Found problems: 85335
2015 Saint Petersburg Mathematical Olympiad, 3
There are weights with mass $1,3,5,....,2i+1,...$ Let $A(n)$ -is number of different sets with total mass equal $n$( For example $A(9)=2$, because we have two sets $9=9=1+3+5$). Prove that $A(n) \leq A(n+1)$ for $n>1$
2003 IMO Shortlist, 8
Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:
(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;
(ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power.
What is the largest possible number of elements in $A$ ?
2023 Taiwan TST Round 1, A
Let $f:\mathbb{N}\to\mathbb{R}_{>0}$ be a given increasing function that takes positive values. For any pair $(m,n)$ of positive integers, we call it [i]disobedient[/i] if $f(mn)\neq f(m)f(n)$. For any positive integer $m$, we call it [i]ultra-disobedient[/i] if for any nonnegative integer $N$, there are always infinitely many positive integers $n$ satisfying that $(m,n), (m,n+1),\ldots,(m,n+N)$ are all disobedient pairs.
Show that if there exists some disobedient pair, then there exists some ultra-disobedient positive integer.
[i]
Proposed by usjl[/i]
Russian TST 2019, P3
Let $H{}$ be the orthocenter of the acute-angled triangle $ABC$. In the triangle $BHC$, the median $HM$ and the symedian $HL$ are drawn. The point $K{}$ is marked on the line $LH$ so that $\angle AKL=90^\circ$. Prove that the circumcircles of the triangles $ABC$ and $KLM$ are tangent.
2015 Middle European Mathematical Olympiad, 1
Prove that for all positive real numbers $a$, $b$, $c$ such that $abc=1$ the following inequality holds:
$$\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.$$
1971 IMO Longlists, 25
Let $ABC,AA_1A_2,BB_1B_2, CC_1C_2$ be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments $A_2B_1,B_2C_1, C_2A_1$ by $P,Q,R$ in this order. Prove that the triangle $PQR$ is equilateral.
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 $x=15$.