Found problems: 85335
2013 Stanford Mathematics Tournament, 13
$\mathbb{R}^2$-tic-tac-toe is a game where two players take turns putting red and blue points anywhere on the $xy$ plane. The red player moves first. The first player to get $3$ of their points in a line without any of their opponent's points in between wins. What is the least number of moves in which Red can guarantee a win? (We count each time that Red places a point as a move, including when Red places its winning point.)
2018 ITAMO, 3
Let $x_1,x_2, ... , x_n$ be positive integers,Asumme that in their decimal representations no $x_i$ "prolongs" $x_j$.For instance , $123$ prolongs $12$ , $459$ prolongs $4$ , but $124$ does not prolog $123$.
Prove that :
$\frac {1}{x_1}+\frac {1}{x_2}+...+\frac {1}{x_n} < 3$.
MathLinks Contest 3rd, 1
Let $a, b, c$ be positive reals. Prove that $$\sqrt{abc}(\sqrt{a} +\sqrt{b} +\sqrt{c}) + (a + b + c)^2 \ge 4 \sqrt{3abc(a + b + c)}.$$
2020 Bangladesh Mathematical Olympiad National, Problem 10
Let $ABCD$ be a convex quadrilateral. $O$ is the intersection of $AC$ and $BD$. $AO=3$ ,$BO=4$, $CO=5$, $DO=6$. $X$ and $Y$ are points in segment $AB$ and $CD$ respectively, such that $X,O,Y$ are collinear. The minimum of $\frac{XB}{XA}+\frac{YC}{YD}$ can be written as $\frac{a\sqrt{c}}{b}$ , where $\frac{a}{b}$ is in lowest term and $c$ is not divisible by any square number greater then $1$. What is the value of $10a+b+c$?
1987 IMO, 3
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.
2016 Iran Team Selection Test, 5
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
2001 Austrian-Polish Competition, 8
The prism with the regular octagonal base and with all edges of the length equal to $1$ is given. The points $M_{1},M_{2},\cdots,M_{10}$ are the midpoints of all the faces of the prism. For the point $P$ from the inside of the prism denote by $P_{i}$ the intersection point (not equal to $M_{i}$) of the line $M_{i}P$ with the surface of the prism. Assume that the point $P$ is so chosen that all associated with $P$ points $P_{i}$ do not belong to any edge of the prism and on each face lies exactly one point $P_{i}$. Prove that \[\sum_{i=1}^{10}\frac{M_{i}P}{M_{i}P_{i}}=5\]
2006 USA Team Selection Test, 4
Let $n$ be a positive integer. Find, with proof, the least positive integer $d_{n}$ which cannot be expressed in the form \[\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},\]
where $a_{i}$ and $b_{i}$ are nonnegative integers for each $i.$
2015 Math Prize for Girls Problems, 3
What is the area of the region bounded by the graphs of $y = |x + 2| - |x - 2|$ and $y = |x + 1| - |x - 3|$?
1995 China Team Selection Test, 1
Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.
1998 Portugal MO, 6
Let $a_0$ be a positive real number and consider the general term sequence $a_n$ defined by $$a_n =a_{n-1} + \frac{1}{a_{n-1}} \,\,\, n=1,2,3,...$$ Prove that $a_{1998} > 63$.
2000 Harvard-MIT Mathematics Tournament, 7
Assume that $a,b,c,d$ are positive integers, and $\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{3}{4}$, $\sqrt{a^2+c^2}-\sqrt{b^2+d^2}=15$. Find $ac+bd-ad-bc$.
2014 Harvard-MIT Mathematics Tournament, 5
[5] If four fair six-sided dice are rolled, what is the probability that the lowest number appearing on any die is exactly $3$?
Russian TST 2018, P1
Let $I{}$ be the incircle of the triangle $ABC$. Let $A_1, B_1$ and $C_1$ be the midpoints of the sides $BC, CA$ and $AB$ respectively. The point $X{}$ is symmetric to $I{}$ with respect to $A_1$. The line $\ell$ parallel to $BC$ and passing through $X{}$ intersects the lines $A_1B_1$ and $A_1C_1$ at $M{}$ and $N{}$ respectively. Prove that one of the excenters of the triangle $ABC$ lies on the $A_1$-excircle of the triangle $A_1MN$.
