Found problems: 85335
Russian TST 2019, P1
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied:
[list=1]
[*] Each number in the table is congruent to $1$ modulo $n$.
[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.
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Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.
2016 Math Prize for Girls Problems, 11
Compute the number of ordered pairs of complex numbers $(u, v)$ such that $uv = 10$ and such that the real and imaginary parts of $u$ and $v$ are integers.
2009 All-Russian Olympiad, 2
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
2023 IFYM, Sozopol, 6
In triangle $ABC$, $\angle ABC = 54^\circ$ and $\angle ACB = 42^\circ$. Point $D$ is the foot of the altitude from vertex $A$ to $BC$, and $I$ is the incenter of $\triangle ABC$. Point $K$ lies on line $AD$, such that $D$ is between $A$ and $K$ and $AK$ is equal to the diameter of the circumcircle of $\triangle ABC$. Find the measure of $\angle KID$.
2017 NIMO Summer Contest, 5
Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten.
[i]Proposed by Evan Chen[/i]
1970 IMO Longlists, 10
In $\triangle ABC$, prove that $1< \sum_{cyc}{\cos A}\le \frac{3}{2}$.
1998 Abels Math Contest (Norwegian MO), 3
Let $n$ be a positive integer.
(a) Prove that $1^5 +3^5 +5^5 +...+(2n-1)^5$ is divisible by $n$.
(b) Prove that $1^3 +3^3 +5^3 +...+(2n-1)^3$ is divisible by $n^2$.
2021 CCA Math Bonanza, T5
We say that a [i]special word[/i] is any sequence of letters [b]containing a vowel[/b]. How many ordered triples of special words $(W_1,W_2,W_3)$ have the property that if you concatenate the three words, you obtain a rearrangement of "aadvarks"?
For example, the number of triples of special words such that the concatenation is a rearrangement of ``adaa" is $6$, and all of the possible triples are:
[center]
(da,a,a),(ad,a,a),(a,da,a),(a,ad,a),(a,a,da),(a,a,ad).
[/center]
[i]2021 CCA Math Bonanza Team Round #5[/i]
2002 China Team Selection Test, 2
There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.
2018 Caucasus Mathematical Olympiad, 3
For $2n$ positive integers a matching (i.e. dividing them into $n$ pairs) is called {\it non-square} if the product of two numbers in each pair is not a perfect square. Prove that if there is a non-square matching, then there are at least $n!$ non-square matchings.
(By $n!$ denote the product $1\cdot 2\cdot 3\cdot \ldots \cdot n$.)
Estonia Open Junior - geometry, 1998.1.3
Two non intersecting circles with centers $O_1$ and $O_2$ are tangent to line $s$ at points $A_1$ and $A_2$, respectively, and lying on the same side of this line. Line $O_1O_2$ intersects the first circle at $B_1$ and the second at $B_2$. Prove that the lines $A_1B_1$ and $A_2B_2$ are perpendicular to each other.
2005 Austrian-Polish Competition, 10
Determine all pairs $(k,n)$ of non-negative integers such that the following inequality holds $\forall x,y>0$:
\[1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.\]
1977 IMO Longlists, 53
Find all pairs of integers $a$ and $b$ for which
\[7a+14b=5a^2+5ab+5b^2\]
2011 IMO Shortlist, 5
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent.
[i]Proposed by Irena Majcen and Kris Stopar, Slovenia[/i]
2022 AMC 12/AHSME, 20
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$
1992 IMTS, 3
Show that there exists an equiangular hexagon in the plane, whose sides measure 5,8,11,14,23, and 29 units in some order.
2017 Azerbaijan Team Selection Test, 2
Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that
[list]
[*]$m = 1$ and $l = 2k$; or
[*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$.
[/list]
2021 MIG, 13
In a restaurant, a meal consists of one sandwich and one optional drink. In other words, a sandwich is necessary for a meal but a drink is not necessary. There are two types of sandwiches and two types of drinks. How many possible meals can be purchased?
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }16$
2011 239 Open Mathematical Olympiad, 5
Prove that there exist 1000 consecutive numbers such that none of them is divisible by its sum of the digits
May Olympiad L2 - geometry, 1995.4
Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?
2008 iTest Tournament of Champions, 1
Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers such that \[x^2-y^2=n.\]
2020 Regional Olympiad of Mexico Southeast, 5
Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.
2017 ASDAN Math Tournament, 8
Let $S=\{1,2,3,4,5,6\}$. Compute the number of functions $f:S\rightarrow S$ such that $f(f(f(s)))=2$ if $s$ is odd and $f(f(f(s)))=1$ if $s$ is even.
1994 AMC 12/AHSME, 10
For distinct real numbers $x$ and $y$, let $M(x,y)$ be the larger of $x$ and $y$ and let $m(x,y)$ be the smaller of $x$ and $y$. If $a<b<c<d<e$, then
\[ M(M(a,m(b,c)),m(d,m(a,e)))= \]
$ \textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ d \qquad\textbf{(E)}\ e $
2023 Macedonian Mathematical Olympiad, Problem 4
Let $ABC$ be a scalene acute triangle with orthocenter $H$. The circle with center $A$ and radius $AH$ meets the circumcircle of $BHC$ at $T_{a} \neq H$. Define $T_{b}$ and $T_{c}$ similarly. Show that $H$ lies on the circumcircle of $T_{a}T_{b}T_{c}$.
[i]Authored by Nikola Velov[/i]