Found problems: 85335
2022 Malaysia IMONST 2, 4
Given a pentagon $ABCDE$ with all its interior angles less than $180^\circ$. Prove that if $\angle ABC = \angle ADE$ and $\angle ADB = \angle AEC$, then $\angle BAC = \angle DAE$.
2022 All-Russian Olympiad, 8
From each vertex of triangle $ABC$ we draw two rays, red and blue, symmetric about the angle bisector of the corresponding angle. The circumcircles of triangles formed by the intersection of rays of the same color. Prove that if the circumcircle of triangle $ABC$ touches one of these circles then it also touches to the other one.
1990 Austrian-Polish Competition, 4
Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$
1994 Tournament Of Towns, (434) 4
A rectangular $1$ by $10$ strip is divided into $10$ $1$ by $1$ squares. The numbers $1$, $2$, $3$,$...$, $10$ are placed in the squares in the following way. First the number $1$ is placed in an arbitrary square, then $2$ is placed in a neighbouring square, then $3$ is placed into a free square neighbouring one of the squares occupied earlier, and so on (up to $10$). How many different permutations of $1$,$2$, $3$,$...$, $10$ can one get in this way?
(A Shen)
1988 Romania Team Selection Test, 5
The cells of a $11\times 11$ chess-board are colored in 3 colors. Prove that there exists on the board a $m\times n$ rectangle such that the four cells interior to the rectangle and containing the four vertices of the rectangle have the same color.
[i]Ioan Tomescu[/i]
2009 F = Ma, 15
A $\text{22.0 kg}$ suitcase is dragged in a straight line at a constant speed of $\text{1.10 m/s}$ across a level airport floor by a student on the way to the 40th IPhO in Merida, Mexico. The individual pulls with a $\text{1.00} \times \text{10}^2 \text{N}$ force along a handle with makes an upward angle of $\text{30.0}$ degrees with respect to the horizontal. What is the coefficient of kinetic friction between the suitcase and the floor?
(A) $\mu_\text{k} = \text{0.013}$
(B) $\mu_\text{k} = \text{0.394}$
(C) $\mu_\text{k} = \text{0.509}$
(D) $\mu_\text{k} = \text{0.866}$
(E) $\mu_\text{k} = \text{1.055}$
2021 Bundeswettbewerb Mathematik, 4
In the Cartesian plane, a line segment is called [i]tame[/i] if it lies parallel to one of the coordinate axes and its distance to this axis is an integer. Otherwise it is called [i]wild[/i].
Let $m$ and $n$ be odd positive integers. The rectangle with vertices $(0,0),(m,0),(m,n)$ and $(0,n)$ is partitioned into finitely many triangles. Let $M$ be the set of these triangles. Assume that
(1) Each triangle from $M$ has at least one tame side.
(2) For each tame side of a triangle from $M$, the corresponding altitude has length $1$.
(3) Each wild side of a triangle from $M$ is a common side of exactly two triangles from $M$.
Show that at least two triangles from $M$ have two tame sides each.
2010 ELMO Shortlist, 3
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that
\begin{align*}
a^2 + b^2 + 3 &= 4ab\\
c^2 + d^2 + 3 &= 4cd\\
4c^3 - 3c &= a
\end{align*}
[i]Travis Hance.[/i]
2011 F = Ma, 6
A child is sliding out of control with velocity $v_\text{c}$ across a frozen lake. He runs head-on into another child, initially at rest, with $3$ times the mass of the first child, who holds on so that the two now slide together. What is the velocity of the couple after the collision?
(A) $2v_\text{c}$
(B) $v_\text{c}$
(C) $v_\text{c}/2$
(D) $v_\text{c}/3$
(E) $v_\text{c}/4$
PEN E Problems, 18
Without using Dirichlet's theorem, show that there are infinitely many primes ending in the digit $9$.
KoMaL A Problems 2022/2023, A. 834
Let $A_1A_2\ldots A_8$ be a convex cyclic octagon, and for $i=1,2\ldots,8$ let $B_i=A_iA_{i+3}\cap A_{i+1}A_{i+4}$ (indices are meant modulo 8). Prove that points $B_1,\ldots, B_8$ lie on the same conic section.
2018 Estonia Team Selection Test, 6
We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)
2020 Bulgaria National Olympiad, P4
Are there positive integers $m>4$ and $n$, such that
a) ${m \choose 3}=n^2;$
b) ${m \choose 4}=n^2+9?$
2006 Rioplatense Mathematical Olympiad, Level 3, 1
(a) For each integer $k\ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ mutually distinct positive divisors of $n$.
