Found problems: 85335
2020-21 KVS IOQM India, 29
Consider a permutation $(a_1,a_2,a_3,a_4,a_5)$ of $\{1,2,3,4,5\}$. We say the $5$-tuple $(a_1,a_2,a_3,a_4,a_5)$ is dlawless if for all $1 \le i<j<k \le 5$, the sequence $(a_i,a_j,a_k)$ is [b]not [/b] an arithmetic progression (in that order). Find the number of flawless $5$-tuples.
2003 AMC 8, 15
A fi gure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a fi gure with the front and side views shown?
[asy]
defaultpen(linewidth(0.8));
path p=unitsquare;
draw(p^^shift(0,1)*p^^shift(1,0)*p);
draw(shift(4,0)*p^^shift(5,0)*p^^shift(5,1)*p);
label("FRONT", (1,0), S);
label("SIDE", (5,0), S);[/asy]
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$
Brazil L2 Finals (OBM) - geometry, 2021.5
Let $ABC$ be an acute-angled triangle. Let $A_1$ be the midpoint of the arc $BC$ which contain the point $A$. Let $A_2$ and $A_3$ be the foot(s) of the perpendicular(s) of the point $A_1$ to the lines $AB$ and $AC$, respectively. Define $B_2,B_3,C_2,C_3$ analogously.
a) Prove that the line $A_2A_3$ cuts $BC$ in the midpoint.
b) Prove that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrents.
2012 ELMO Shortlist, 4
Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$.
[i]Ray Li.[/i]
1980 Austrian-Polish Competition, 8
Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$
2012 AMC 10, 9
Two integers have a sum of $26$. When two more integers are added to the first two integers the sum is $41$. Finally when two more integers are added to the sum of the previous four integers the sum is $57$. What is the minimum number of even integers among the $6$ integers?
${{ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4}\qquad\textbf{(E)}\ 5} $
2005 Turkey Team Selection Test, 2
Let $N$ be midpoint of the side $AB$ of a triangle $ABC$ with $\angle A$ greater than $\angle B$. Let $D$ be a point on the ray $AC$ such that $CD=BC$ and $P$ be a point on the ray $DN$ which lies on the same side of $BC$ as $A$ and satisfies the condition $\angle PBC =\angle A$. The lines $PC$ and $AB$ intersect at $E$, and the lines $BC$ and $DP$ intersect at $T$. Determine the value of $\frac{BC}{TC} - \frac{EA}{EB}$.
2019 Saudi Arabia Pre-TST + Training Tests, 3.1
Let $ABC$ be a triangle inscribed in a circle ($\omega$) and $I$ is the incenter. Denote $D,E$ as the intersection of $AI,BI$ with ($\omega$). And $DE$ cuts $AC,BC$ at $F,G$ respectively. Let $P$ be a point such that $PF \parallel AD$ and $PG \parallel BE$. Suppose that the tangent lines of ($\omega$) at $A,B$ meet at $K$. Prove that three lines $AE,BD,KP$ are concurrent or parallel.
2006 Iran MO (3rd Round), 3
Suppose $(u,v)$ is an inner product on $\mathbb R^{n}$ and $f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is an isometry, that $f(0)=0$.
1) Prove that for each $u,v$ we have $(u,v)=(f(u),f(v)$
2) Prove that $f$ is linear.
2014 ASDAN Math Tournament, 7
Two math students play a game with $k$ sticks. Alternating turns, each one chooses a number from the set $\{1,3,4\}$ and removes exactly that number of sticks from the pile (so if the pile only has $2$ sticks remaining the next player must take $1$). The winner is the player who takes the last stick. For $1\leq k\leq100$, determine the number of cases in which the first player can guarantee that he will win.
2011 LMT, 6
Define a sequence by $a_1=a_2=1, a_3=2,$ and
$$a_n+a_{n-3}=a_{n-1}+a_{n-2}$$
for all $n>3.$ What is the value of $a_7?$
2024 Stars of Mathematics, P2
A positive integer is called [i]cool[/i] if it is divisible by the square of each of its prime divisors. Prove that $n$ and $n+1$ are simultaneously cool for infinitely many $n$.
