Found problems: 85335
1981 IMO Shortlist, 13
Let $P$ be a polynomial of degree $n$ satisfying
\[P(k) = \binom{n+1}{k}^{-1} \qquad \text{ for } k = 0, 1, . . ., n.\]
Determine $P(n + 1).$
2002 Spain Mathematical Olympiad, Problem 4
Denote $n$ as a natural number, and $m$ as the result of writing the digits of $n$ in reverse order. Determine, if they exist, the numbers of three digits which satisfy $2m + S = n$, $S$ being the sum of the digits of $n$.
2023 Malaysian IMO Training Camp, 4
Do there exist infinitely many triples of positive integers $(a, b, c)$ such that $a$, $b$, $c$ are pairwise coprime, and $a! + b! + c!$ is divisible by $a^2 + b^2 + c^2$?
[i]Proposed by Anzo Teh Zhao Yang[/i]
2020 HK IMO Preliminary Selection Contest, 2
Let $x$, $y$, $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$. Find the greatest possible value of $x+y+z$.
2003 Serbia Team Selection Test, 2
Let M and N be the distinct points in the plane of the triangle ABC such that AM : BM : CM = AN : BN : CN. Prove that the line MN contains the circumcenter of â–³ABC.
2002 District Olympiad, 2
A group of $67$ students pass their examination consisting of $6$ questions, labeled with the numbers $1$ to $6$. A correct answer to question $n$ is quoted $n$ points and for an incorrect answer to the same question a student loses $n$ point.
a) Find the least possible positive difference between any $2$ final scores
b) Show that at least $4$ participants have the same final score
c) Show that at least $2$ students gave identical answer to all six questions.
1951 AMC 12/AHSME, 12
At $ 2: 15$ o'clock, the hour and minute hands of a clock form an angle of:
$ \textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 5^{\circ} \qquad\textbf{(C)}\ 22\frac {1}{2}^{\circ} \qquad\textbf{(D)}\ 7\frac {1}{2} ^{\circ} \qquad\textbf{(E)}\ 28^{\circ}$
2010 Contests, 2
A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.
2020 Purple Comet Problems, 16
Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$.
PEN D Problems, 7
Somebody incorrectly remembered Fermat's little theorem as saying that the congruence $a^{n+1} \equiv a \; \pmod{n}$ holds for all $a$ if $n$ is prime. Describe the set of integers $n$ for which this property is in fact true.
1987 IMO Longlists, 72
Is it possible to cover a rectangle of dimensions $m \times n$ with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter $\text L$) if:
[b](a)[/b] $m \times n = 1985 \times 1987;$
[b](b)[/b] $m \times n = 1987 \times 1989 \quad ?$
2016 Taiwan TST Round 3, 1
Let $n$ be a positive integer. Find the number of odd coefficients of the polynomial $(x^2-x+1)^n$.
2014 Albania Round 2, 2
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its
sides.
2009 Stars Of Mathematics, 1
Let $x_1, x_2, ... , x_n$ and $y_1, y_2, ..., y_n$ be positive real numbers so that
$$x_1 + x_2 + ...+ x_n \ge x_1y_1 + x_2y_2 + ... + x_ny_n.$$
Show that for any non-negative integer $p$ the following inequality holds
$$\frac{x_1}{y_1^p} +\frac{ x_2}{y_2^p} + ...+ \frac{x_n}{y_n^p} \ge x_1 + x_2 + ...+ x_n.$$
2018 China National Olympiad, 1
Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying
$$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$
are both integers that are relatively prime to $p$. If $A_n$ is finite, let $f(n)$ denote $|A_n|$.
a) Prove that $A_n$ is finite if and only if $n \not = 2$.
b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that
$$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$
2018 AMC 10, 10
In the rectangular parallelpiped shown, $AB = 3, BC= 1,$ and $CG = 2.$ Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$?
[asy]
size(250);
defaultpen(fontsize(10pt));
pair A =origin;
pair B = (4.75,0);
pair E1=(0,3);
pair F = (4.75,3);
pair G = (5.95,4.2);
pair C = (5.95,1.2);
pair D = (1.2,1.2);
pair H= (1.2,4.2);
pair M = ((4.75+5.95)/2,3.6);
draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H);
draw(B--C);
draw(F--G);
draw(A--D--H--C--D,dashed);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",E1,W);
label("$F$",F,SW);
label("$G$",G,NE);
label("$H$",H,NW);
label("$M$",M,N);
dot(A);
dot(B);
dot(E1);
dot(F);
dot(G);
dot(C);
dot(D);
dot(H);
dot(M);
label("3",A/2+B/2,S);
label("2",C/2+G/2,E);
label("1",C/2+B/2,SE);[/asy]
$\textbf{(A) } 1 \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{5}{3} \qquad \textbf{(E) } 2$
2004 Germany Team Selection Test, 2
Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.
2011 AIME Problems, 14
Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
2017 Bosnia and Herzegovina Junior BMO TST, 2
Let $A$ be a set $A=\{1,2,3,...,2017\}$. Subset $S$ of set $A$ is [i]good [/i] if for all $x\in A$ sum of remaining elements of set $S$ has same last digit as $x$. Prove that [i]good[/i] subset with $405$ elements is not possible.
2020 Tuymaada Olympiad, 6
An isosceles triangle $ABC$ ($AB = BC$) is given. Circles $\omega_1$ and $\omega_2$ with centres $O_1$ and $O_2$ lie in the angle $ABC$ and touch the sides $AB$ and $CB$ at $A$ and $C$ respectively, and touch each other externally at point $X$. The side $AC$ meets the circles again at points $Y$ and $Z$. $O$ is the circumcenter of the triangle $XYZ$. Lines $O_2 O$ and $O_1 O$ intersect lines $AB$ and $BC$ at points $C_1$ and $A_1$ respectively. Prove that $B$ is the circumcentre of the triangle $A_1 OC_1$.
2012 AMC 12/AHSME, 17
Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$?
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8 $
2001 Baltic Way, 15
Let $a_0,a_1,a_2,\ldots $ be a sequence of positive real numbers satisfying $i\cdot a_2\ge (i + 1)\cdot a_{i_1}a_{i+1}$ for $i=1, 2, \ldots $ Furthermore, let $x$ and $y$ be positive reals, and let $b_i=xa_i+ya_{i-1}$ for $i=1, 2, \ldots $
Prove that the inequality $i\cdot b_2\ge (i + 1)\cdot b_{i-1}b_{i+1}$ holds for all integers $i\ge 2$.
2016 Harvard-MIT Mathematics Tournament, 13
A right triangle has side lengths $a$, $b$, and $\sqrt{2016}$ in some order, where $a$ and $b$ are positive integers. Determine the smallest possible perimeter of the triangle.
2015 IFYM, Sozopol, 8
Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.
2019 MIG, 8
Greg plays a game in which he is given three random $1$ digit numbers, each between $0$ and $9$, inclusive, with repeats allowed. He is to put these three numbers into any order. Exactly one ordering of the three numbers is correct, and if he guesses the correct ordering, he wins $\$150$. What are Greg's expected winnings for this game, given that he randomly guesses one valid ordering when he plays?