Found problems: 85335
2024 Miklos Schweitzer, 10
Let $A > 0$ and $B = (3 + 2\sqrt{2})A$. Prove that in the finite sequence $a_k = \lfloor k / \sqrt{2} \rfloor$ for $k \in (A, B) \cap \mathbb{Z}$, the number of even and odd terms differs by at most $2$.
2003 Croatia Team Selection Test, 1
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
2010 Purple Comet Problems, 8
The diagram below shows some small squares each with area $3$ enclosed inside a larger square. Squares that touch each other do so with the corner of one square coinciding with the midpoint of a side of the other square. Find integer $n$ such that the area of the shaded region inside the larger square but outside the smaller squares is $\sqrt{n}$.
[asy]
size(150);
real r=1/(2sqrt(2)+1);
path square=(0,1)--(r,1)--(r,1-r)--(0,1-r)--cycle;
path square2=(0,.5)--(r/sqrt(2),.5+r/sqrt(2))--(r*sqrt(2),.5)--(r/sqrt(2),.5-r/sqrt(2))--cycle;
defaultpen(linewidth(0.8));
filldraw(unitsquare,gray);
filldraw(square2,white);
filldraw(shift((0.5-r/sqrt(2),0.5-r/sqrt(2)))*square2,white);
filldraw(shift(1-r*sqrt(2),0)*square2,white);
filldraw(shift((0.5-r/sqrt(2),-0.5+r/sqrt(2)))*square2,white);
filldraw(shift(0.5-r/sqrt(2)-r,-(0.5-r/sqrt(2)-r))*square,white);
filldraw(shift(0.5-r/sqrt(2)-r,-(0.5+r/sqrt(2)))*square,white);
filldraw(shift(0.5+r/sqrt(2),-(0.5+r/sqrt(2)))*square,white);
filldraw(shift(0.5+r/sqrt(2),-(0.5-r/sqrt(2)-r))*square,white);
filldraw(shift(0.5-r/2,-0.5+r/2)*square,white);
[/asy]
2017 QEDMO 15th, 8
For a function $f: R\to R $ , $ f (2017)> 0$ as well as $f (x^2 + yf (z)) = xf (x) + zf (y)$ for all $x,y,z \in R$ is known. What is the value of $f (-42)$?
Ukrainian TYM Qualifying - geometry, 2015.21
Let $CH$ be the altitude of the triangle $ABC$ drawn on the board, in which $\angle C = 90^o$, $CA \ne CB$. The mathematics teacher drew the perpendicular bisectors of segments$ CA$ and $CB$, which cut the line CH at points $K$ and $M$, respectively, and then erased the drawing, leaving only the points $C, K$ and $M$ on the board. Restore triangle $ABC$, using only a compass and a ruler.
JOM 2015 Shortlist, C5
Let $G$ be a simple connected graph. Each edge has two phases, which is either blue or red. Each vertex are switches that change the colour of every edge that connects the vertex. All edges are initially red. Find all ordered pairs $(n,k)$, $n\ge 3$, such that:
a) For all graph $G$ with $n$ vertex and $k$ edges, it is always possible to perform a series of switching process so that all edges are eventually blue.
b) There exist a graph $G$ with $n$ vertex and $k$ edges and it is possible to perform a series of switching process so that all edges are eventually blue.
2009 Junior Balkan Team Selection Tests - Romania, 2
Consider a rhombus $ABCD$. Point $M$ and $N$ are given on the line segments $AC$ and $BC$ respectively, such that $DM = MN$. Lines $AC$ and $DN$ meet at point $P$ and lines $AB$ and $DM$ meet at point $R$. Prove that $RP = PD$.
1996 China Team Selection Test, 3
Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?
2016 Saudi Arabia IMO TST, 3
Given two circles $(O_1)$ and $(O_2)$ intersect at $A$ and $B$. Let $d_1$ and $d_2$ be two lines through $A$ and be symmetric with respect to $AB$. The line $d_1$ cuts $(O_1)$ and $(O_2)$ at $G, E$ ($\ne A$), respectively, the line $d_2$ cuts $(O_1)$ and $(O_2)$ at $F, H$ ($\ne A$), respectively, such that $E$ is between $A, G$ and $F$ is between $A, H$. Let $J$ be the intersection of $EH$ and $FG$. The line $BJ$ cuts $(O_1), (O_2)$ at $K, L$ ($\ne B$), respectively. Let $N$ be the intersection of $O_1K$ and $O_2L$. Prove that the circle $(NLK)$ is tangent to $AB$.
2005 Junior Balkan Team Selection Tests - Romania, 5
On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$.
Prove that the triangles $QCP$ and $MCN$ have the same area.
2000 Moldova Team Selection Test, 4
Let $S{}$ be the set of nonnegative integers, which cointain only digits $0$ and $1$ in base $4$ numeral system.
a) Show that if $x\in S, y\in S, x\neq y,$ then $\frac{x+y}{2}\notin S$.
b) Let $T$ be a set of nonnegative integers such that $S\subset T, T\neq S$. Show that there exist $x\in T, y\in T, x\neq y,$ such that $\frac{x+y}{2} \in T$.
