Found problems: 85335
2025 Polish MO Finals, 1
Find all $(a, b, c, d)\in \mathbb{R}$ satisfying
\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]
PEN J Problems, 1
Let $n$ be an integer with $n \ge 2$. Show that $\phi(2^{n}-1)$ is divisible by $n$.
2002 AIME Problems, 3
It is given that $\log_{6}a+\log_{6}b+\log_{6}c=6,$ where $a,$ $b,$ and $c$ are positive integers that form an increasing geometric sequence and $b-a$ is the square of an integer. Find $a+b+c.$
2023 Chile Junior Math Olympiad, 2
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
2021 XVII International Zhautykov Olympiad, #4
Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$.
Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$
1988 Spain Mathematical Olympiad, 1
A sequence of integers $(x_n)_{n=1}^{\infty}$ satisfies $x_1 = 1$ and $x_n < x_{n+1} \le 2n$ for all $n$.
Show that for every positive integer $k$ there exist indices $r, s$ such that $x_r-x_s = k$.
2020 Purple Comet Problems, 12
Let $a$ and $b$ be positive integers such that $(a^3 - a^2 + 1)(b^3 - b^2 + 2) = 2020$. Find $10a + b$.
2011 Saudi Arabia BMO TST, 3
Let $ABCDE$ be a convex pentagon such that $\angle BAC = \angle CAD = \angle DAE$ and $\angle ABC = \angle ACD = \angle ADE$. Diagonals $BD$ and $CE$ meet at $P$. Prove that $AP$ bisects side $CD$.
2015 Moldova Team Selection Test, 3
The tangents to the inscribed circle of $\triangle ABC$, which are parallel to the sides of the triangle and do not coincide with them, intersect the sides of the triangle in the points $M,N,P,Q,R,S$ such that $M,S\in (AB)$, $N,P\in (AC)$, $Q,R\in (BC)$. The interior angle bisectors of $\triangle AMN$, $\triangle BSR$ and $\triangle CPQ$, from points $A,B$ and respectively $C$ have lengths $l_{1}$ , $l_{2}$ and $l_{3}$ .\\
Prove the inequality: $\frac {1}{l^{2}_{1}}+\frac {1}{l^{2}_{2}}+\frac {1}{l^{2}_{3}} \ge \frac{81}{p^{2}}$ where $p$ is the semiperimeter of $\triangle ABC$ .
2008 China Girls Math Olympiad, 8
For positive integers $ n$, $ f_n \equal{} \lfloor2^n\sqrt {2008}\rfloor \plus{} \lfloor2^n\sqrt {2009}\rfloor$. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $ f_1,f_2,\ldots$.
2019 USAMTS Problems, 3
Circle $\omega$ is inscribed in unit square $PLUM$, and points $I$ and $E$ lie on $\omega$ such that $U,I,$ and $E$ are collinear. Find, with proof, the greatest possible area for $\triangle PIE$.
2008 Tournament Of Towns, 6
Do there exist positive integers $a,b,c$ and $d$ such that $$\begin{cases} \dfrac{a}{b} + \dfrac{c}{d} = 1\\ \\ \dfrac{a}{d} + \dfrac{c}{b} = 2008\end{cases}$$ ?
2016-2017 SDML (Middle School), 7
Point $P$ is selected at random from the interior of the pentagon with vertices $A = (0, 2), B = (4, 0), C = (2\pi + 1, 0), D = (2\pi + 1, 4),$ and $E = (0, 4)$. What is the probability that $\angle ABP$ is obtuse? Express your answer as a common fraction.
2006 China Northern MO, 7
Can we put positive integers $1,2,3, \cdots 64$ into $8 \times 8$ grids such that the sum of the numbers in any $4$ grids that have the form like $T$ ( $3$ on top and $1$ under the middle one on the top, this can be rotate to any direction) can be divided by $5$?
2015 Dutch IMO TST, 3
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$
Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
2019 Harvard-MIT Mathematics Tournament, 1
Let $ABCD$ be a parallelogram. Points $X$ and $Y$ lie on segments $AB$ and $AD$ respectively, and $AC$ intersects $XY$ at point $Z$. Prove that
\[\frac{AB}{AX} + \frac{AD}{AY} = \frac{AC}{AZ}.\]
1993 Irish Math Olympiad, 3
If $ 1 \le r \le n$ are integers, prove the identity:
$ \displaystyle\sum_{d\equal{}1}^{\infty}\binom {n\minus{}r\plus{}1}{d} \binom {r\minus{}1} {d\minus{}1}\equal{}\binom {n}{r}.$
2010 Today's Calculation Of Integral, 523
Prove the following inequality.
\[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]
2016 Taiwan TST Round 2, 2
Let $x,y$ be positive real numbers such that $x+y=1$.
Prove that$\frac{x}{x^2+y^3}+\frac{y}{x^3+y^2}\leq2(\frac{x}{x+y^2}+\frac{y}{x^2+y})$.
2004 IMC, 2
Let $f_1(x)=x^2-1$, and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real roots does the polynomial $f_{2004}$ have?
2023 Stars of Mathematics, 1
A convex polygon is dissected into a finite number of triangles with disjoint interiors, whose sides have odd integer lengths. The triangles may have multiple vertices on the boundary of the polygon and their sides may overlap partially.
[list=a]
[*]Prove that the polygon's perimeter is an integer which has the same parity as the number of triangles in the dissection.
[*]Determine whether part a) holds if the polygon is not convex.
[/list]
[i]Proposed by Marius Cavachi[/i]
[i]Note: the junior version only included part a), with an arbitrary triangle instead of a polygon.[/i]
2004 Purple Comet Problems, 6
How many different positive integers divide $10!$ ?
2021 Philippine MO, 6
A certain country wishes to interconnect $2021$ cities with flight routes, which are always two-way, in the following manner:
• There is a way to travel between any two cities either via a direct flight or via a sequence of connecting flights.
• For every pair $(A, B)$ of cities that are connected by a direct flight, there is another city $C$ such that $(A, C)$ and $(B, C)$ are connected by direct flights.
Show that at least $3030$ flight routes are needed to satisfy the two requirements.
2008 Tournament Of Towns, 3
A $30$-gon $A_1A_2\cdots A_{30}$ is inscribed in a circle of radius $2$. Prove that one can choose a point $B_k$ on the arc $A_kA_{k+1}$ for $1 \leq k \leq 29$ and a point $B_{30}$ on the arc $A_{30}A_1$, such that the numerical value of the area of the $60$-gon $A_1B_1A_2B_2 \dots A_{30}B_{30}$ is equal to the numerical value of the perimeter of the original $30$-gon.
1999 Mediterranean Mathematics Olympiad, 4
In triangle $\triangle ABC$ we have $BC=a,CA=b,AB=c$ and $\angle B=4\angle A$ Show that \[ab^2c^3=(b^2-a^2-ac)((a^2-b^2)^2-a^2c^2)\]