_2, y_1), (x_2, y_2), (x_2, y_3), (x_3, y_2) \in A.$$
Determine the largest possible number of elements in $A$.
1969 IMO Shortlist, 42
$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.
2017 Morocco TST-, 2
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
2019 Nordic, 4
Let $n$ be an integer with $n\geq 3$ and assume that $2n$ vertices of a regular $(4n + 1)-$gon are coloured. Show that there must exist three of the coloured vertices forming an isosceles triangle.
2020 IMO Shortlist, G1
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
2010 LMT, 34
A [i]prime power[/i] is an integer of the form $p^k,$ where $p$ is a prime and $k$ is a nonnegative integer. How many prime powers are there less than or equal to $10^6?$ Your score will be $16-80|\frac{\textbf{Your Answer}}{\textbf{Actual Answer}}-1|$ rounded to the nearest integer or $0,$ whichever is higher.
Indonesia Regional MO OSP SMA - geometry, 2006.1
Suppose triangle $ABC$ is right-angled at $B$. The altitude from $B$ intersects the side $AC$ at point $D$. If points $E$ and $F$ are the midpoints of $BD$ and $CD$, prove that $AE \perp BF$.
1993 Tournament Of Towns, (378) 7
In a handbook of plants each plant is characterized by $100$ attributes (each attribute may either be present in a plant or not). Two plants are called [i]dissimilar [/i] if they differ by no less than $51$ attributes.
(a) Prove that the handbook cannot describe more than $50$ pair-wise dissimilar plants.
(b) Can it describe $50$ pairwise dissimilar plants?
(Dima Tereshin)
MOAA Accuracy Rounds, 2021.4
Compute the number of two-digit numbers $\overline{ab}$ with nonzero digits $a$ and $b$ such that $a$ and $b$ are both factors of $\overline{ab}$.
[i]Proposed by Nathan Xiong[/i]
2011 Tuymaada Olympiad, 4
Prove that, among $100000$ consecutive $100$-digit positive integers, there is an integer $n$ such that the length of the period of the decimal expansion of $\frac1n$ is greater than $2011$.
1988 China Team Selection Test, 1
Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.
2000 All-Russian Olympiad Regional Round, 9.6
Among $2000$ outwardly indistinguishable balls, wines - aluminum weighing 1$0$ g, and the rest - duralumin weighing $9.9$ g. It is required to select two piles of balls so that the masses of the piles are different, and the number of balls in them - the same. What is the smallest number of weighings on a cup scale without weights that can be done?
2023 Lusophon Mathematical Olympiad, 5
Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\angle XAY$.
2011 International Zhautykov Olympiad, 2
Let $n$ be integer, $n>1.$ An element of the set $M=\{ 1,2,3,\ldots,n^2-1\}$ is called [i]good[/i] if there exists some element $b$ of $M$ such that $ab-b$ is divisible by $n^2.$ Furthermore, an element $a$ is called [i]very good[/i] if $a^2-a$ is divisible by $n^2.$ Let $g$ denote the number of [i]good[/i] elements in $M$ and $v$ denote the number of [i]very good[/i] elements in $M.$ Prove that
\[v^2+v \leq g \leq n^2-n.\]