Found problems: 85335
2020 Latvia Baltic Way TST, 14
Prove that there are infinitely many different triangles in coordinate plane satisfying:
1) their vertices are lattice points
2) their side lengths are consecutive integers
[b]Remark[/b]: Triangles that can be obtained by rotation or translation or shifting the coordinate system are considered as equal triangles
2021 Princeton University Math Competition, A7
Consider the following expression
$$S = \log_2 \left( \sum^{2019}_{k=1}\sum^{2020}_{j=2}\log_{2^{1/k}} (j) \log_{j^2} \left(\sin \frac{\pi k}{2020}\right) \right).$$
Find the smallest integer $n$ which is bigger than $S$ (i.e. find $\lceil S \rceil$).
2014 Baltic Way, 13
Let $ABCD$ be a square inscribed in a circle $\omega$ and let $P$ be a point on the shorter arc $AB$ of $\omega$. Let $CP\cap BD = R$ and $DP \cap AC = S.$
Show that triangles $ARB$ and $DSR$ have equal areas.
2022 Junior Balkan Team Selection Tests - Romania, P1
Let $M,N$ and $P$ be the midpoints of sides $BC,CA$ and $AB$ respectively, of the acute triangle $ABC.$ Let $A',B'$ and $C'$ be the antipodes of $A,B$ and $C$ in the circumcircle of triangle $ABC.$ On the open segments $MA',NB'$ and $PC'$ we consider points $X,Y$ and $Z$ respectively such that \[\frac{MX}{XA'}=\frac{NY}{YB'}=\frac{PZ}{ZC'}.\][list=a]
[*]Prove that the lines $AX,BY,$ and $CZ$ are concurrent at some point $S.$
[*]Prove that $OS<OG$ where $O$ is the circumcenter and $G$ is the centroid of triangle $ABC.$
[/list]
2023 Romania Team Selection Test, P5
Let $ABCDEF$ be a convex hexagon. The diagonals $AC$ and $BD$ cross at $P,$ the diagonals $AE{}$ and $DF$ cross at $Q,$ and the line $PQ$ crosses the sides $BC$ and $EF$ at $X$ and $Y,{}$ respectively. Prove that the length of the segment $XY$ does not exceed the sum of the lengths of one of the diagonals through $P{}$ and one of the diagonals through $Q{}$.
[i]The Problem Selection Committee[/i]
1958 Miklós Schweitzer, 3
[b]3.[/b] Let $n$ be a positive integer having at least one prime factor with expoente $\geq 2$. Show that $n$ has as many factorizations into an odd number of factors as into an even number of factors. (Factorizations into the same factors arranged in different order are considered different.)[b](N. 10)[/b]
2003 Belarusian National Olympiad, 8
Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$.
(I.Voronovich)
2003 Iran MO (3rd Round), 8
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
2010 Contests, 2
A polynomial $f$ with integer coefficients is written on the blackboard. The teacher is a mathematician who has $3$ kids: Andrew, Beth and Charles. Andrew, who is $7$, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains:
[list]$f(7) = 77$
$f(b) = 85$, where $b$ is Beth's age,
$f(c) = 0$, where $c$ is Charles' age.[/list]
How old is each child?
2012 China Western Mathematical Olympiad, 1
$O$ is the circumcenter of acute $\Delta ABC$, $H$ is the Orthocenter. $AD \bot BC$, $EF$ is the perpendicular bisector of $AO$,$D,E$ on the $BC$. Prove that the circumcircle of $\Delta ADE$ through the midpoint of $OH$.
2020 Saint Petersburg Mathematical Olympiad, 3.
$BB_1$ is the angle bisector of $\triangle ABC$, and $I$ is its incenter. The perpendicular bisector of segment $AC$ intersects the circumcircle of $\triangle AIC$ at $D$ and $E$. Point $F$ is on the segment $B_1C$ such that $AB_1=CF$.Prove that the four points $B, D, E$ and $F$ are concyclic.
1979 Swedish Mathematical Competition, 1
Solve the equations:
\[\left\{ \begin{array}{l}
x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n \\
2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1 \\
3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2 \\
\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot \\
(n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2 \\
n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1
\end{array} \right.
\]
2014 Balkan MO Shortlist, A3
$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$
2006 China Team Selection Test, 2
Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$.
Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.
LMT Guts Rounds, 2022 F
[u]Round 6 [/u]
[b]p16.[/b] Let $a$ be a solution to $x^3 -x +1 = 0$. Find $a^6 -a^2 +2a$.
[b]p17.[/b] For a positive integer $n$, $\phi (n)$ is the number of positive integers less than $n$ that are relatively prime to $n$. Compute the sum of all $n$ for which $\phi (n) = 24$.
[b]p18.[/b] Let $x$ be a positive integer such that $x^2 \equiv 57$ (mod $59$). Find the least possible value of $x$.
[u]Round 7[/u]
[b]p19.[/b] In the diagram below, find the number of ways to color each vertex red, green, yellow or blue such that no two vertices of a triangle have the same color.
