Found problems: 85335
1990 AMC 8, 1
What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $ 4,5,6,7,8,9 $ in one of the six boxes in this addition problem?
[asy]
unitsize(12);
draw((0,0)--(10,0)); draw((-1.5,1.5)--(-1.5,2.5)); draw((-1,2)--(-2,2));
draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4)--(3,6)--(1,6)--cycle);
draw((4,1)--(6,1)--(6,3)--(4,3)--cycle); draw((4,4)--(6,4)--(6,6)--(4,6)--cycle);
draw((7,1)--(9,1)--(9,3)--(7,3)--cycle); draw((7,4)--(9,4)--(9,6)--(7,6)--cycle);[/asy]
$ \text{(A)}\ 947\qquad\text{(B)}\ 1037\qquad\text{(C)}\ 1047\qquad\text{(D)}\ 1056\qquad\text{(E)}\ 1245 $
2016 APMC, 2
Let $ABC$ be a triangle with incenter $I$, and suppose that $AI$, $BI$, and $CI$ intersect $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Let the circumcircles of $BDF$ and $CDE$ intersect at $D$ and $P$, and let $H$ be the orthocenter of $DEF$. Prove that $HI=HP$.
2021 Durer Math Competition Finals, 2
Find the number of integers $n$ between $1$ and $2021$ such that $2^n+2^{n+3}$ is a perfect square.
2008 China Team Selection Test, 3
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.
1999 Romania National Olympiad, 2
Let $a, b, c$ be non zero integers,$ a\ne c$ such that $$\frac{a}{c}=\frac{a^2+b^2}{c^2+b^2}$$
Prove that $a^2 +b^2 +c^2$ cannot be a prime number.
1991 China Team Selection Test, 3
All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.
2002 HKIMO Preliminary Selection Contest, 11
Find the 2002nd positive integer that is not the difference of two square integers
2020 MBMT, 23
Let $ABCD$ be a cyclic quadrilateral so that $\overline{AC} \perp \overline{BD}$. Let $E$ be the intersection of $\overline{AC}$ and $\overline{BD}$, and let $F$ be the foot of the altitude from $E$ to $\overline{AB}$. Let $\overline{EF}$ intersect $\overline{CD}$ at $G$, and let the foot of the perpendiculars from $G$ to $\overline{AC}$ and $\overline{BD}$ be $H, I$ respectively. If $\overline{AB} = \sqrt{5}, \overline{BC} = \sqrt{10}, \overline{CD} = 3\sqrt{5}, \overline{DA} = 2\sqrt{10}$, find the length of $\overline{HI}$.
[i]Proposed by Timothy Qian[/i]
2025 All-Russian Olympiad, 11.3
A pair of polynomials \(F(x, y)\) and \(G(x, y)\) with integer coefficients is called $\emph{important}$ if from the divisibility of both differences \(F(a, b) - F(c, d)\) and \(G(a, b) - G(c, d)\) by $100$, it follows that both \(a - c\) and \(b - d\) are divisible by 100. Does there exist such an important pair of polynomials \(P(x, y)\), \(Q(x, y)\), such that the pair \(P(x, y) - xy\) and \(Q(x, y) + xy\) is also important?
2016 Regional Olympiad of Mexico Center Zone, 1
The grid shown below is completed by choosing nine of the following numbers without repeating: $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$. If the sum of the five rows are equal to each other and the sum of the three columns are equal to each other, in how many different ways is it possible to fill the grid?
$ \[\begin {array} {| c | c | c |} \hline 10 & & \\ \hline & & 9 \\ \hline & 3 & \\ \hline 11 & & 17 \\ \hline & 20 & \\ \hline \end {array} \] $
Note: The sum of the rows and the sum of the columns are not necessarily equal.
2014 Math Prize For Girls Problems, 11
Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$. Compute the area of region $R$. Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$.