LMT Theme Rounds, 2023F 3A
A rectangular tea bag $PART$ has a logo in its interior at the point $Y$ . The distances from $Y$ to $PT$ and $PA$ are $12$ and $9$ respectively, and triangles $\triangle PYT$ and $\triangle AYR$ have areas $84$ and $42$ respectively. Find the perimeter of pentagon $PARTY$.
[i]Proposed by Muztaba Syed[/i]
[hide=Solution]
[i]Solution[/i]. $\boxed{78}$
Using the area and the height in $\triangle PYT$, we see that $PT = 14$, and thus $AR = 14$, meaning the height from $Y$ to $AR$ is $6$. This means $PA = TR = 18$. By the Pythagorean Theorem $PY=\sqrt{12^2+9^2} = 15$ and $YT =\sqrt{12^2 +5^2} = 13$. Combining all of these gives us an answer of $18+14+18+13+15 = \boxed{78}$.
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2016 BMT Spring, 12
What is the number of nondecreasing positive integer sequences of length $7$ whose last term is at most $9$?
2007 Polish MO Finals, 6
6. Sequence $a_{0}, a_{1}, a_{2},...$ is determined by $a_{0}=-1$ and
$a_{n}+\frac{a_{n-1}}{2}+\frac{a_{n-2}}{3}+...+\frac{a_{1}}{n}+\frac{a_{0}}{n+1}=0$ for $n\geq 1$
Prove that $a_{n}>0$ for $n\geq 1$
1992 Tournament Of Towns, (335) 3
The numbers $$\frac{1}{i+j-1} \,\,\,\,\,\,\, (i = 1,2,...,n; j = 1,2,...,n)$$ are written in an $n$ by $n$ table: the number $1/(i + j - 1)$ stands at the intersection of the $i$-th row and $j$-th column. Chose any $n$ squares of the table so that no two of them stand in the same row and no two of them stand in the same column. Prove that the sum of the numbers in these $n$ squares is not less than $1$.
(Sergey Ivanov, St Petersburg)
2006 IMO, 1
Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that $AP \geq AI$, and that equality holds if and only if $P=I$.
2009 Purple Comet Problems, 2
Let $p_1 = 2, p_2 = 3, p_3 = 5 ...$ be the sequence of prime numbers. Find the least positive even integer $n$ so that $p_1 + p_2 + p_3 + ... + p_n$ is not prime.
2023 HMNT, 17
Let $ABC$ be an equilateral triangle of side length $15.$ Let $A_b$ and $B_a$ be points on side $AB,$ $A_c$ and $C_a$ be points on $AC,$ and $B_c$ and $C_b$ be points on $BC$ such that $\triangle{AA_bA_c}, \triangle{BB_cB_a},$ and $\triangle{CC_aC_b}$ are equilateral triangles with side lengths $3,4,$ and $5,$ respectively. Compute the radius of the circle tangent to segments $\overline{A_bA_c}, \overline{B_aB_c},$ and $\overline{C_aC_b}.$
2016 Postal Coaching, 5
A real polynomial of odd degree has all positive coefficients. Prove that there is a (possibly trivial) permutation of the coefficients such that the resulting polynomial has exactly one real zero.
2003 Estonia National Olympiad, 5
Is it possible to cover an $n \times n$ chessboard which has its center square cut out with tiles shown in the picture (each tile covers exactly $4$ squares, tiles can be rotated and turned around) if
a) $n = 5$,
b) $n = 2003$?
[img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]
2008 Sharygin Geometry Olympiad, 18
(A.Abdullayev, 9--11) Prove that the triangle having sides $ a$, $ b$, $ c$ and area $ S$ satisfies the inequality
\[ a^2\plus{}b^2\plus{}c^2\minus{}\frac12(|a\minus{}b|\plus{}|b\minus{}c|\plus{}|c\minus{}a|)^2\geq 4\sqrt3 S.\]
2018 Sharygin Geometry Olympiad, 6
Let $ABCD$ be a circumscribed quadrilateral. Prove that the common point of the diagonals, the incenter of triangle $ABC$ and the centre of excircle of triangle $CDA$ touching the side $AC$ are collinear.