Found problems: 85335
Ukraine Correspondence MO - geometry, 2020.11
The diagonals of the cyclic quadrilateral $ABCD$ intersect at the point $E$. Let $P$ and $Q$ are the centers of the circles circumscribed around the triangles $BCE$ and $DCE$, respectively. A straight line passing through the point $P$ parallel to $AB$, and a straight line passing through the point $Q$ parallel to $AD$, intersect at the point $R$. Prove that the point $R$ lies on segment $AC$.
2024 CMIMC Team, 6
Cyclic quadrilateral $ABCD$ has circumradius $3$. Additionally, $AC = 3\sqrt{2}$, $AB/CD = 2/3$, and $AD = BD$. Find $CD$.
[i]Proposed by Justin Hsieh[/i]
2014 Indonesia MO Shortlist, G2
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.
1962 Putnam, A5
Evaluate
$$ \sum_{k=0}^{n} \binom{n}{k}k^{2}.$$
2024 Sharygin Geometry Olympiad, 11
Let $M, N$ be the midpoints of sides $AB, AC$ respectively of a triangle $ABC$. The perpendicular bisector to the bisectrix $AL$ meets the bisectrixes of angles $B$ and $C$ at points $P$ and $Q$ respectively. Prove that the common point of lines $PM$ and $QN$ lies on the tangent to the circumcircle of $ABC$ at $A$.
1976 IMO Shortlist, 3
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
2020 Baltic Way, 13
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $\ell$ be the tangent line to $\omega$ at $A$. Let $X$ and $Y$ be the projections of $B$ onto lines $\ell$ and $AC$, respectively. Let $H$ be the orthocenter of $BXY$. Let $CH$ intersect $\ell$ at $D$. Prove that $BA$ bisects angle $CBD$.
2009 Mexico National Olympiad, 2
In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules:
$\bullet$ If $p$ is a prime number, we place it in box $1$.
$\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$.
Find all positive integers $n$ that are placed in the box labeled $n$.
2020 CMIMC Team, 4
Given $n=2020$, sort the $6$ values $$n^{n^2},\,\, 2^{2^{2^n}},\,\, n^{2^n},\,\, 2^{2^{n^2}},\,\, 2^{n^n},\,\,\text{and}\,\, 2^{n^{2^2}}$$ from [b]least[/b] to [b]greatest[/b]. Give your answer as a 6 digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.
1997 Estonia Team Selection Test, 2
A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB,BC,CD,DA$ one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to $CD,DA,AB,BC,$ respectively. Prove that the centers of these four rectangles are vertices of a rectangle.
KoMaL A Problems 2021/2022, A. 813
Let $p$ be a prime number and $k$ be a positive integer. Let \[t=\sum_{i=0}^\infty\bigg\lfloor\frac{k}{p^i}\bigg\rfloor.\]a) Let $f(x)$ be a polynomial of degree $k$ with integer coefficients such that its leading coefficient is $1$ and its constant is divisible by $p.$ prove that there exists $n\in\mathbb{N}$ for which $p\mid f(n),$ but $p^{t+1}\nmid f(n).$
b) Prove that the statement above is sharp, i.e. there exists a polynomial $g(x)$ of degree $k,$ integer coefficients, leading coefficient $1$ and constant divisible by $p$ such that if $p\mid g(n)$ is true for a certain $n\in\mathbb{N},$ then $p^t\mid g(n)$ also holds.
[i]Proposed by Kristóf Szabó, Budapest[/i]
2020-2021 OMMC, 11
In equilateral triangle $XYZ$ with side length $10$, define points $A, B$ on $XY,$ points $C, D$ on $YZ,$ and points $E, F$ on $ZX$ such that $ABDE$ and $ACEF$ are rectangles, $XA<XB,$ $YC<YD,$ and $ZE<ZF$. The area of hexagon $ABCDEF$ can be written as $\sqrt{x}$ for some positive integer $x$. Find $x$.