2022 Israel National Olympiad, P2
Real nonzero numbers $a,b,c,d,e,f,k,m$ satisfy the equations
\[\frac{a}{b}+\frac{c}{d}+\frac{e}{f}=k\]
\[\frac{b}{c}+\frac{d}{e}+\frac{f}{a}=m\]
\[ad=be=cf\]
Express $\frac{a}{c}+\frac{c}{e}+\frac{e}{a}+\frac{b}{d}+\frac{d}{f}+\frac{f}{b}$ using $m$ and $k$.
PEN O Problems, 1
Suppose all the pairs of a positive integers from a finite collection \[A=\{a_{1}, a_{2}, \cdots \}\] are added together to form a new collection \[A^{*}=\{a_{i}+a_{j}\;\; \vert \; 1 \le i < j \le n \}.\] For example, $A=\{ 2, 3, 4, 7 \}$ would yield $A^{*}=\{ 5, 6, 7, 9, 10, 11 \}$ and $B=\{ 1, 4, 5, 6 \}$ would give $B^{*}=\{ 5, 6, 7, 9, 10, 11 \}$. These examples show that it's possible for different collections $A$ and $B$ to generate the same collections $A^{*}$ and $B^{*}$. Show that if $A^{*}=B^{*}$ for different sets $A$ and $B$, then $|A|=|B|$ and $|A|=|B|$ must be a power of $2$.
1993 Flanders Math Olympiad, 3
For $a,b,c>0$ we have:
\[ -1 < \left(\dfrac{a-b}{a+b}\right)^{1993} + \left(\dfrac{b-c}{b+c}\right)^{1993} + \left(\dfrac{c-a}{c+a}\right)^{1993} < 1 \]
MOAA Gunga Bowls, 2021.5
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
2024 Polish Junior MO Finals, 5
Let $S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}$. Show that the $2S$-digit number
\[\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}\]
is a multiple of $19$.
2010 Rioplatense Mathematical Olympiad, Level 3, 1
Let $r_2, r_3,\ldots, r_{1000}$ denote the remainders when a positive odd integer is divided by $2,3,\ldots,1000$, respectively. It is known that the remainders are pairwise distinct and one of them is $0$. Find all values of $k$ for which it is possible that $r_k = 0$.
2008 Switzerland - Final Round, 4
Consider three sides of an $n \times n \times n$ cube that meet at one of the corners of the cube. For which $n$ is it possible to use this completely and without overlapping to cover strips of paper of size $3 \times 1$? The paper strips can also do this glued over the edges between these cube faces.
1989 AIME Problems, 15
Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$.
[asy]
size(200);
pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C);
draw(A--B--C--A--D^^C--F^^B--E);
pair point=P;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));
label("$P$", P, dir(0));[/asy]
2017 Pakistan TST, Problem 2
There are $n$ students in a circle, one behind the other, all facing clockwise. The students have heights $h_1 <h_2 < h_3 < \cdots < h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or lesss, the two students are permitted to switch places Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.
2024 Saint Petersburg Mathematical Olympiad, 6
Inscribed hexagon $AB_1CA_1BC_1$ is given. Circle $\omega$ is inscribed in both triangles $ABC$ and $A_1B_1C_1$ and touches segments $AB$ and $A_1B_1$ at points $D$ and $D_1$ respectively. Prove that if $\angle ACD = \angle BCD_1$, then $\angle A_1C_1D_1 = \angle B_1C_1D$.
2002 AIME Problems, 2
Three vertices of a cube are $P=(7,12,10),$ $Q=(8,8,1),$ and $R=(11,3,9).$ What is the surface area of the cube?
2004 Regional Olympiad - Republic of Srpska, 2
The positive real numbers $x,y,z$ satisfy $x+y+z=1$. Show that
\[\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+
\frac{4xyz}{(1-x)(1-y)(1-z)}.\]
2024 ITAMO, 1
Let $x_0=2024^{2024}$ and $x_{n+1}=|x_n-\pi|$ for $n \ge 0$. Show that there exists a value of $n$ such that $x_{n+2}=x_n$.
JBMO Geometry Collection, 2018
Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.