2021 May Olympiad, 3
In a year that has $365$ days, what is the maximum number of "Tuesday the $13$th" there can be?
Note: The months of April, June, September and November have $30$ days each, February has $28$ and all others have $31$ days.
2020 Dutch IMO TST, 1
For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$.
Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.
1947 Kurschak Competition, 2
Show that any graph with $6$ points has a triangle or three points which are not joined to each other.
2014 Taiwan TST Round 1, 5
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
VMEO III 2006 Shortlist, N9
Assume the $m$ is a given integer greater than $ 1$. Find the largest number $C$ such that for all $n \in N$ we have
$$\sum_{1\le k \le m ,\,\, (k,m)=1}\frac{1}{k}\ge C \sum_{k=1}^{m}\frac{1}{k}$$
2023 Baltic Way, 17
Find all pairs of positive integers $(a, b)$, such that $S(a^{b+1})=a^b$, where $S(m)$ denotes the digit sum of $m$.
2009 Indonesia TST, 3
Let $ ABC$ be an isoceles triangle with $ AC\equal{}BC$. A point $ P$ lies inside $ ABC$ such that \[ \angle PAB \equal{} \angle PBC, \angle PAC \equal{} \angle PCB.\] Let $ M$ be the midpoint of $ AB$ and $ K$ be the intersection of $ BP$ and $ AC$. Prove that $ AP$ and $ PK$ trisect $ \angle MPC$.
1960 Putnam, B3
The motion of the particles of a fluid in the plane is specified by the following components of velocity
$$\frac{dx}{dt}=y+2x(1-x^2 -y^2),\;\; \frac{dy}{dt}=-x.$$
Sketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as $t\to \infty$, and justify your conclusion.
2009 India IMO Training Camp, 4
Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly.
Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.
2021 Iran MO (3rd Round), 2
Given an acute triangle $ABC$ let $M$ be the midpoint of $AB$. Point $K$ is given on the other side of line $AC$ from that of point $B$ such that $\angle KMC = 90 ^ \circ $ and $\angle KAC = 180^\circ - \angle ABC$. The tangent to circumcircle of triangle $ABC$ at $A$ intersects line $CK$ at $E$. Prove that the reflection of line $BC$ with respect to $CM$ passes through the midpoint of line segment $ME$.
2012 Singapore MO Open, 1
The incircle with centre $I$ of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ at $D, E, F$ respectively. The line $ID$ intersects the segment $EF$ at $K$. Proof that $A, K, M$ collinear, where $M$ is the midpoint of $BC$.
LMT Team Rounds 2010-20, 2018 Spring
[b]p1[/b]. Points $P_1,P_2,P_3,... ,P_n$ lie on a plane such that $P_aP_b = 1$,$P_cP_d = 2$, and $P_eP_f = 2018$ for not necessarily distinct indices $a,b,c,d,e, f \in \{1, 2,... ,n\}$. Find the minimum possible value of $n$.
[b]p2.[/b] Find the coefficient of the $x^2y^4$ term in the expansion of $(3x +2y)^6$.
[b]p3.[/b] Find the number of positive integers $n < 1000$ such that $n$ is a multiple of $27$ and the digit sum of $n$ is a multiple of $11$.
[b]p4.[/b] How many times do the minute hand and hour hand of a $ 12$-hour analog clock overlap in a $366$-day leap year?
[b]p5.[/b] Find the number of ordered triples of integers $(a,b,c)$ such that $(a +b)(b +c)(c + a) = 2018$.
[b]p6.[/b] Let $S$ denote the set of the first $2018$ positive integers. Call the score of a subset the sum of its maximal element and its minimal element. Find the sum of score $(x)$ over all subsets $s \in S$
[b]p7.[/b] How many ordered pairs of integers $(a,b)$ exist such that $1 \le a,b \le 20$ and $a^a$ divides $b^b$?
[b]p8.[/b] Let $f$ be a function such that for every non-negative integer $p$, $f (p)$ equals the number of ordered pairs of positive integers $(a,n)$ such that $a^n = a^p \cdot n$. Find $\sum^{2018}_{p=0}f (p)$.
[b]p9.[/b] A point $P$ is randomly chosen inside a regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$. What is the probability that the projections of $P$ onto the lines $\overleftrightarrow{A_i A_{i+1}}$ for $i = 1,2,... ,8$ lie on the segments $\overline{A_iA_{i+1}}$ for $i = 1,2,... ,8$ (where indices are taken $mod \,\, 8$)?
[b]p10. [/b]A person keeps flipping an unfair coin until it flips $3$ tails in a row. The probability of it landing on heads is $\frac23$ and the probability it lands on tails is $\frac13$ . What is the expected value of the number of the times the coin flips?
PS. You had better use hide for answers.
2014 Thailand TSTST, 3
Let $O$ be the incenter of a tangential quadrilateral $ABCD$. Prove that the orthocenters of $\vartriangle AOB$, $\vartriangle BOC$, $\vartriangle COD$, $\vartriangle DOA$ lie on a line.
2008 Tournament Of Towns, 2
Space is dissected into congruent cubes. Is it necessarily true that for each cube there exists another cube so that both cubes have a whole face in common?