Found problems: 85335
2020/2021 Tournament of Towns, P4
There is a row of $100N$ sandwiches with ham. A boy and his cat play a game. In one action the boy eats the first sandwich from any end of the row. In one action the cat either eats the ham from one sandwich or does nothing. The boy performs 100 actions in each of his turns, and the cat makes only 1 action each turn; the boy starts first. The boy wins if the last sandwich he eats contains ham. Is it true that he can win for any positive integer $N{}$ no matter how the cat plays?
[i]Ivan Mitrofanov[/i]
2003 Germany Team Selection Test, 2
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.
1998 Harvard-MIT Mathematics Tournament, 6
In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?
[asy]
size(340);
int i, j;
for(i = 0; i<10; i = i+1) {
for(j = 0; j<5; j = j+1) {
if(10*j + i == 11 || 10*j + i == 12 || 10*j + i == 14 || 10*j + i == 15 || 10*j + i == 18 || 10*j + i == 32 || 10*j + i == 35 || 10*j + i == 38 ) { }
else{ label("$*$", (i,j));}
}}
label("$\leftarrow$"+"Dec. 31", (10.3,0));
label("Jan. 1"+"$\rightarrow$", (-1.3,4));[/asy]
2012 Gulf Math Olympiad, 1
Let $X,\ Y$ and $Z$ be the midpoints of sides $BC,\ CA$, and $AB$ of the triangle $ABC$, respectively. Let $P$ be a point inside the triangle. Prove that the quadrilaterals $AZPY,\ BXPZ$, and $CYPX$ have equal areas if, and only if, $P$ is the centroid of $ABC$.
2019 Jozsef Wildt International Math Competition, W. 69
Denote $\overline{w_a}, \overline{w_b}, \overline{w_c}$ the external angle-bisectors in triangle $ABC$, prove that $$\sum \limits_{cyc} \frac{1}{w_a}\leq \sqrt{\frac{(s^2 - r^2 - 4Rr)(8R^2 - s^2 - r^2 - 2Rr)}{8s^2R^2r}}$$
2021 Brazil Undergrad MO, Problem 4
For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$.
Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$
2010 Tournament Of Towns, 1
There are $100$ points on the plane. All $4950$ pairwise distances between two points have been recorded.
$(a)$ A single record has been erased. Is it always possible to restore it using the remaining records?
$(b)$ Suppose no three points are on a line, and $k$ records were erased. What is the maximum value of $k$ such that restoration of all the erased records is always possible?
2024 Canadian Mathematical Olympiad Qualification, 7a
In triangle $ABC$, let $I$ be the incentre. Let $H$ be the orthocentre of triangle $BIC$. Show that $AH$ is parallel to $BC$ if and only if $H$ lies on the circle with diameter $AI$.
2012 Dutch BxMO/EGMO TST, 4
Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$.
Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.
1993 Cono Sur Olympiad, 2
Prove that there exists a succession $a_1, a_2, ... , a_k, ...$, where each $a_i$ is a digit ($a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)$ ) and $a_0=6$, such that, for each positive integrer $n$, the number $x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1}$ verify that $x_n^2-x_n$ is divisible by $10^n$.
2007 Junior Balkan Team Selection Tests - Romania, 4
Find all integer positive numbers $n$ such that:
$n=[a,b]+[b,c]+[c,a]$, where $a,b,c$ are integer positive numbers and $[p,q]$ represents the least common multiple of numbers $p,q$.
2014 Flanders Math Olympiad, 1
(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals.
(b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that
$$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$$
2008 ITest, 43
Alexis notices Joshua working with Dr. Lisi and decides to join in on the fun. Dr. Lisi challenges her to compute the sum of all $2008$ terms in the sequence. Alexis thinks about the problem and remembers a story one of her teahcers at school taught her about how a young Karl Gauss quickly computed the sum \[1+2+3+\cdots+98+99+100\] in elementary school. Using Gauss's method, Alexis correctly finds the sum of the $2008$ terms in Dr. Lisi's sequence. What is this sum?