[img]https://cdn.artofproblemsolving.com/attachments/1/e/01418af242c7e2c095a53dd23e997b8d1f3686.png[/img]
[b]p20.[/b] In a set with $n$ elements, the sum of the number of ways to choose $3$ or $4$ elements is a multiple of the sumof the number of ways to choose $1$ or $2$ elements. Find the number of possible values of $n$ between $4$ and $120$ inclusive.
[b]p21.[/b] In unit square $ABCD$, let $\Gamma$ be the locus of points $P$ in the interior of $ABCD$ such that $2AP < BP$. The area of $\Gamma$ can be written as $\frac{a\pi +b\sqrt{c}}{d}$ for integers $a,b,c,d$ with $c$ squarefree and $gcd(a,b,d) = 1$. Find $1000000a +10000b +100c +d$.
[u]Round 8 [/u]
[b]p22.[/b] Ephram, GammaZero, and Orz walk into a bar. Each write some permutation of the letters “LMT” once, then concatenate their permutations one after the other (i.e. LTMTLMTLM would be a possible string, but not LLLMMMTTT). Suppose that the probability that the string “LMT” appears in that order among the new $9$-character string can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$.
[b]p23.[/b] In $\vartriangle ABC$ with side lengths $AB = 27$, $BC = 35$, and $C A = 32$, let $D$ be the point at which the incircle is tangent to $BC$. The value of $\frac{\sin \angle C AD }{\sin\angle B AD}$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$.
[b]p24.[/b] Let $A$ be the greatest possible area of a square contained in a regular hexagon with side length $1$. Let B be the least possible area of a square that contains a regular hexagon with side length $1$. The value of $B-A$ can be expressed as $a\sqrt{b}-c$ for positive integers $a$, $b$, and $c$ with $b$ squarefree. Find $10000a +100b +c$.
[u]Round 9[/u]
[b]p25.[/b] Estimate how many days before today this problem was written. If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{2} \right| \right \rfloor , 0 \right)$ points.
[b]p26.[/b] Circle $\omega_1$ is inscribed in unit square $ABCD$. For every integer $1 < n \le 10,000$, $\omega_n$ is defined as the largest circle which can be drawn inside $ABCD$ that does not overlap the interior of any of $\omega_1$,$\omega_2$, $...$,$\omega_{n-1}$ (If there are multiple such $\omega_n$ that can be drawn, one is chosen at random). Let r be the radius of ω10,000. Estimate $\frac{1}{r}$ . If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{200} \right| \right \rfloor , 0 \right)$ points.
[b]p27.[/b] Answer with a positive integer less than or equal to $20$. We will compare your response with the response of every other team that answered this problem. When two equal responses are compared, neither team wins. When two unequal responses $A > B$ are compared, $A$ wins if $B | A$, and $B$ wins otherwise. If your team wins n times, you will receive $\left \lfloor \frac{n}{2} \right \rfloor$ points.
PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167135p28823324]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2006 ISI B.Math Entrance Exam, 1
Bishops on a chessboard move along the diagonals ( that is , on lines parallel to the two main diagonals ) . Prove that the maximum number of non-attacking bishops on an $n*n$ chessboard is $2n-2$. (Two bishops are said to be attacking if they are on a common diagonal).
2018 Polish Junior MO First Round, 6
Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$. Show that $m \le n$.
2015 Princeton University Math Competition, A8
Let $P(x)$ be a polynomial with positive integer coefficients and degree 2015. Given that there exists some $\omega \in \mathbb{C}$ satisfying
$$\omega^{73} = 1\quad \text{and}$$
$$P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0,$$
what is the minimum possible value of $P(1)$?
1953 Moscow Mathematical Olympiad, 246
a) On a plane, $11$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?
b) On a plane, $n$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?
2001 Tuymaada Olympiad, 8
Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h.
2023 Assam Mathematics Olympiad, 6
What is the remainder when $128^{2023}$ is divided by $126$?
2024 CMIMC Combinatorics and Computer Science, 2
Robert has two stacks of five cards numbered 1--5, one of which is randomly shuffled while the other is in numerical order. They pick one of the stacks at random and turn over the first three cards, seeing that they are 1, 2, and 3 respectively. What is the probability the next card is a 4?
[i]Proposed by Connor Gordon[/i]
1990 IMO Longlists, 56
For positive integers $n, p$ with $n \geq p$, define real number $K_{n, p}$ as follows:
$K_{n, 0} = \frac{1}{n+1}$ and $K_{n, p} = K_{n-1, p-1} -K_{n, p-1}$ for $1 \leq p \leq n.$
(i) Define $S_n = \sum_{p=0}^n K_{n,p} , \ n = 0, 1, 2, \ldots$ . Find $\lim_{n \to \infty} S_n.$
(ii) Find $T_n = \sum_{p=0}^n (-1)^p K_{n,p} , \ n = 0, 1, 2, \ldots$.
2002 AMC 10, -1
This test and the matching AMC 12P were developed for the use of a group of Taiwan schools, in early January of 2002. When Taiwan had taken the contests, the AMC released the questions here as a set of practice questions for the 2002 AMC 10 and AMC 12 contests.
2004 Gheorghe Vranceanu, 4
Given a natural prime $ p, $ find the number of integer solutions of the equation $ p+xy=p(x+y). $