Found problems: 85335
2019 CMIMC, 1
The figure below depicts two congruent triangles with angle measures $40^\circ$, $50^\circ$, and $90^\circ$. What is the measure of the obtuse angle $\alpha$ formed by the hypotenuses of these two triangles?
[asy]
import olympiad;
size(80);
defaultpen(linewidth(0.8));
draw((0,0)--(3,0)--(0,4.25)--(0,0)^^(0,3)--(4.25,0)--(3,0)^^rightanglemark((0,3),(0,0),(3,0),10));
pair P = intersectionpoint((3,0)--(0,4.25),(0,3)--(4.25,0));
draw(anglemark((4.25,0),P,(0,4.25),10));
label("$\alpha$",P,2 * NE);
[/asy]
2024 Harvard-MIT Mathematics Tournament, 7
Let $P(n)=(n-1^3)(n-2^3)\ldots (n-40^3)$ for positive integers $n$. Let $d$ be the largest positive integer such that $d \mid P(n)$ for any $n>2023$. If $d$ is product of $m$ not necessarily distinct primes, find $m$.
2016 Azerbaijan National Mathematical Olympiad, 1
Find the perimeter of the convex polygon whose coordinates of the vertices are the set of pairs of the integer solutions of the equation $x^2+xy = x + 2y + 9$.
2013 Tournament of Towns, 3
Assume that $C$ is a right angle of triangle $ABC$ and $N$ is a midpoint of the semicircle, constructed on $CB$ as on diameter externally. Prove that $AN$ divides the bisector of angle $C$ in half.
2016 Vietnam Team Selection Test, 6
Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$. For each polynomial $P$, denote \[ V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}). \] Prove that there is a monic polynomial $Q$, $\deg Q=8$ satisfying:
i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$.
ii) $Q$ has $8$ real roots (including multiplicity).
1992 IMO Longlists, 68
Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy
\[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]
2020 LMT Fall, 21
A sequence with first term $a_0$ is defined such that $a_{n+1}=2a_n^2-1$ for $n\geq0.$ Let $N$ denote the number of possible values of $a_0$ such that $a_0=a_{2020}.$ Find the number of factors of $N.$
[i]Proposed by Alex Li[/i]
2018 Taiwan APMO Preliminary, 7
$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$. (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action:
First, if your number is divisible by $3$ then turn around.
Then, if your number is divisible by $5$ then turn around.
Finally, if your number is divisible by $7$ then turn around.
(a) How many students are face to coach now?
(b) What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?
2008 Harvard-MIT Mathematics Tournament, 30
Triangle $ ABC$ obeys $ AB = 2AC$ and $ \angle{BAC} = 120^{\circ}.$ Points $ P$ and $ Q$ lie on segment $ BC$ such that
\begin{eqnarray*}
AB^2 + BC \cdot CP = BC^2 \\
3AC^2 + 2BC \cdot CQ = BC^2
\end{eqnarray*}
Find $ \angle{PAQ}$ in degrees.
2014 IMS, 7
Let $G$ be a finite group such that for every two subgroups of it like $H$ and $K$, $H \cong K$ or $H \subseteq K$ or $K \subseteq H$. Prove that we can produce each subgroup of $G$ with 2 elements at most.
2015 India Regional MathematicaI Olympiad, 2
Let $P_1(x) = x^2 + a_1x + b_1$ and $P_2(x) = x^2 + a_2x + b_2$ be two quadratic polynomials with integer coeffcients. Suppose $a_1 \ne a_2$ and there exist integers $m \ne n$ such that $P_1(m) = P_2(n), P_2(m) = P_1(n)$. Prove that $a_1 - a_2$ is even.
2023 Greece Junior Math Olympiad, 1
Solve in real numbers the system:
$$\begin{cases} a+b+c=0 \\ ab^3+bc^3+ca^3=0 \end{cases}$$
2004 Purple Comet Problems, 19
There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction $m/n$ in lowest terms, find $m + n$.
2009 Italy TST, 3
Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules:
i) Any incantation can appear no more than once;
ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it;
iii)The first person who cannot spell an incantation loses the contest. Answer the following questions:
a) If A says '$STAGEPREIMO$' first, then who will win?
b) Let $M$ be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are $2009$ and containing only four letters $A,B,C,D$, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in $M$ such that A has a winning strategy by starting with it.
1996 Mexico National Olympiad, 6
In a triangle $ABC$ with $AB < BC < AC$, points $A' ,B' ,C'$ are such that $AA' \perp BC$ and $AA' = BC, BB' \perp CA$ and $BB'=CA$, and $CC' \perp AB$ and $CC'= AB$, as shown on the picture. Suppose that $\angle AC'B$ is a right angle. Prove that the points $A',B' ,C' $ are collinear.
2023 District Olympiad, P3
Let $x,y{}$ and $z{}$ be positive real numbers satisfying $x+y+z=1$. Prove that
[list=a]
[*]\[1-\frac{x^2-yz}{x^2+x}=\frac{(1-y)(1-z)}{x^2+x};\]
[*]\[\frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leqslant 0.\]
[/list]
1980 IMO Shortlist, 2
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way:
\[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \]
Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]
2007 Germany Team Selection Test, 2
Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that \[ p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.\]
2021 Kyiv Mathematical Festival, 3
Is it true that for every $n\ge 2021$ there exist $n$ integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal? (O. Rudenko)
1969 IMO Longlists, 38
$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$
1999 Harvard-MIT Mathematics Tournament, 7
Let $\frac{1}{1-x-x^2-x^3} =\sum^{\infty}_{i=0} a_nx^n$, for what positive integers $n$ does $a_{n-1} = n^2$?
2005 AIME Problems, 5
Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5$, $2 \leq a \leq 2005$, and $2 \leq b \leq 2005$.
2006 Princeton University Math Competition, 3
Let $r_1, \dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r^2_1+\dots+r^2_5$?
2019 Dutch IMO TST, 1
Let $P(x)$ be a quadratic polynomial with two distinct real roots.
For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$.
Show that at least one of the roots of $P$ is negative.
1982 IMO Longlists, 34
Let $M$ be the set of all functions $f$ with the following properties:
[b](i)[/b] $f$ is defined for all real numbers and takes only real values.
[b](ii)[/b] For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$
[b](iii)[/b] $f(0) \neq 0.$
Determine all functions $f \in M$ such that
[b](a)[/b] $f(1)=\frac 52$,
[b](b)[/b] $f(1)= \sqrt 3$.