(b) Suppose that $n$ can be expressed as the sum of exactly $k$ mutually distinct positive divisors of $n$ for some $k\ge 3$. Let $p$ be the smallest prime divisor of $n$. Show that \[\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.\]
2010 Switzerland - Final Round, 10
Let $ n\geqslant 3$ and $ P$ a convex $ n$-gon. Show that $ P$ can be, by $ n \minus{} 3$ non-intersecting diagonals, partitioned in triangles such that the circumcircle of each triangle contains the whole area of $ P$. Under which conditions is there exactly one such triangulation?
2005 IMAR Test, 2
Let $P$ be an arbitrary point on the side $BC$ of triangle $ABC$ and let $D$ be the tangency point between the incircle of the triangle $ABC$ and the side $BC$. If $Q$ and $R$ are respectively the incenters in the triangles $ABP$ and $ACP$, prove that $\angle QDR$ is a right angle.
Prove that the triangle $QDR$ is isosceles if and only if $P$ is the foot of the altitude from $A$ in the triangle $ABC$.
2022 BMT, 25
For triangle $\vartriangle ABC$, define its $A$-excircle to be the circle that is externally tangent to line segment $BC$ and extensions of $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, and define the $B$-excircle and $C$-excircle likewise.
Then, define the $A$-[i]veryexcircle [/i] to be the unique circle externally tangent to both the $A$-excircle as well as the extensions of $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, but that shares no points with line $\overleftrightarrow{BC}$, and define the $B$-veryexcircle and $C$-veryexcircle likewise.
Compute the smallest integer $N \ge 337$ such that for all $N_1 \ge N$, the area of a triangle with lengths $3N^2_1$ , $3N^2_1 + 1$, and $2022N_1$ is at most $\frac{1}{22022}$ times the area of the triangle formed by connecting the centers of its three veryexcircles.
If your submitted estimate is a positive number $E$ and the true value is $A$, then your score is given by $\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^3\right\rfloor \right)$.
1963 AMC 12/AHSME, 7
Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad
\textbf{(2)}\ -2x-3y=10 \qquad
\textbf{(3)}\ 3y+2x=12 \qquad
\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is:
$\textbf{(A)}\ \text{(1) and (4)} \qquad
\textbf{(B)}\ \text{(1) and (3)} \qquad
\textbf{(C)}\ \text{(1) and (2)} \qquad
\textbf{(D)}\ \text{(2) and (4)} \qquad
\textbf{(E)}\ \text{(2) and (3)}$
2006 Sharygin Geometry Olympiad, 20
Four points are given $A, B, C, D$. Points $A_1, B_1, C_1,D_1$ are orthocenters of the triangles $BCD, CDA, DAB, ABC$ and $A_2, B_2, C_2,D_2$ are orthocenters of the triangles $B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1$ etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.
2006 Grigore Moisil Intercounty, 1
Let $ABC$ be a triangle with $b\neq c$. Points $D$ is the midpoint of $BC$ and let $E$ be the foot of angle $A$ bisector. In the exterior of the triangle we construct the similar triangles $AMB$ and $ANC$ . Prove:
a) $MN\bot AD \Longleftrightarrow MA \bot AB$
b) $MN\bot AE \Longleftrightarrow M,A,N$ are colinear.
LMT Theme Rounds, 9
A function $f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\}$ is called [i]good[/i] if the function $g(n)=|f(n)-n|$ is injective. Furthermore, a good function $f$ is called [i]excellent[/i] if there exists another good function $f'$ such that $f(n)-f'(n)$ is nonzero for exactly one value of $n$. Let $N$ be the number of good functions that are not excellent. Find the remainder when $N$ is divided by $1000$.
[i]Proposed by Nathan Ramesh
1998 AMC 8, 25
Three generous friends, each with some money, redistribute the money as follow:
Amy gives enough money to Jan and Toy to double each amount has.
Jan then gives enough to Amy and Toy to double their amounts.
Finally, Toy gives enough to Amy and Jan to double their amounts.
If Toy had 36 dollars at the beginning and 36 dollars at the end, what is the total amount that all three friends have?
$\textbf{(A)}\ 108 \qquad
\textbf{(B)}\ 180 \qquad
\textbf{(C)}\ 216 \qquad
\textbf{(D)}\ 252 \qquad
\textbf{(E)}\ 288$
2017 BMT Spring, 3
Compute $\int^9_{-9}17x^3 \cos (x^2) dx.$
1957 AMC 12/AHSME, 32
The largest of the following integers which divides each of the numbers of the sequence $ 1^5 \minus{} 1,\, 2^5 \minus{} 2,\, 3^5 \minus{} 3,\, \cdots, n^5 \minus{} n, \cdots$ is:
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 120\qquad \textbf{(E)}\ 30$
1990 AMC 8, 4
Which of the following could not be the unit's digit [one's digit] of the square of a whole number?
$ \text{(A)}\ 1\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 8 $