Found problems: 85335
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.
1993 Poland - First Round, 4
Given is a circle with center $O$, point $A$ inside the circle and a chord $PQ$ which is not a diameter and passing through $A$. The lines $p$ and $q$ are tangent to the given circle at $P$ and $Q$ respectively. The line $l$ passing through $A$ and perpendicular to $OA$ intersects the lines $p$ and $q$ at $K$ and $L$ respectively. Prove that $|AK| = |AL|$.
1998 Bosnia and Herzegovina Team Selection Test, 1
Let $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$ be five different points which are inside $D$ or on the border of figure $D$. Let $M=min\left\{P_iP_j \mid i \neq j\right\}$ be minimal distance between different points $P_i$. For which configuration of points $P_i$, value $M$ is at maximum, if :
$a)$ $D$ is unit square
$b)$ $D$ is equilateral triangle with side equal $1$
$c)$ $D$ is unit circle, circle with radius $1$
2019 Denmark MO - Mohr Contest, 2
Two distinct numbers a and b satisfy that the two equations $x^{2019} + ax + 2b = 0$ and $x^{2019}+ bx + 2a = 0$ have a common solution. Determine all possible values of $a + b$.
2008 IMO Shortlist, 5
Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that
\[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\]
[i]Proposed by Pavel Novotný, Slovakia[/i]
2017 Danube Mathematical Olympiad, 4
Determine all triples of positive integers $(x,y,z)$ such that $x^4+y^4 =2z^2$ and $x$ and $y$ are relatively prime.
2014 Contests, 3
Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$.
Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.