2025 Serbia Team Selection Test for the BMO 2025, 5
In Mexico, there live $n$ Mexicans, some of whom know each other. They decided to play a game. On the first day, each Mexican wrote a non-negative integer on their forehead. On each following day, they changed their number according to the following rule: On day $i+1$, each Mexican writes on their forehead the smallest non-negative integer that did not appear on the forehead of any of their acquaintances on day $i$. It is known that on some day every Mexican wrote the same number as on the previous day, after which they decided to stop the game. Determine the maximum number of days this game could have lasted.
[i]Proposed by Pavle Martinović[/i]
2015 Taiwan TST Round 2, 1
Let $ABC$ be a triangle with incircle $\omega$, incenter $I$ and circumcircle $\Gamma$. Let $D$ be the tangency point of $\omega$ with $BC$, let $M$ be the midpoint of $ID$, and let $A'$ be the diametral opposite of $A$ with respect to $\Gamma$. If we denote $X=A'M\cap \Gamma$, then prove that the circumcircle of triangle $AXD$ is tangent to $BC$.
2020 LMT Fall, A12
Richard comes across an infinite row of magic hats, $H_1, H_2, \dots$ each of which may contain a dollar bill with probabilities $p_1, p_2, \dots$. If Richard draws a dollar bill from $H_i$, then $p_{i+1} = p_i$, and if not, $p_{i+1}=\frac{1}{2}p_i$. If $p_1 = \frac{1}{2}$ and $E$ is the expected amount of money Richard makes from all the hats, compute $\lfloor 100E \rfloor$.
[i]Proposed by Alex Li[/i]
2011 Romanian Master of Mathematics, 3
A triangle $ABC$ is inscribed in a circle $\omega$.
A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$).
Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$.
Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$.
[i](Russia) Vasily Mokin and Fedor Ivlev[/i]
2024 Junior Macedonian Mathematical Olympiad, 2
It is known that in a group of $2024$ students each student has at least $1011$ acquaintances among the remaining members of the group. What is more, there exists a student that has at least $1012$ acquaintances in the group. Prove that for every pair of students $X, Y$, there exist students $X_0 = X, X_1, ..., X_{n - 1}, X_n = Y$ in the group such that for every index $i = 0, ..., n - 1$, the students $X_i$ and $X_{i + 1}$ are acquaintances.
[i]Proposed by Mirko Petruševski[/i]
1998 National Olympiad First Round, 18
Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum _{i\equal{}1}^{24}p_{i}^{99!} \equiv a\, \, \left(mod\, 100\right)$.
$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 99$
1998 Austrian-Polish Competition, 2
For n points \[ P_1;P_2;...;P_n \] in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points \[ P_i;P_{i+1} (i=1;2;...n-1) \] is the same color or 2 points with at least one of 2 points are colored white. How many ways acceptable color?
2021 Science ON all problems, 2
There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\
In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\
For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations.
$\textit{(Andrei Bâra)}$
Gheorghe Țițeica 2024, P2
Prove that the number $$\bigg\lfloor\frac{2024}{1}\bigg\rfloor+\bigg\lfloor\frac{2023}{2}\bigg\rfloor+\bigg\lfloor\frac{2022}{3}\bigg\rfloor+\dots +\bigg\lfloor\frac{1013}{1012}\bigg\rfloor$$ is even.
1993 Tournament Of Towns, (358) 1
Let $M$ be a point on the side $AB$ of triangle $ABC$. The length $AB = c$ and $\angle CMA=\phi$ are given. Find the distance between the orthocentres (intersection points of altitudes) of the triangles $AMC$ and $BMC$.
(IF Sharygin)
1995 Baltic Way, 13
Consider the following two person game. A number of pebbles are situated on the table. Two players make their moves alternately. A move consists of taking off the table $x$ pebbles where $x$ is the square of any positive integer. The player who is unable to make a move loses. Prove that there are infinitely many initial situations in which the second player can win no matter how his opponent plays.
2007 Stanford Mathematics Tournament, 6
$x\equiv\left(\sum_{k=1}^{2007}k\right)\mod{2016}$, where $0\le x\le 2015$. Solve for $x$.
2015 ASDAN Math Tournament, 34
Compute the number of natural numbers $1\leq n\leq10^6$ such that the least prime divisor of $n$ is $17$. Your score will be given by $\lfloor26\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor$, where $A$ is your answer and $C$ is the actual answer.