2000 Moldova National Olympiad, Problem 1
What is the greatest possible number of Fridays by the date $13$ in a year?
2021 Girls in Mathematics Tournament, 2
Let $\vartriangle ABC$ be a triangle in which $\angle ACB = 40^o$ and $\angle BAC = 60^o$ . Let $D$ be a point inside the segment $BC$ such that $CD =\frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\angle CMD$ in degrees?
KoMaL A Problems 2022/2023, A. 841
Find all non-negative integer solutions of the equation $2^a+p^b=n^{p-1}$, where $p$ is a prime number.
Proposed by [i]Máté Weisz[/i], Cambridge
2008 Miklós Schweitzer, 3
A bipartite graph on the sets $\{ x_1,\ldots, x_n \}$ and $\{ y_1,\ldots, y_n\}$ of vertices (that is the edges are of the form $x_iy_j$) is called tame if it has no $x_iy_jx_ky_l$ path ($i,j,k,l\in\{ 1,\ldots, n\}$) where $j<l$ and $i+j>k+l$. Calculate the infimum of those real numbers $\alpha$ for which there exists a constant $c=c(\alpha)>0$ such that for all tame graphs $e\le cn^{\alpha}$, where $e$ is the number of edges and $n$ is half of the number of vertices.
(translated by Miklós Maróti)
2012 IFYM, Sozopol, 8
In a non-isosceles $\Delta ABC$ with angle bisectors $AL_a$, $BL_b$, and $CL_c$ we have that $L_aL_c=L_bL_c$. Prove that $\angle C$ is smaller than $120^\circ$.
1970 IMO Longlists, 38
Find the greatest integer $A$ for which in any permutation of the numbers $1, 2, \ldots , 100$ there exist ten consecutive numbers whose sum is at least $A$.
1999 Harvard-MIT Mathematics Tournament, 6
Evaluate $\dfrac{d}{dx}\left(\sin x - \dfrac{4}{3}\sin^3 x\right)$ when $x=15$.
2024 India Iran Friendly Math Competition, 6
Let $ABC$ be a triangle with midpoint $M$ of $BC$. A point $X$ is called [i]immaculate[/i] if the perpendicular line from $X$ to line $MX$ intersects lines $AB$ and $AC$ at two points that are equidistant from $M$. Suppose $U, V, W$ are three immaculate points on the circumcircle of triangle $ABC$. Prove that $M$ is the incentre of $\triangle UVW$.
[i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]
2012 Dutch IMO TST, 5
Let $\Gamma$ be the circumcircle of the acute triangle $ABC$. The angle bisector of angle $ABC$ intersects $AC$ in the point $B_1$ and the short arc $AC$ of $\Gamma$ in the point $P$. The line through $B_1$ perpendicular to $BC$ intersects the short arc $BC$ of $\Gamma$ in $K$. The line through $B$ perpendicular to $AK$ intersects $AC$ in $L$. Prove that $K, L$ and $P$ lie on a line.
1985 AIME Problems, 13
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
1980 IMO, 7
Prove that $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x,y$ with $x\le 1980$.
[i] Variant: [/i] Prove the equation $4x^3-3x+1=2y^2$ has infinitely many solutions in positive integers x,y.
1985 Bulgaria National Olympiad, Problem 5
Let $P$ be a point on the median $CM$ of a triangle $ABC$ with $AC\ne BC$ and the acute angle $\gamma$ at $C$, such that the bisectors of $\angle PAC$ and $\angle PBC$ intersect at a point $Q$ on the median $CM$. Determine $\angle APB$ and $\angle AQB$.
2024 Princeton University Math Competition, 13
Consider the square with vertices $(0, 0),(1, 0),(1, 1),(0, 1).$ The line segments from $(t, 0)$ to $(0, 1 - t)$ are drawn for $0 \le t \le 1.$ The set of points inside the square but not on one of these line segments has area $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ Find $m + n.$