2024 Bulgaria MO Regional Round, 12.2
Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$ for all positive integers $n>N$. Show that there exists a constant $c>0$, such that $x_n \leq \frac{n} {2}+c$ for all positive integers $n$.
2006 ISI B.Math Entrance Exam, 2
Prove that there is no non-constant polynomial $P(x)$ with integer coefficients such that $P(n)$ is a prime number for all positive integers $n$.
2011 Saudi Arabia IMO TST, 3
In acute triangle $ABC$, $\angle A = 20^o$. Prove that the triangle is isosceles if and only if $$\sqrt[3]{a^3 + b^3 + c^3 -3abc} = \min\{b, c\}$$, where $a,b, c$ are the side lengths of triangle $ABC$.
2017 Bulgaria EGMO TST, 3
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
1985 National High School Mathematics League, 1
In rectangular coordinate system $xOy, A(x_1,y_1), B(x_2,y_2)$, where $x_1,y_1,x_2,y_2$ are 1-digit-numbers. Intersection angle between $OA$ and $x$-axis positive direction is larger than $\frac{\pi}{4}$, intersection angle between $OB$ and $x$-axis positive direction is smaller than $\frac{\pi}{4}$. Projection of $A$ on $y$-axis is $A'$, projection of $B$ on $x$-axis is $B'$. Area of $\triangle OBB'$ is $33.5$ larger than $\triangle OAA'$. Find all 4-digit-number $\overline{x_1x_2y_1y_2}$.
Kyiv City MO Juniors 2003+ geometry, 2016.9.5
On the sides $BC$ and $AB$ of the triangle $ABC$ the points ${{A} _ {1}}$ and ${{C} _ {1}} $ are selected accordingly so that the segments $A {{A} _ {1}}$ and $C {{C} _ {1}}$ are equal and perpendicular. Prove that if $\angle ABC = 45 {} ^ \circ$, then $AC = A {{A} _ {1}} $.
(Gogolev Andrew)
2016 Online Math Open Problems, 23
Let $\mathbb N$ denote the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function such that the following conditions hold:
(a) For any $n\in \mathbb N$, we have $f(n) | n^{2016}$.
(b) For any $a,b,c\in \mathbb N$ satisfying $a^2+b^2=c^2$, we have $f(a)f(b)=f(c)$.
Over all possible functions $f$, determine the number of distinct values that can be achieved by $f(2014)+f(2)-f(2016)$.
[i]Proposed by Vincent Huang[/i]
2021 Iranian Geometry Olympiad, 1
Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point
$E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that
$BE \perp HD$.
[i]Proposed by Tran Quang Hung - Vietnam[/i]
2024 Belarusian National Olympiad, 8.6
For each number $x$ we denote by $S(x)$ the sum of digits from its decimal representation. Find all positive integers $m$ for each of which there exists a positive integer $n$, such that $$S(n^2-2n+10)=m$$
[i]Chernov S.[/i]
2004 Italy TST, 2
Let $\mathcal{P}_0=A_0A_1\ldots A_{n-1}$ be a convex polygon such that $A_iA_{i+1}=2^{[i/2]}$ for $i=0, 1,\ldots ,n-1$ (where $A_n=A_0$). Define the sequence of polygons $\mathcal{P}_k=A_0^kA_1^k\ldots A_{n-1}^k$ as follows: $A_i^1$ is symmetric to $A_i$ with respect to $A_0$, $A_i^2$ is symmetric to $A_i^1$ with respect to $A_1^1$, $A_i^3$ is symmetric to $A_i^2$ with respect to $A_2^2$ and so on. Find the values of $n$ for which infinitely many polygons $\mathcal{P}_k$ coincide with $\mathcal{P}_0$.
2003 USAMO, 1
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
1992 Tournament Of Towns, (356) 5
The bisector of the angle $A$ of triangle $ABC$ intersects its circumscribed circle at the point $D$. Suppose $P$ is the point symmetric to the incentre of the triangle with respect to the midpoint of the side $BC$, and $M$ is the second intersection point of the line $PD$ with the circumscribed circle. Prove that one of the distances $AM$, $BM$, $CM$ is equal to the sum of two other distances.
(VO Gordon)