Found problems: 85335
2018 Iran MO (1st Round), 22
There are eight congruent $1\times 2$ tiles formed of one blue square and one red square. In how many ways can we cover a $4\times 4$ area with these tiles so that each row and each column has two blue squares and two red squares?
1986 IMO, 3
To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.
2008 Stanford Mathematics Tournament, 3
Give the positive root(s) of $ x^3 \plus{} 2x^2 \minus{} 2x \minus{} 4$.
2013 IMAR Test, 2
For every non-negative integer $n$ , let $s_n$ be the sum of digits in the decimal expansion of $2^n$. Is the sequence $(s_n)_{n \in \mathbb{N}}$ eventually increasing ?
2002 National High School Mathematics League, 9
Points $P_1,P_2,P_3,P_4$ are vertexes of a regular triangular pyramid, and $P_5,P_6,P_7,P_8,P_9,P_{10}$ midpoints of edges. The number of groups $(P_1,P_i,P_j,P_k)(1<i<j<k\leq10)$ that $P_1,P_i,P_j,P_k$ are coplane is________.
2009 Ukraine National Mathematical Olympiad, 4
[b]а)[/b] Prove that for any positive integer $n$ there exist a pair of positive integers $(m, k)$ such that
\[{k + m^k + n^{m^k}} = 2009^n.\]
[b]b)[/b] Prove that there are infinitely many positive integers $n$ for which there is only one such pair.
2019 South Africa National Olympiad, 6
Determine all pairs $(m, n)$ of non-negative integers that satisfy the equation
$$
20^m - 10m^2 + 1 = 19^n.
$$
2014 Online Math Open Problems, 6
For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure a multiple of $10^{\circ}$. She doesn't want her triangle to have any special properties, so none of the angles can measure $30^{\circ}$ or $60^{\circ}$, and the triangle should definitely not be isosceles.
How many different triangles can Tina draw? (Similar triangles are considered the same.)
[i]Proposed by Evan Chen[/i]
2001 Switzerland Team Selection Test, 2
If $a,b$, and $c$ are the sides of a triangle, prove the inequality $\sqrt{a+b-c}+\sqrt{c+a-b}+\sqrt{b+c-a } \le \sqrt{a}+\sqrt{b}+\sqrt{c}$.
When does equality occur?
1995 Tournament Of Towns, (468) 2
The first five terms of a sequence are $1, 2, 3, 4$ and $5$. From the sixth term on, each term is $1$ less than the product of all the proceeding ones. Prove that the product of the first$ 70$ terms is equal to the sum of their squares.
(LD Kurliandchik)
2005 Harvard-MIT Mathematics Tournament, 1
The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal?
2012 Portugal MO, 1
A five-digit positive integer $abcde_{10}$ ($a\neq 0$) is said to be a [i]range[/i] if its digits satisfy the inequalities $a<b>c<d>e$. For example, $37452$ is a range. How many ranges are there?
2020 Brazil Team Selection Test, 3
Let $ABC$ be a triangle such that $AB > BC$ and let $D$ be a variable point on the line segment $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$, lying on the opposite side of $BC$ from $A$ such that $\angle BAE = \angle DAC$. Let $I$ be the incenter of triangle $ABD$ and let $J$ be the incenter of triangle $ACE$. Prove that the line $IJ$ passes through a fixed point, that is independent of $D$.
[i]Proposed by Merlijn Staps[/i]
2014 District Olympiad, 4
Let $ABCD$ be a square and consider the points $K\in AB, L\in BC,$ and $M\in CD$ such that $\Delta KLM$ is a right isosceles triangle, with the right angle at $L$. Prove that the lines $AL$ and $DK$ are perpendicular to each other.
2006 Iran Team Selection Test, 1
Suppose that $p$ is a prime number.
Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have
\[ n|a^{\frac{\varphi(n)}{p}}-1 \]
1968 Swedish Mathematical Competition, 5
Let $a, b$ be non-zero integers. Let $m(a, b)$ be the smallest value of $\cos ax + \cos bx$ (for real $x$).
Show that for some $r$, $m(a, b) \le r < 0$ for all $a, b$.
MathLinks Contest 3rd, 2
Let $k \ge 1$ be an integer and $a_1, a_2, ... , a_k, b1, b_2, ..., b_k$ rational numbers with the property that for any irrational numbers $x_i >1$, $i = 1, 2, ..., k$, there exist the positive integers $n_1, n_2, ... , n_k, m_1, m_2, ..., m_k$ such that $$a_1\lfloor x^{n_1}_1\rfloor + a_2 \lfloor x^{n_2}_2\rfloor + ...+ a_k\lfloor x^{n_k}_k\rfloor=b_1\lfloor x^{m_1}_1\rfloor +2_1\lfloor x^{m_2}_2\rfloor+...+b_k\lfloor x^{m_k}_k\rfloor $$
Prove that $a_i = b_i$ for all $i = 1, 2, ... , k$.
1981 USAMO, 1
The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).
2019 AMC 10, 17
A child builds towers using identically shaped cubes of different color. How many different towers with a height $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)
$\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,260 \qquad\textbf{(E) } 40,320$
Geometry Mathley 2011-12, 3.4
A triangle $ABC$ is inscribed in the circle $(O,R)$. A circle $(O',R')$ is internally tangent to $(O)$ at $I$ such that $R < R'$. $P$ is a point on the circle $(O)$. Rays $PA, PB, PC$ meet $(O')$ at $A_1,B_1,C_1$. Let $A_2B_2C_2$ be the triangle formed by the intersections of the line symmetric to $B_1C_1$ about $BC$, the line symmetric to $C_1A_1$ about $CA$ and the line symmetric to $A_1B_1$ about $AB$. Prove that the circumcircle of $A_2B_2C_2$ is tangent to $(O)$.
Nguyễn Văn Linh
2006 Australia National Olympiad, 1
In a square $ABCD$, $E$ is a point on diagonal $BD$. $P$ and $Q$ are the circumcentres of $\triangle ABE$ and $\triangle ADE$ respectively. Prove that $APEQ$ is a square.
2022 South Africa National Olympiad, 4
Let $ABC$ be a triangle with $AB < AC$. A point $P$ on the circumcircle of $ABC$ (on the same side of $BC$ as $A$) is chosen in such a way that $BP = CP$. Let $BP$ and the angle bisector of $\angle BAC$ intersect at $Q$, and let the line through $Q$ and parallel to $BC$ intersect $AC$ at $R$. Prove that $BR = CR$.
India EGMO 2024 TST, 3
Find all functions $f: \mathbb{N} \mapsto \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0+a_1x+\dots+a_nx^n$ has at least one integer root, so does \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\]
[i]Proposed by Sutanay Bhattacharya[/i]
2012 Argentina National Olympiad Level 2, 3
Let $ABC$ be a triangle with $\angle A= 105^\circ$ and $\angle B= 45^\circ$. Let $L$ be a point on side $BC$ such that $AL$ is the bisector of angle $\angle BAC$ and let $M$ be the midpoint of side $AC$. Suppose that lines $AL$ and $BM$ intersect at point $P$. Calculate the ratio $\dfrac{AP}{AL}$.
PEN G Problems, 2
Prove that for any positive integers $ a$ and $ b$
\[ \left\vert a\sqrt{2}\minus{}b\right\vert >\frac{1}{2(a\plus{}b)}.\]